London Number Theory Seminar: Previous Talks

London Number Theory Seminar Previous Seminars

The London Number Theory Seminar is hosted in rotation for one semester each by UCL, IC and KCL.

Spring 2010

This term, the seminar was held at University College and was organised by Andrei Yafaev.

The seminar program for Spring 2010

13 January		Michael Schein "On families of irreducible supersingular mod p representations of GL_2(F)"
20 January		Mathieu Florence (Paris) "Equivariant birational geometry of Grassmannians" Abstract: Let k be a field, and A a finite-dimensional k-algebra. Let d be an integer. Denote by Gr(d,A) the Grassmannian of d-subspaces of A (viewed as a k-vector space), and by GL_1(A) the algebraic k-group whose points are invertible elements of A. The group GL_1(A) acts naturally on Gr(d,A) (by the formula g.E=gE). My aim is to study some birational properties of this action. More precisely, let r be the gcd of d and dim(A). Under some hypothesis on A (satisfied if A/k is etale), I will show that the variety Gr(d,A) is birationally and GL_1(A)-equivariantly isomorphic to the product of Gr(r,A) by an affine space (on which GL_1(A) acts trivially). By twisting, this result has a few corollaries in the theory of central simple algebras. For instance, let B and C be two central simple algebras over k, of coprime degrees. Then the Severi-Brauer variety SB(B \otimes C) is birational to the product of SB(B) \times SB(C) by an affine space of the correct dimension. These corollaries are in the spirit of Krashen's generalized version of Amitsur's conjecture.
27 January		Fernando Villegas (Texas) "Hypergeometric motives"
3 February		Toby Gee (Harvard) "The Sato-Tate conjecture for Hilbert modular forms" Abstract: I will discuss the Sato-Tate conjecture for Hilbert modular forms, which I recently proved in collaboration with Thomas Barnet-Lamb and David Geraghty.
10 February		Jonathan Pila (Bristol) "A model-theoretic approach to problems of Manin-Mumford-Andre-Oort-type" Abstract: I will describe a result, joint with Alex Wilkie, about the distribution of rational points on certain non-algebraic sets in real space. The natural setting is an 'o-minimal structure over the real numbers', a notion from model-theory. A surprising strategy, proposed by Umberto Zannier, uses this result to approach diophantine problems in the Manin-Mumford-Andre-Oort circle of conjectures. I will describe some implementations of this strategy, including an unconditional proof of the Andre-Oort conjecture for products of modular curves.
17 February		Frank Neumann (Leicester) "Moduli stacks of vector bundles on curves and Frobenius morphisms" Abstract: After giving a brief introduction into moduli problems and moduli stacks I will indicate how to calculate the l-adic cohomology ring of the moduli stack of vector bundles on an algebraic curve in positive characteristic and explicitly describe the actions of the various geometric and arithmetic Frobenius morphisms on the cohomology ring. It turns out that using the language of algebraic stacks instead of geometric invariant theory this becomes surprisingly easy. If time permits I will indicate how to prove some analogues of the classical Weil conjectures for the moduli stack. This is work in progress with Ulrich Stuhler (Goettingen).
24 February		Wansu Kim (Imperial) "Galois deformation theory for norm fields"
3 March		Lawrence Breen (Paris) "Non-abelian and partly abelian cohomology theories"
10 March		Sarah Zerbes (Exeter) "Wach modules and Iwasawa theory for modular forms"
17 March		Cecile Armana (Paris/Barcelona) "Coefficients of Drinfeld modular forms and Hecke operators" Abstract: We will talk about Drinfeld modular forms, which are analogues, over the function field $\mathbf{F}_{q}(T)$, of classical modular forms. Given a classical cusp form $f$, there exists a simple formula relating the $n$-th Fourier coefficient of $f$ to the first coefficient of $T_{n}(f)$ ($T_n$ denotes the $n$-th Hecke operator). This property has several consequences, for instance the multiplicity one theorem. Drinfeld modular forms possess series expansion and Hecke operators acting on them. The aim of the talk is to present a formula giving, for any Hecke Drinfeld eigenform, some of its coefficients in terms of its eigenvalues.

Autumn 2009

This term, the seminar was held at Imperial College and was organised by Alexei Skorobogatov.

The seminar program for Autumn 2009

7 October		Samir Siksek (Warwick) "Explicit Chabauty over Number Fields" Abstract: Let $C$ be a curve of genus at least $2$ over a number field $K$ of degree $d$. Let $J$ be the Jacobian of $C$ and $r$ the rank of the Mordell-Weil group $J(K)$. Chabauty is a practical method for explicitly computing $C(K)$ provided $r \leq g-1$. In unpublished work, Wetherell suggested that Chabauty's method should still be applicable provided the weaker bound $r \leq d(g-1)$ is satisfied. We give details of this and use it to solve the Diophantine equation $x^2+y^3=z^{10}$ by reducing the problem to determining the $K$-rational points on several genus $2$ curves over $K=\Q(\sqrt[3]{2})$.
14 October		Florian Pop (University of Pennsylvania and the Newton Institute) "On the Ihara/Oda-Matsumoto Conjecture" Abstract: In his "Esquisse d'un programme", Grothendieck suggested that one should be able to give a non-tautological description of the absolute Galois group of the rationals via its action on the geometric fundamental group of "interesting" varieties. Similar was suggested/asked by Ihara, and a precise conjecture was made by Oda-Matsumoto. In my talk I plan to report on the status of the art of this problem.
21 October		Imperial Commemoration day (no seminar)
28 October		Lassina Dembele (Warwick) "Nonsolvable Galois number fields ramified at 2, 3 and 5 only" Abstract: In the mid 90s, Dick Gross proposed the following conjecture. Conjecture: For every prime p, there is a nonsolvable Galois number field K ramified at p only. For p>=11, this conjecture is a consequence of results of Serre and Deligne (using classical modular forms). In this talk, we will show that the conjecture is true for p=2, 3 and 5. The extensions K we constructed in those cases are obtained by using Galois representations attached to Hilbert modular forms. We will also outline a strategy to tackle the case p=7 using automorphic forms on U(3).
4 November		Roger Heath-Brown (Oxford) "Counting points on cubic curves" Abstract: Given a smooth plane cubic curve C defined over the rationals, we are interested in upper bounds for the number of rational points of height at most B, say, which are uniform in the curve C. Two previous approaches will be described, along with a new hybrid version.
11 November		Don Blasius (UCLA) "Asymptotic Fullness of Automorphic Galois Representations" Abstract: On a reductive group G over a number field, limit multiplicity theorems give the growth rate, as a function of suitably growing level, for the number of cusp forms $\pi$ which have given discrete series type at infinity. In this talk we look at some finer structure arising from the existence of Galois representations attached to such forms. Specifically, we ask whether the subset of those with largest Zariski closure has density one among all the forms. For some simple cases we prove the conjecture, or provide a positive density result. One proof of the latter uses a result about the asymptotic distribution of Hecke eigenvalues at a fixed unramified finite place, namely that this distribution is Plancherel measure.
16 November		The London-Paris Number Theory Seminar speakers: M. Emerton, A. Skorobogatov, S. David
18 November		Herbert Gangl (Durham) "Double zeta values and periods of modular forms" Abstract: We give new relations among double zeta values \zeta(r,s)=\sum_{m>n>0} m^{-r} n^{-s} and show that the structure of the Q-vector space of all relations among double zeta values of weight k is connected in several different ways with the structure of the space of modular forms of weight k on the full modular group. (Joint work with M.Kaneko and D.Zagier.)
25 November		Fabien Trihan (Nottingham) "On the p-parity conjecture in the function field case" Abstract: Let F be a function field in one variable with field of constants a finite field of characteristic p>0. Let E/F be an elliptic curve over F. We show that the order of the Hasse-Weil L-function of E/F at s=1 and the corank of the p-Selmer group of E/F have the same parity (joint work with C. Wuthrich).
2 December		Behrang Noohi (King's) "Galois cohomology of crossed-modules and cohomology of reductive groups" Abstract: A 2-group (or a crossed-module) is a categorified version of a group. Line bundles over a scheme, for instance, form the Picard 2-group. Galois cohomology of 2-groups can be used to give information about Galois cohomology of ordinary groups (via, say, certain long exact sequences). We discuss the basics of the theory and give some simple examples involving Picard and Brauer groups. We then explain Borovoi's application of these ideas to the study of Galois cohomology of reductive groups.
9 December		Javier Lopez (Queen Mary) "Torified schemes and geometry over the field with one element" Abstract: In this talk we introduce the notion of torified variety as a reduced scheme X of finite type that admits a decomposition $T = \{T_i\}_{i\in I}$ by split tori. This is a general concept that includes toric varieties, homogeneous spaces and Chevalley group schemes among others. We will show some of the main properties of torified varieties, show how the torifications define geometries over the field with one element. We also show how a torification provides an easy way to compute the counting function of $X$, which can be immediately applied to compute the corresponding zeta functions over $\mathbb{F}_1$.

Summer 2009

This term, the seminar was held at King's College and was organised by Fred Diamond.

The seminar program for Summer 2009

8 April		Mehmet Haluk Sengun (Duisburg-Essen) "Computing With Bianchi Modular Forms" Abstract: Bianchi modular forms are modular forms over imaginary quadratic fields. In this talk, we present an algorithm to compute these forms and the Hecke action on them. Then we discuss their conjectural connections with mod p Galois representations, presenting certain results and calculations.
29 April		Francis Brown (Paris-Jussieu) "Feynman graphs, moduli spaces and multiple zeta values" Abstract: I will begin by explaining how Feynman graphs in perturbative quantum field theory define interesting periods in the sense of algebraic geometry. Extensive computations by physicists suggest that these evaluate numerically to multiple zeta values in all known cases, but recent work of Belkale and Brosnan leads one to expect that the underlying motives may be of general type. After giving an overview of recent work on the subject, I will try to give a geometric and combinatorial explanation for these observations.
6 May		Alexander Stasinski (Southampton) "Unramified and Regular Representations" Abstract: The talk will be about two rather different constructions of smooth (complex) representations of certain compact $p$-adic groups. The first is a cohomological construction of so called unramified representations of reductive groups over finite local rings, and is a generalization of the classical construction of Deligne and Lusztig. This gives in particular a family of representations of any compact $p$-adic group of the form $G(\mathfrak{o})$, where $G$ is a reductive group over the ring of integers $\mathfrak{o}$ in a local non-Archimedean field. The second construction is a purely algebraic approach to the regular representations of $GL_N(\mathfrak{o})$, which is formally similar to the Bushnell-Kutzko construction of supercuspidal representations. We shall describe the main features of the constructions, and discuss some open questions regarding their overlap, that is, to what extent representations given by one construction are also given by the other.
13 May		Alex Bartel (Cambridge) "On class number relations in dihedral extensions of number fields" Abstract: In 1950, Brauer and Kuroda independently considered relations of class numbers and of regulators of intermediate fields in Galois extensions. These relations arise from the analytic class number formula and Artin formalism for L-functions and allow one to express certain quotients of class numbers in terms of corresponding quotients of regulators and of numbers of roots of unity. In some special cases, the regulator quotient can then be interpreted as a unit index. For extensions with dihedral Galois group of order 2p for p an odd prime, this was first done by Halter-Koch over Q and more recently by Lemmermeyer over arbitrary fields but under a very restrictive assumption. I will show how to derive a formula for arbitrary dihedral extensions of order 2p^n. The technique, which is purely representation theoretic, comes from the theory of elliptic curves, where one can consider a similar compatibility of the Birch and Swinnerton-Dyer conjecture with Artin formalism.
20 May		Andreas Langer (Exeter) "Torsion zero cycles and p-adic integration theory" Abstract: We study the Chow-group of zero-cycles on the self-product of a CM-elliptic curve over the field of p-adic numbers and prove that its p-primary torsion subgroup is finite, provided that p is an ordinary good reduction prime and the p-adic L-function L_p(E,s) does not vanish at s=0. In the course of the proof we construct a new indecomposable element in K_1 which is integral at p, by using Coleman's p-adic integration theory and Besser's computation of syntomic regulators for K_2 of curves and K_1 of surfaces.
27 May two talks room 2B08		2:00-3:00, room 2B08 Nigel Boston (UC-Dublin, Wisconsin) "The fewest primes ramifying in a G-extension of Q" Abstract: If G is a finite group, what is the smallest number of primes ramifying in a G-extension of the rationals? We give evidence for a conjectural answer, together with a conjectural density for such n-tuples. [Parts are joint work with Ellenberg-Venkatesh and Markin.] 3:30-4:30, room 2B08 Jean-Pierre Serre "Variation with p of the number of solutions mod p of a given family of equations"
3 June - 4 June		The London-Paris Number Theory Seminar at King's College London, room 2B08 theme: p-adic modular forms speakers: Buzzard, Fargues, Loeffler, Colmez, Mokrane, Dimitrov, Panchishkin
10 June cancelled		cancelled The previously announced talk by Lassina Dembele is cancelled because of delays in getting his visa.
17 June		Christopher Deninger (Muenster) "Vector bundles on p-adic curves and p-adic representations" Abstract: The classical Narasimhan-Seshadri correspondence gives a bijection between stable vector bundles of degree zero on a compact Riemannian surface and irreducible unitary representations of its fundamental group. In joint work with Annette Werner we have transferred this correspondence to some extent to a p-adic setting. We will report on recent progress and the main open questions in this area. There is related work of Faltings on p-adic Higgs bundles.

Spring 2009

This term, the seminar was held at University College and was organised by Andrei Yafaev.

The seminar program for Spring 2009

14 January		Victor Abrashkin (Durham) "p-adic semistable representations and generalization of the Shafarevich Conjecture" Abstract: Breuil's theory of semistable p-adic representations is applied to prove the following property: if X is a projective variety over Q with semistable reduction modulo 3 and good reduction at all other primes then its Hodge number h^{2,0} = 0.
21 January		no seminar (Minhyong Kim's inaugural lecture "On numbers and figures" in room 505, Mathematics Department, UCL at 4.30pm)
28 January		Anna Cadoret (Bordeaux) "A uniform open image theorem for p-adic representations of etale fundamental groups of curves"
4 February		Tim Browning (Bristol) "Rational points on cubic hypersurfaces" Abstract: Given a cubic hypersurface X defined over Q, the circle method furnishes a method for establishing the existence of Q-rational points on the hypersurface, provided that the dimension is sufficiently large. Thanks to work of Davenport, and more recently of Heath-Brown, we can now treat cubic hypersurfaces of dimension at least 12. In this talk I show how this can be improved to dimension 11 when the underlying cubic form can be written as the sum of two forms without any variables in common.
11 February		Pierre Parent (Bordeaux) "Method of Runge and modular curves"
18 February		Adrian Diaconu (Nottingham) "Trace formulas and moments of automorphic L-functions"
25 February TWO TALKS		2:00-3:00 Hershy Kisilevsky (Concordia) "Critical values of derivatives of (twisted) elliptic L-functions" Abstract: Let $L(E/\Q,s) be the $L$-function of an elliptic curve $E$ defined over the rational field $\Q.$ If $\chi$ is a Dirichlet character of odd prime order such that $L(E,1,\chi)=0,$ we examine the special values of the derivative. If $L'(E,1,\chi)$ is non-zero, we provide computational evidence for an "explicit formula" for its value. We also have some cases of higher order special values in the case that $\ord_{s=1}L(E,s,\chi)>1.$ 4:00-5:00 Christian Wuthrich (Nottingham) "Self-points on Elliptic Curves" Abstract: Let $E$ be an elliptic curve of conductor $N$. Given a cyclic subgroup $C$ of order $N$ in $E$, we construct a modular point $P_C$ on $E$, called self-point, as the image of $(E,C)$ on $X_0(N)$ under the modular parametrisation $X_0(N)\to E$. In many cases (e.g. when E is semi-stable), one can prove that the point is of infinite order in the Mordell-Weil group of $E$ over the field of definition of $C$. The study of these points in the $PGL_2(\mathbb{Z}_p)$-tower inside $\mathbb{Q}(E[p^\infty])$ continues earlier work of Harris. It is also possible to construct ``derivatives'' \`a la Kolyvagin.
4 March TWO TALKS		2:00-3:00 Pierre Debes (Lille) "Specializations of Galois covers" Abstract: The motivation is to investigate the specializations of a Galois cover over some field. The rational points on some twisted cover provide a key to the problem. Good behaviour of these twists with respect to reduction leads to some concrete answer over "big" fields. Our results relate to some questions in inverse Galois theory, to some works of Fried, Colliot-Thelene, Ekhedal on Hilbert's irreducibility theorem and to some classical theorems of Grunwald and Neukirch. (This is a joint work with Nour Ghazi). 4:00-5:00 Gunter Harder (Bonn) "Denominators of Eisenstein classes"
9 March (Monday)		London Number Theory Seminar, Special Lectures Bao Chau Ngo (Institute for Advanced Study) Lecture 1: "Fundamental lemma and Hitchin fibration" at 3:00 pm, room 706 Lecture 2: "Symmetry of Hitchin fibration and endoscopy" at 4:30 pm, room 706
11 March		Ambrus Pal (Imperial) "Rational points on genus one curves"
13 March (Friday)		London Number Theory Seminar, Special Lectures Bao Chau Ngo (Institute for Advanced Study) Lecture 3: "Decomposition theorem in the case of the Hitchin fibration" at 4:00 pm, room 500
18 March		Richard Hill (UCL) "Residually infinite extensions of arithmetic groups"
25 March		Victor Snaith (Sheffield) "Computing the Borel regulator" in room 505

Autumn 2008

This term, the seminar was held at Imperial College and was organised by Kevin Buzzard.

The seminar program for Autumn 2008

1 October 14.00-15.00 room 342		Thomas Zink (Bielefeld) "p-divisible groups over regular local rings of mixed characteristics"
8 October		Go Yamashita "Upper bounds for the dimensions of p-adic multiple zeta values"
15 October		Gihan Marasingha (Bristol) "A degree 4 del Pezzo surface: Manin's conjecture and almost primes"
22 October		Cecilia Busuioc (Imperial) "Milnor K-theory and Modular Symbols"
29 October		Tejaswi Navilarekallu (Vrije Universiteit Amsterdam) "Equivariant p-adic L-values"
5 November		Owen Jones (Imperial) "Analytically induced representations and generalised Verma modules"
12 November		Seidai Yasuda (Kyoto) "Diagonal periods of GL(n) over the rational function field"
17 November (Monday)		The London-Paris Number Theory Seminar at the Insitut Henri Poincaré in Paris
19 November		Jeanine van Order (Cambridge) "Analogues of Rohrlich's theorem"
26 November		Kevin McGerty (Imperial) "A gentle introduction to the geometric Langlands program"
3 December		Mohamed Saidi (Exeter) "On Grothendieck's anabelian section conjecture for curves"
10 December		Gaetan Chenevier (ENS Paris) "The infinite fern of Galois representations of type U(3)"
17 December		Payman Kassaei (King's) "Geometry of Hilbert modular varieties and canonical subgroups of abelian varieties with real multiplication"

Summer 2008

This term, the seminar was held at King's College and was organised by David Solomon.

The seminar program for Summer 2008

23 April		Henri Johnston (Oxford) "Non-existence and splitting theorems for normal integral bases" Abstract: This is joint work with Cornelius Greither. We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower Q \subset K \subset L forces the tower to be split in a very strong sense.
30 April		Richard Hill (UCL and Heilbronn Institute) "Vanishing theorems for p-adic automorphic forms"
7 May - 8 May		The London-Paris Number Theory Seminar
13 May Tuesday 3:30-4:30 room 436		Cristian Popescu (UCSD) "On the Coates-Sinnott Conjectures"
14 May		Andrew Booker (Bristol) "Computing automorphic forms on GL(3)" Abstract: My student, Ce Bian, announced the computation of a few "generic" rank 3 automorphic forms (meaning they are not lifts from lower rank examples) at the AIM workshop "Computing arithmetic spectra" in March. I will give a brief introduction to the theme of the workshop and describe Bian's computations. I'll also say a few words about the bewildering amount of attention that the work received subsequently.
21 May		Ambrus Pal (Imperial College) "The Manin constant of elliptic curves over function fields" Abstract: We study the p-adic valuation of the values of normalized Hecke eigenforms attached to non-isotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic p. Under certain assumptions we derive lower and upper bounds on the smallest attained valuation in terms of the minimal discriminant. As a consequence we show that the former can be arbitrarily small. We also use our results to prove for the first time the analogue of the degree conjecture unconditionally for infinite families of strong Weil curves defined over rational function fields.
28 May		Gautam Chinta (City College of New York) "Sums of two squares and sums of three squares" Abstract: I will begin by describing three results -- Gauss's three squares theorem, Hamburger's converse theorem, and Maass's evaluation of a sum over Heegner points of the Eisenstein series for the modular group. I then will describe conjectural generalizations of these results to GL(3). The unifying theme is a conjecture of Jacquet on orthogonal periods of automorphic forms on GL(r). This is a joint work with Omer Offen.
4 June		Doug Ulmer (U. Arizona at Tucson, and Paris) "On Mordell-Weil groups of abelian varieties over function fields" Abstract: I will sketch a construction which, among other things, relates CM of certain abelian varieties over a field k to Mordell- Weil groups of certain abelian varieties over K=k(t). The construction yields completely explicit Mordell-Weil groups of arbitrarily large rank for finite k and a less explicit, but new, construction of abelian varieties over K of moderately large rank when k is the field of algebraic numbers.
11 June		Steven Galbraith (Royal Holloway) "Applications of the Frobenius map in elliptic curve cryptography" Abstract: Elliptic curves over finite fields provide groups for which the discrete logarithm problem seems to be hard. Hence, elliptic curves have applications in public key cryptography. The Frobenius map has been used to speed up arithmetic on elliptic curves. The talk will survey some of these ideas. We will also discuss security implications of using Frobenius maps and present a new algorithm for solving the "Frobenius expansion discrete logarithm problem"
18 June		Mahesh Kakde (Cambridge) "On the non-commutative Main Conjecture for totally real number fields"
25 June		Jonathan Pila (Bristol) "Rational points of definable sets and the Manin-Mumford conjecture" Abstract: I will discuss problems and results concerning the distribution of rational points on certain non-algebraic sets. More specifically, definable sets in o-minimal structures. I will describe a result, joint with Wilkie, that such a set X can have only ``few'' rational points in a suitable sense, that do not lie on some connected semi-algebraic subset of X of positive dimension. I will describe some further results and conjectures, connections with transcendence theory, and a new proof (with Zannier) of the Manin-Mumford conjecture by combining these ideas with a result of Masser.

Spring 2008

This term, the seminar was held at University College and was organised by Minhyong Kim.

The seminar program for Spring 2008

9 January		Pierre Debes (Lille) "Inverse Galois theory, Abelian varieties and modular towers" Abstract: This is a joint work with Anna Cadoret. We show a new constraint in constructing Galois covers of $\P^1$ over $\Q$ with a given Galois group $G$. If for some prime $p$, the order of the abelianization $P^{ab}$ of the $p$-Sylow subgroups $P$ of $G$ is suitably large, compared to the index $[G:P]$ and the number $r$ of branch points, then the branch points must coalesce modulo small primes. This is related to some rationality questions on the torsion of abelian varieties. This connection also provides a new viewpoint and new results on the Modular Tower program.
16 January		Makis Dousmanis (Paris 13) "Reductions of some families of two-dimensional crystalline representations"
23 January		Konstantin Ardakov (Nottingham) "Reflexive ideals in Iwasawa algebras"
30 January		Ben Green (Cambridge) "Distribution of Polynomials over finite fields" Abstract: Let F be a finite field and consider polynomials P : F^n -> F in n variables. What can we say about polynomials which are not equi-distributed, i.e. for which the proportion of x for which P(x) = c is not roughly 1/\|F\|? We introduce a notion of "rank" for multi-variable polynomials and show that such polynomials must have low rank. We apply this result to the study of so-called Gowers norms. Let P : F^n -> F be any function. It is well-known that P is a polynomial of degree d if the polynomial Q obtained by differencing d+1 times is identically zero. What if Q is not identically zero, but merely biased towards zero? The inverse conjecture for the Gowers norms predicted that, in this case, P correlates with a degree d polynomial. Using the result described above we establish this in certain cases. We will also discuss an example which shows that the conjecture can fail in very low characteristic. Joint work with T. Tao.
6 February two talks 4 pm-6 pm		4 pm-5 pm: Takako Fukaya (Keio and Cambridge) "Root numbers, Selmer groups, and non-commutative Iwasawa theory" 5 pm-6 pm: Kazuya Kato (Kyoto and Cambridge) "Classifying spaces of mixed Hodge (resp. p-adic Hodge) structures"
13 February		Nick Shepherd-Barron (Cambridge) "Thomae's formula for non-hyperelliptic curves" Abstract: In 1857 Thomae gave formulae for the theta-constants of a hyperelliptic curve in terms of projective data of the curve. In this talk we explain what this means in terms of moduli spaces and extend it to non-hyperelliptic curves.
20 February		Fabien Trihan (Nottingham) "Crystalline representations and F-crystals"
27 February		Andreas Doering (Imperial) "Topos theory in the foundations of physics"
5 March		Burt Totaro (Cambridge) "Moving codimension-one subvarieties over finite fields"
12 March		Tamas Hausel (Oxford) "Arithmetic harmonic analysis on character and quiver varieties"

Autumn 2007

This term, the seminar was held at Imperial College and was organised by Ambrus Pal.

The seminar program for Autumn 2007

3 October		David Loeffler (ICL) "Overconvergent p-adic automorphic forms and eigenvarieties for compact reductive groups" Abstract: I shall describe a construction of an eigenvariety parametrising p-adic automorphic forms for any reductive group G over Q that is split at p and compact at infinity. The construction generalises the work of Chenevier for compact forms of GL_n and Buzzard for quaternion algebras. The method gives a space of automorphic forms for each standard parabolic subgroup P of G; in this gives a hierarchy of "semi-classical" automorphic forms intermediate between the space of classical forms (corresponding to P = G) and the spaces constructed by Chenevier in the unitary case (which correspond to P = Borel). If there is time, I shall also mention ongoing work on classicality criteria and connections to Galois representations.
10 October		Yiannis Petridis (UCL) "On the distribution of modular symbols"
17 October		Andreas Schweizer (University of Exeter) "On the torsion of elliptic curves over sufficiently general function fields" Abstract: If K varies over all complex function fields and E varies over all elliptic curves over K with j(E) not in C, it is known that the size of the torsion group E(K)_{tors} can be uniformly bounded by a number depending only on the genus of K. Moreover, if one restricts to function fields K that are ``special'', for example hyperelliptic, one can even give absolute bounds (not depending on the genus of K) for the size of E(K)_{tors}. We will discuss what happens if K varies over all function fields that are ``sufficiently general''.
24 October		David Solomon (King's College) "Stickelberger's Theorem Revisited" Abstract: Stickelberger's Theorem (from 1890) gives an explicit ideal in the Galois group-ring which annihilates the imaginary part of the class group of an abelian field. In the 1980s Tate and Brumer proposed a generalisation (the "Brumer-Stark conjecture" ) for in any abelian extension of number fields K/k with K CM and k totally real. Both the theorem and the conjecture leave certain questions unanswered: Is the (generalised) Stickelberger ideal the full annihilator, the Fitting ideal or what? And, at a more basic level, what can we say in the plus part, eg for a real abelian field? (In the latter case, Stickelberger's theorem amounts to little more than 0=0!) I shall discuss possible answers, some still conjectural, to pieces of these puzzles, using two new p-adic ideals of the group ring. There are interesting connections with Iwasawa Theory, the Equivariant Tamagawa Number Conjecture etc.
31 October		Samir Siksek (University of Warwick) "Chabauty for Symmetric Powers of Curves" Abstract: Chabauty is a classical method for computing the rational points of curves of higher genus. In this talk, we explain an adaptation of Chabauty which allows us in many cases to compute all rational points on the d-th symmetric power of a curve provided the rank of the Mordell-Weil group of the Jacobian is at most g-d (where g is the genus). We illustrate this by giving two examples of genus 3, one hyperelliptic and the other plane quartic.
7 November		Urs Hartl (University of Münster) "Period Spaces for Hodge-Structures in Equal Characteristic" Abstract: We construct period spaces for Hodge structures in equal characteristic. These Hodge structures were invented by Pink. The period spaces are analogues of the Rapoport-Zink period spaces for Fontaine's filtered isocrystals in mixed characteristic. For our period spaces we determine the image of the period morphism as a Berkovich open subspace. We prove the analogue of a conjecture of Rapoport Zink stating the existence of interesting local systems on this image. Moreover, we prove the analogue of the Colmez-Fontaine Theorem that "weakly admissible implies admissible". As a consequence the Berkovich open subspace mentioned above contains every classical rigid analytic point of the period space.
12 November		The London-Paris Number Theory Seminar
14 November		Alberto Minguez (University of East Anglia) "On the Howe correspondence" Abstract: The aim of this talk is to introduce the audience to the theory of local Howe correspondence. For the dual pair of type (Gl(n), Gl(m)) we will show a new proof which allows us to describe the correspondence in terms of Langlands parameters. At the end, we will discus about the possibility of having such a correspondence for l-modular representations.
21 November 4 pm-5:30 pm		Eyal Goren (McGill University, Montréal) "Class invariants for CM fields of degree 4" Abstract: The problem of effective construction of units in abelian extensions of number fields is at present out of reach. Notwithstanding conjectural constructions, the only exceptions are the constructions for abelian extensions of Q and of a quadratic imaginary field, where the units are the cyclotomic and elliptic units respectively. One reason one seeks such constructions is to find Stark units which appear in Stark's conjectures on special values of L functions. In this talk, after explaining what is the source of the difficulty, I shall survey what we know at present about the case of CM fields of degree 4, focusing on my work with Ehud de Shalit, Kristin Lauter and Daniel Vallieres. Time allowing, I shall try and put the results in the perspective of the work of Jan Bruinier and Tonghai Yang, indicate some proofs of our results and discuss work in progress.
28 November		Carlos Castano-Bernard (ICTP, Trieste) "On the subgroup generated by the traces of Heegner points on elliptic curves" Abstract: Consider an elliptic curve E over Q and assume its L-function has a simple zero at s = 1. In particular, there is a non-constant morphism X0(N)/wN --> E defined over Q, where wN is the Fricke involution and N is the conductor of E. So the trace of each Heegner point on E is Q-rational. Moreover, it is well-known that E(Q)/E(Q)tors is isomorphic to Z, and in fact the images of the traces in E(Q)/E(Q)tors generate a subgroup of finite index I. In this talk we shall discuss a conjecture that predicts that whenever N is prime and the index I > 1, then the real locus (X0(N)/wN)(R) has more than one connected component or--less likely--the Tate-Shafarevich group of E is non-trivial.
5 December		Laurent Fargues (Université Paris-Sud, Orsay) "Ramification of Lubin-Tate groups and the Bruhat-Tits building" Abstract: One of the purposes of this talk is to give a description of the isomorphism between the p-adic Lubin-Tate and Drinfeld towers at the level of their skeletons. For the Drinfeld space, its skeleton is the Bruhat-Tits building of the linear group. For example, we can describe explicitly the pull-back of this simplicial structure on the open p-adic ball associated to the Lubin-Tate space. We also study in detail the different ramification filtrations (upper and lower) associated to Lubin-Tate groups. We give applications to generalized canonical subgroups and fundamental domains for Hecke correspondences.
12 December		Sarah Zerbes (ICL) "Formulae for the higher Hilbert pairing"

Summer 2007

This term, the seminar was held at University College and was organised by Richard Hill.

The seminar program for Summer 2007

25 April		Shaun Stevens (UEA) "Supercuspidal representations of p-adic classical groups"
2 May		The London-Paris Number Theory Seminar 11 am - 4:30 pm, Imperial College London
9 May		Ben Smith (Royal Holloway) "Computing Explicit Isogenies" Abstract: Isogenies --- surjective homomorphisms of algebraic groups with finite kernel --- are basic objects in number theory. Algorithms for computing with isogenies of elliptic curves are well-known; in higher dimensions, however, the situation is more complicated, and few explicit non-trivial examples of isogenies are known. We will describe some interesting examples of explicit isogenies of Jacobians of low-genus curves, discuss some of the computational issues, and give some applications in modern cryptography.
16 May		Matthias Strauch "Potentially crystalline representations and associated p-adic representations of GL_2" Abstract: This is about joint work in progress with C. Breuil. For a certain family of potentially crystalline but not semistable two-dimensional representations of the absolute Galois group of Q_p we construct locally analytic representations of GL_2(Q_p), naturally parameterised by the Galois representations.
23 May		Manuel Breuning (KCL) "Determinant functors and Euler characteristics"
30 May		Jayanta Manoharmayum (Sheffield) "Lifting Galois representations"

Spring 2007

This term, the seminar was held at King's College and was organised by David Burns.

The seminar program for Spring 2007

17 January		Kevin Buzzard (Imperial) "Mod p Galois representations and modular forms"
24 January		Richard Hill (UCL) "Singular cohomology of modular curves" Abstract: Let \Gamma be an arithmetic group acting on the upper half-plane H, either with cusps or cocompact. Let \Gamma' be a normal subgroup of \Gamma. Then the quotient group G=\Gamma/\Gamma' acts on the cohomology of \Gamma'. I'll describe the structure of H^1(\Gamma',\Z) as a \ZG-module.
31 January		David Burns (King's) "Iwasawa theory of elliptic curves in p-adic Lie extensions"
7 February		Alex Paulin (Imperial) "Local to Global Compatibility on the Eigencurve" Abstract: To any classical cuspidal eigenform one can attach both a smooth irreducible representation of GL_2(Q_l) and a two dimensional Frob-semisimple Weil-Deligne representation. Classical Local-Global compatibility ensures that these agree under the (correctly normalised) local langlands correspondence. I will discuss ways of attaching such objects to overconvergent p-adic eigenforms across the eigencurve and to what extent local-global compatibility remains valid.
14 February		Tom Fisher (Cambridge) "Finding rational points on elliptic curves using 6-descent and 12-descent" Abstract: Descent on an elliptic curve E is used to obtain partial information about both the group of rational points (the Mordell-Weil group) and the failure of the Hasse principle for certain twists of E (the Tate-Shafarevich group). The Selmer group elements computed may be represented as n-coverings of E. Traditionally one takes n to be a prime power. Breaking with this tradition, I explain how to combine the data of an m-covering and an n-covering, for m and n coprime, to obtain an mn-covering. This technique improves the search for rational points on E. In particular using 6-descent and 12-descent, I was recently able to find all the "missing" generators for the elliptic curves of analytic rank 2 in the Stein-Watkins database.
21 February		Daniel Delbourgo (Nottingham) "Non-abelian congruences between L-values of elliptic curves"
28 February		Olivier Brinon (Paris 13) "Overconvergence of $p$-adic representations: the relative case (joint work with F. Andreatta)"
7 March		Jan Kohlhaase (Muenster) "The Cohomology of locally analytic Representations" Abstract: Starting with smooth representations of a p-adic reductive group, we will recall what is meant by a supercuspidal representation and the role such representations play in the local Langlands correspondence. We will then pass to locally analytic representations in the sense of Schneider/Teitelbaum, sketch the construction of locally analytic cohomology and generalize the above notion of supercuspidality to locally analytic representations. In the end, we will compute the (higher) Jacquet modules of locally analytic principal series representations and indicate why this is of importance for the p-adic Langlands correspondence.
14 March		Colin Bushnell (King's) "Characters and constants"
21 March		Tobias Berger (Cambridge) "Congruences between modular forms over imaginary quadratic fields" Abstract: We present two applications of congruences involving Harder's Eisenstein cohomology classes. We first prove a lower bound for the size of the Selmer group of certain Galois characters of imaginary quadratic fields coinciding with the value given by the Bloch-Kato conjecture. We further show how to obtain instances of the Fontaine-Mazur conjecture for imaginary quadratic fields in the residually reducible case. The latter is joint work in progress with Kris Klosin.

Autumn 2006

This term, the seminar was held at Imperial and was organised by Alexei Skorobogatov.

The seminar program for Autumn 2006

4 October		Andrei Yafaev (UCL) "On the triviality of rational points on certain Atkin-Lehner quotients of Shimura curves" Abstract: This is a joint work with Pierre Parent. We use a modification of Mazur's method to prove that the only possible rational points on certain Atkin-Lehner quotients of Shimura curves come from CM points.
11 October		Alexei Skorobogatov (Imperial College) "A finiteness theorem for the Brauer group of K3 surfaces" Abstract: Let k be a field finitely generated over the rationals, and let X be a K3 surface over k. We prove that Br(X)/Br(k) is finite.
18 October		Tim Dokchitser (Cambridge) "Parity of ranks for elliptic curves with a cyclic isogeny" Abstract: This is a joint work with Vladimir Dokchitser. Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny and semistable at primes above p. Then one can determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves (with an extra mild assumption for p=2).
25 October		no seminar (Imperial's Commemoration Day)
1 November		Jan Nekovar (Paris 7) "Parity of ranks of Selmer groups in p-adic families"
8 November		Sarah Zerbes (Imperial) "Higher-dimensional logarithmic derivatives" Abstract: In my talk, I will explain how to construct logarithmic derivative maps for n-dimensional local fields of mixed characteristic (0,p). The main ingredients for this construction are higher-dimensional rings of overconvergent series and Tony Scholl's work on general fields of norms. As an application of the logarithmic derivative, I will give a new construction of Kato's dual exponential map for K_n.
13 November		Séminaire de théorie des nombres Londres-Paris à l'Institut Henri Poincaré programme Orateurs: Fred Diamond, Toby Gee, Florian Herzig.
15 November		Andres Helfgott (Bristol) "How small must ill-distributed sets be?" (joint with A Venkatesh) Abstract: Consider a set $S\subset \mathbb{Z}^n$. Suppose that, for many primes $p$, the distribution of $S$ in congruence classes $\mo p$ is far from uniform. How sparse is $S$ forced to be thereby? A clear dichotomy appears: it seems that $S$ must either be very small or possess much algebraic structure. We show that, if $S\subset \mathbb{Z}^2 \cap \lbrack 0, N\rbrack^2$ occupies few congruence classes $\mo p$ for many $p$, then either $S$ has fewer than $N^{\epsilon}$ elements or most of $S$ is contained in an algebraic curve of degree $O_{\epsilon}(1)$. Similar statements are conjectured for $S\subset \mathbb{Z}^n$, $n\neq 2$. We follow an approach that combines ideas from the larger sieve of Gallagher \cite{Ga} and from the work of Bombieri and Pila \cite{BP}. All techniques used are elementary.
22 November		Fred Diamond (King's) "The weight part of Serre's conjecture for Hilbert modular forms" Abstract: I will explain the statement of a generalization of Serre's conjecture on mod p Galois representations to the context of Hilbert modular forms. The emphasis will be on the recipe for the set of possible weights (formulated by Buzzard, Jarvis and myself, and partly proved by Gee) and its behavior in some special cases.
29 November		Ambrus Pal (Imperial) "On a conjecture about the cohomology of arithmetic groups"
6 December		Jean Gillibert (Manchester) "Geometric Galois module structure and abelian varieties of higher dimension" Abstract: The so-called class-invariant homomorphism $\psi_n$, introduced by M. J. Taylor, measures the Galois module structure of (rings of integers of) extensions of the form $K(\frac{1}{n}P)/K$, where $K$ is a number field, $P$ is a $K$-rational point on an abelian variety $A$, and $n>1$ is an integer. When $A$ is an elliptic curve and $n$ is coprime to 6, then $\psi_n$ vanishes on torsion points. We explain here how, using Weil restrictions of elliptic curves, it is possible to construct abelian varieties of higher dimension for which this vanishing result is no longer true.
13 December		Payman Kassaei (King's) "A ``subgroup-free" approach to Canonical Subgroups" Abstract: I will discuss joint work with E. Goren in which we present a ``subgroup-free'' approach to canonical subgroups, which in particular extends all aspects of the classical theory of canonical subgroups of elliptic curves to many various Shimura curves of interest.

Summer 2006

This term, the seminar was held at KCL and was organised by David Burns.

The seminar program for Summer 2006

3 May		Alan Lauder (Oxford) "Zeta functions of surfaces over finite fields"
10 May		Carlos Castano-Bernard (Cambridge) "Two approaches in the study of $X_0^+(N)(Q)$" Abstract: In this talk we discuss two approaches in the study of the rational points of $X_0^+(N)$. The first was suggested by certain geometric properties of rational points of the canonical embedding of $X_0^+(N)$ in projective space, while the second approach was suggested by questions raised by Shimura on the connected components of the real locus of a given variety defined over the rational numbers.
17 May		Ben Smith (Royal Holloway) "Examples of explicit isogenies" Abstract: In "Factorisations explicites de g(y) - h(z)", Cassou-Nogues and Couveignes give a classification of the pairs of irreducible polynomials g and h such that g(y) - h(z) is reducible. We describe their results, and show how they may be used to derive families of hyperelliptic Jacobians with an explicit isogeny.
24 May		Rob de Jeu (Durham) "The Beilinson conjectures for $K_2$ of curves" Abstract: The Birch-Swinnerton-Dyer conjecture relates the value of the L-function of an elliptic curve defined over a number field at 1 with its codimension 1 Chow group. The latter is part of $K_0$ of the curve. Beilinson formulated conjectures of a similar flavour for all K-groups of any smooth projective variety defined over number fields, and we'll discuss those conjectures for $K_2$ of curves.
31 May		Herbert Gangl (Durham) "Double zeta values and modular forms" Abstract: We give new relations among double zeta values \zeta(r,s) = \sum_{m>n>0} m^{-r}n^{-s} and show that the structure of the Q-vector space of all relations among double zeta values of weight k is connected in several different ways with the structure of the space of modular forms of weight k on the full modular group.
7 June		Guy Henniart (Paris) "Representations of p-adic GL(2), old and new" Abstract: Classical modular forms have their local counterpart, which are irreducible representations of GL(2,Q_p) on complex vector spaces. Congruences between modular forms lead to study continuous representations of GL(2,F), where F is a p-adic field, on non-Archimedean Banach spaces,or on fields of positive characteristic. I shall review recent work and conjectures, with special emphasis on the case of mod-p representations.
14 June		Otmar Venjakob (Bonn) "On the leading terms of p-adic L-functions in non-commutative Iwasawa theory" Abstract: In the $GL_2$-main conjecture for elliptic curves without complex multiplication formulated by Coates, Fukaya, Kato, Sujatha and the speaker the existence of a (non-commutative) $p$-adic $L$-function $\mathcal{L}$ is crucial satisfying an interpolation formula related to the value at $s=1$ of certain twists of the Hasse-Weil $L$-function of $E$ twisted by certain Artin representations. By the work of Fukaya and Kato the interpolation property and existence of $\mathcal{L}$ is a consequence of their non-commutative Tamagawa number conjecture as well as a local analogue of the non-commutative Tamagawa number conjecture. The aim of this talk is to report on recent joint work with David Burns: We introduce leading terms of the $p$-adic $L$-function $\mathcal{L}$ at Artin representations and extend Fukaya and Kato's interpolation formula now also involving the leading terms of the twisted complex Hasse-Weil $L$-functions.
21 June 2.30pm		Minhyong Kim (Purdue) "Fundamental groups and Diophantine Geometry"
21 June 4.00pm		Martin Taylor (Manchester) "Galois structure and Riemann-Roch theorems" Abstract: The talk will begin with a general discussion of the importance of Euler characteristics the study of arithmetic modules. We shall then consider the use of Riemann Roch theorems in the determination of Euler characteristics: firstly, in a fairly general situation of varieties over fields; then for various torsors over arithmetic varieties.
26 July 2.00pm		Doug Ulmer (U. Arizona, Tucson) "Abelian varieties of large rank over function fields" Abstract: For every prime p and every integer g>0 we construct explicit abelian varieties of dimension g over F_p(t) for which the Birch and Swinnerton-Dyer conjecture holds and with arbitrarily large Mordell-Weil rank.

Spring 2006

This term, the seminar was held at KCL and was organised by David Solomon.

The seminar programme for Spring 2006

18 January		Tim Dokchitser (Cambridge) "Ranks of elliptic curves in cubic extensions" Abstract: For an elliptic curve E over a number field K, I shall prove that the algebraic rank of E goes up in infinitely many extensions of K obtained by adjoining a cube root of an element of K. I shall also discuss how this relates to root numbers and Iwasawa theory, with E=X_1(11) over Q as a specific example.
25 January		Andrei Yafaev (UCL) "The Andre-Oort conjecture" Abstract: This is a joint work with Bruno Klingler. We explain a proof of the Andre-Oort conjecture under the assumption of the generalised Riemann Hypothesis.
1 February		David Burns (KCl) "Algebraic p-adic L-functions in non-commutative Iwasawa theory" Abstract: There have been several important developments in non-commutative Iwasawa theory over the last few years. We discuss a natural construction of algebraic p-adic L-functions in this setting and discuss some interesting consequences for the main conjectures of non-commutative Iwasawa theory formulated by Coates, Fukaya, Kato, Sujatha and Venjakob and by Fukaya and Kato.
8 February		Daniel Caro (Durham) "Towards a good p-adic cohomology" Abstract: First, we will trace the history of the search for a good p-adic cohomology over schemes in characteristic p. We arrive at the construction of Berthelot's arithmetical D-modules. We will explain why these objects now represent the only possibility of obtaining a good p-adic cohomology. An important result which inspires trust in this theory is the following: for every overholonomic F-complex E of arithmetical D-modules over a variety X of characteristic p, there exists a splitting of X into locally closed subvarieties X _i such that the restrictions of E on X _i become much simpler (i.e., come from overconvergent F-isocrystals). In this talk, we will recall basic definitions and explain the meaning of this splitting.
15 February		Ivan Horozov (Durham) "Euler characteristics of arithmetic groups" Abstract: The talk will be about Euler characterisitcs of the general linear group, the special linear group, and the symplectic group over rings of algebraic integers. I will present a method for computing the homological Euler characteristic, as well as some applications to values of Dedekind zeta function at -1 and at -3 and to Kummer and Greenberg criteria for divisibility of certain class numbers by a prime.
22 February		Avner Ash (Boston College) "Symmetries of Algebraic Numbers" Abstract: The Absolute Galois Group of Q, notated G_Q acts on the solution sets of systems of polynomial equations with rational coefficients. Linear representations of G_Q play a leading role in getting information about the solution sets, e.g. in Wiles's proof of Fermat's Last Theorem. One influential conjecture in this area is that of Serre, concerning 2-dimensional representations of G_Q over finite fields. I will briefly review Serre's conjecture and discuss generalizations to higher dimensional representations.
1 March		Emmanuel Kowalski (Bordeaux) "The algebraic principle of the large sieve" Abstract: Linnik's original large sieve gives upper bounds for the number of integers in an interval with reductions modulo primes restricted to fall in fairly small sets. The talk will describe an abstract sieve setting which can lead to such results in more general situations. Then applications to the average distribution of Frobenius elements in families of algebraic varieties over finite fields will be discussed, and some work in progress concerning arithmetic properties of integral unimodular matrices. In one case the Riemann Hypothesis of Deligne is the crucial ingredient, in the other the spectral theory of automorphic forms appears naturally.
8 March		Misha Gavrilovich (Oxford) "Model theory, Z-extensions of C*, the exponential function, and a homotopy-theory viewpoint on some arithmetic issues"
15 March		James McKee (Royal Holloway) "Salem numbers, Pisot numbers, graphs, and Mahler measure" Abstract: This is joint work with Chris Smyth.We use graphs to define sets of Salem and Pisot numbers, and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the n-th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measure of a graph, and find all graphs of Mahler measure less than (1+sqrt5)/2. We start the task of extending this work from graph adjacency matrices to all integer symmetric matrices by classifying all such matrices having all eigenvalues in the interval [-2, 2].
22 March		Graham Everest (UEA) "Descent and Divisibility" Abstract: In 2001, Bilu, Hanrot and Voutier proved that for every n>30, the nth term of a Lucas or Lehmer sequence must have a primitive divisor. This is a remarkable result because of its uniform nature and the smallness of the bound 30. I will report on an elliptic analogue of their theorem. Also, I will report on the apparently harder statement about prime values.
5 April		Werner Bley (University of Kassel) "Computation of class groups"
12 April		Andrew Jones (KCL) "Dirichlet $L$-functions at $s=1$ and Fitting invariants of ideal class groups" Abstract: We show that a special case of the equivariant Tamagawa number conjecture (ETNC) of Burns and Flach implies a refinement of a `$p$-adic integrality conjecture' of Solomon concerning leading terms at $s = 0$ of certain `Twisted Zeta-functions'. In fact, the ETNC implies that an ideal defined by Solomon belongs to the initial Fitting invariant of a certain ideal class group. Since the relevant case of the ETNC is known to be valid for absolutely abelian fields we thereby obtain new results about the structure of ideal class groups. In particular, we obtain an analogue of Stickelberger's Theorem.

Autumn 2005

This term, the seminar was held at Imperial and was organised by Toby Gee.

The seminar programme for Autumn 2005

5 October		Toby Gee (Imperial) "On the weights of mod p modular forms"
12 October		Manuel Breuning (KCL) "A reinterpretation of the Chinburg conjectures"
19 October		Adam Joyce (Imperial) "Stable models of modular curves" Abstract: Using the Drinfeldian notion of level structures, one can obtain integral moduli spaces over rings of cyclotomic integers, whose generic fibres are the well-known algebraic curves over Q, X_0(N) and X_1(N). These moduli spaces are not smooth in characteristics dividing the level. For applications to arithmetic geometry, it is advantageous to have stable models (in the sense of Deligne-Mumford) for algebraic curves. The moduli spaces above are often not stable models for their generic fibres. We describe certain stable models, using tools from algebraic geometry.
26 October		David Solomon (KCL) "Stark Units, Hilbert Symbols and a Stickelberger Ideal at s=1" Abstract: Let M/k be an abelian extension of totally real number fields. For appropriately chosen finite sets of places S of k, Stark's conjecture predicts the existence of (an element of an exterior power of the) S -units of M whose equivariant regulator gives the leading term at s=0 of Artin L-functions L_S(s,\chi) for all characters \chi of G=Gal(M/k). These `Stark Units' are cyclotomic units if k=Q but otherwise are not known to exist. If instead M is of CM type (and k is still totally real) the Brumer-Stark conjecture predicts that the values at s=0 of the L_S(s,\chi) now define an ideal of ZG that generalises the Stickelberger ideal (the case k=Q). In particular, it annihilates the `odd' part of Cl(M). I shall discuss two p-adic conjectures that, in a sense, tie these two other conjectures together. For all odd \chi, the functional equation relates L_S(0,\chi) to L_S(1,\chi) . For each odd prime p, the latter values allow us to define a map the exterior power of the p-semilocal units of M into Q_pG . I conjecture firstly that the image of this map lies in Z_pG and secondly (when M contains p^nth roots of unity) that the map is congruent modulo p^n to one defined using Hilbert symbols and the (conjectural) Stark units coming from the maximal real subextension M^+/k.
2 November		Cornelius Greither (Munich) "Fitting ideals of class groups via the Equivariant Tamagawa Number conjecture"
9 November		Samir Siksek (Warwick) "Classical and modular approaches to exponential Diophantine problems"
16 November		Denis Benois (Besan n) "Iwasawa theory of crystalline representations and $(\phi,\Gamma)$-modules"
23 November		Tony Scholl (Cambridge) "Higher fields of norms and (phi,Gamma)-modules"
30 November 2.30pm		Don Zagier (Bonn) "Double zeta values and modular forms"
30 November 4.00pm		Richard Pinch "The distribution of Carmichael numbers"
7 December		David Whitehouse (Caltech) "The twisted weighted fundamental lemma for the transfer of automorphic forms from GSp(4) to GL(4)"
14 December		Tim Browning (Bristol) "Density of rational points on smooth hypersurfaces" Abstract: Let $X$ be a non-singular projective hypersurface of degree $d>1$ and dimension $k$. It has been conjectured that the number of rational points on $X$, which have height at most $B$, should be $O(B^{k+\eps})$ for any $\varepsilon>0$. The implied constant here should be allowed to depend at most upon $d,k$ and the choice of $\eps$. In this talk, which comprises joint work with Heath-Brown, we discuss the final resolution of this conjecture.

Summer 2005

This term, the seminar was held at Imperial and was organised by Alexei Skorobogatov.

The seminar programme for Summer 2005

20 April		Toby Gee (Imperial) "New results for companion forms over totally real fields"
27 April		Kazuhiro Fujiwara (Nagoya and Cambridge) "Galois representations over cyclotomic towers"
4 May		Peter Swinnerton-Dyer (Cambridge) "Counting rational points and the Manin conjecture"
11 May		Vassily Golyshev (Independent Moscow University) "Differential equations of quantum cohomology and higher Apery recurrences." Abstract: We say that a linear polynomial recurrence with integer coefficients is of Apery type if it has two solutions, the quotient of which b_n/a_n tends to an irrational zeta (or L-function) value at an integer point. We consider recurrences that are Mellin transforms of differential equations of quantum cohomology for Grassmannians. We conjecture that these recurrences are of Apery type. We prove certain cases of this conjecture for Grassmannians of classical groups by reducing the DEs in question to modular ones.
18 May		Daniel Delbourgo (Nottingham) "Euler products over C_p" Abstract: Standard Euler products converge in some right half-plane Re(s)>constant. If one tries the same trick replacing the complex numbers with the p-adics, things quickly go wrong. We first explain a way of making Euler products converge over C_p, the Tate field. Fortunately, these products converge to the values of classical L-functions with appropriate modifications to the Euler factor at p. The proof uses fractional calculus and something resembling an explicit reciprocity law. If we've got time, we'll mention some convergence calculations for p-adic L-functions of elliptic curves.
25 May		Chris Skinner (Michigan) "L-values, congruences, and Selmer groups"
1 June		Alexei Skorobogatov (Imperial) "Global points on Shimura curves" Abstract: It is an open question whether all counterexamples to the Hasse principle on smooth projective curves over number fields are due to the Manin obstruction. In the 1980-s Bruce Jordan proved that global points don't exist on certain Shimura curves, producing counterexamples to the Hasse principle. I'll show how these and other known counterexamples are explained by the Manin obstruction.
8 June		Andrei Yafaev (UCL) "Recent progress on the Andre-Oort conjecture"
15 June		Teruyoshi Yoshida (Harvard and Nottingham) Compatibility of local and global Langlands correspondences" (with R.Taylor) Abstract: The work of Harris-Taylor, which proved the local Langlands correspondence for GLn, included the construction of l-adic Galois representations attached to certain class of automorphic representations of GLn over CM fields, compatible with the local Langlands correspondence up to semisimplification at all places outside l. By studying the semistable reduction of the relevant Shimura varieties, we strengthen this result to show that the local monodromies are also the correct ones. The irreducibility of global Galois representations follows.
22 June		Tom Fisher (Cambridge) "Computing models for visible elements of the Tate-Shafarevich group"
29 June		Bjorn Poonen (Berkeley and Cambridge) "Multiples of subvarieties in algebraic groups over finite fields"

Spring 2005

This term, the seminar was held at KCL and was organised by David Solomon.

The seminar programme for Spring 2005

12 January		Michael Harris (Universite Paris VII) "Deformations of automorphic Galois representations" Abstract: The l-adic cohomology of Shimura varieties attached to certain unitary groups provide compatible systems of n-dimensional l-adic representations of the absolute Galois group of a CM field. These representations are necessarily polarized (self-dual, more or less) and their Hodge-Tate weights have multiplicity one. In joint work with Taylor we have proved, under the usual restrictions, that any representation with these properties arises in this way, provided (a) the reduction mod l has at least one modular lifting and (b) the representation is a minimal lifting of its reduction mod l. Work of R. Mann allows us to remove the minimality condition (b) under a precise conjecture on mod l modular forms known as Ihara's Lemma.
19 January		Prof E.V. Flynn (Liverpool) "The Brauer-Manin Obstruction on Curves" Abstract: When a variety violates the Hasse principle, this can be due to an obstruction known as the Brauer-Manin obstruction. It is an unsolved problem whether all violations of the Hasse principle on curves are due to this obstruction. I shall describe work in progress which tests a wide selection of curves, and tries to decide for these examples whether the Brauer-Manin obstruction is the cause of all violations of the Hasse principle. This has involved the development of several new techniques, exploiting the embedding of a curve in its Jacobians via a rational divisor class of degree 1, and has produced examples of certain new types (in response to a request of Alexei Skorobogatov). If there is time, I shall also discuss the loosely related question of annihilation and visualisation of members of the Shafarevich-Tate group of Jacobians.
26 January		Toby Gee (Imperial) "A new proof of an old theorem on companion forms" Abstract: Results on companion forms over Q were obtained in the early 1990s by Gross and Coleman-Voloch. I gave a generalisation to totally real fields last year. In this talk I will discuss a new and much more conceptual proof of the results of Gross and Coleman-Voloch, and indicate the possibilities for further generalisations to totally real fields.
2 February		Prof Francis Johnson (UCL) "Orders in quaternion algebras and recent developments in algebraic homotopy theory" Abstract: It is a fundamental question in non-simply connected homotopy theory (and thereby also in combinatorial group theory) to decide whether, over a given fundamental group G, every algebraic 2-complex is geometrically realizable. Although for most finite G this problem is increasingly well understood, the quaternion groups Q_4n have proved to be exceptional. Using Swan's work on non-cancellation phenomena for modules over orders in quaternion algebras over number fields, we describe families of algebraic 2-complexes over Q_4n for which no geometric realisations are currently known (or, in terms of combinatorial group theory, which do not correspond to any known group presentation)
9 February		Vladimir Dokchitser (Cambridge) "Root numbers and the rank of elliptic curves" Abstract: Fix an elliptic curve E over Q. I will discuss the behaviour of the sign in the functional equation of the L-function L(E/K,s), where K varies over different number fields. When the sign is -1, the L-function (assuming it is exists and is analytic) has a zero at s=1, and the Birch-Swinnerton-Dyer conjecture predicts that E should have a point of infinite order over the number field K. It is then possible to obtain examples of elliptic curves over Q which, while not having any rational points of infinite order, must conjecturally have points of infinite order over all the fields Q(m^{1/3}) for every (non-cube) m>1. I will discuss this, and similar phenomena.
16 February		Andreas Langer (Exeter) "Gauss-Manin connection via Witt-differentials" Abstract: For a scheme X that is smooth over a p-adic base we show an equivalence of categories between the category of locally free crystals and the category of Witt-connections on X. The proof uses the relative de Rham-Witt complex and generalizes a recent result of Bloch. As an application we realize the Gauss-Manin connection in the de Rham-Witt complex.
23 February		Kyu-Hwan Lee (U. Toronto) "Spherical Hecke algebras of GL_n over 2-dimensional local fields" Abstract: At the beginning of the talk, we will briefly review the classical Satake isomorphism, which plays an important role in the Langlands program. Then we will try to generalize the theory to the 2-dimensional local field case. More precisely, we will construct spherical Hecke algebras of GL_n over 2-dimensional local fields and prove the Satake somorphism for the algebras. We will use Fesenko's measure to define the Satake isomorphism. A connection to Kac-Moody groups will also briefly discussed. This is a joint work with Henry Kim.
2 March		Jayanta Manoharmayum (Sheffield) "Minimal deformations of Galois representations" Abstract: I will describe how one can get, in some cases, lifts of residual representation which are minimally ramified.
9 March		Ivan Fesenko (Nottingham) "Poles of the Hasse zeta function" Abstract: The talk will try to discuss some of the applications of the study of the Hasse zeta function of elliptic curves over global fields via 2d zeta integrals to: (a) Riemann hypothesis for the zeta function; (b) location of poles of the zeta function on the critical line; (c) an extension of the class of zeta functions all whose motivic L-factors are automorphic, using Laplace-Carleman transforms of odd mean periodic functions, and importance of this for the Langlands programme; (d) the rank part of the BSD conjecture.
16 March		Jan Nekovar (Jussieu) "The Euler system of CM points" Abstract: We shall discuss a generalization of Kolyvagin's results on Heegner points.
23 March		Steven Galbraith (Royal Holloway College) "Pairings on abelian varieties and cryptography" Abstract: The talk will survey some applications of elliptic curves over finite fields to cryptography. In particular, applications of the Weil and Tate pairings will be described and some new results on efficient computation of these pairings will be presented.

Autumn 2004

This term the seminar was held at Imperial College and was organised by Toby Gee

The seminar programme for Autumn 2004

27 October		No seminar
3 November		Pierre Parent (Bordeaux) "On the triviality of X₀⁺ (p^r) (Q), r>1" Let E be an elliptic curve over Q, without complex multiplication over \overline{Q}. For p a prime number, consider the representation Gal(\overline{Q} /Q )--> GL₂ (F_p) induced by the Galois action on the group of p-torsion points of E. A theorem of Serre, published in 1972, asserts that there exists an integer B_E such that the above representation is surjective if p is larger than B_E. Serre then asked the following question: can B_E be chosen independently of E? This boils down to proving the triviality, for large enough p, of the sets of rational points of four families of modular curves, namely X₀ (p), X_split(p), X_non-split(p) and X_A₄(p) (we say that a point of one of these curves is trivial if it is either a cusp, or the underlying isomorphism class of elliptic curves has complex multiplication over \overline{Q}). The (so-called exceptional) case of X_A₄(p) was ruled out by Serre. The fact that X₀ (p)(Q ) is made of only cusps for p>163 is a well-known theorem of Mazur. In this talk we will discuss the case of X_split (p)(Q). Slightly more generally (because one has a Q-isomorphism between X_split (p) and X₀⁺ (p² )), we will in fact give a criterion for the triviality of X₀⁺ (p^r ) (Q) (with r>1), and show it is verified by a positive density of primes (satisfying explicit congruences).
10 November		Nick Shepherd-Baron (Cambridge) "Perfect forms and moduli of abelian varieties" Perfect quadratic forms lead to a compactification of A_g whose geometry is particularly accessible. The ample classes are characterized by one inequality, generalizing the existence of the discriminant when g = 1. Over a field of char. zero, it is canonical (in the sense of Reid and Mori, not Shimura...).
17 November		Toby Gee (Imperial) "Companion Forms" This will be a different talk to the one I gave last year - it will hopefully be a very relaxed introduction to the Fontaine-Mazur and Serre conjectures, and a discussion of some issues arising from these conjectures.
24 November		Ian Grojnowski (Cambridge) "Geometric Satake for local fields of dimension 2" I'll explain the usual Satake isomorphism, its central role in the Langlands programme, and a generalisation of all this to fields like Q_p((t)), C((s))((t)). Should be of interest to geometers also--- this can be respelled as theorems about the moduli of G-bundles on an algebraic surface--- "Donaldson theory".
1 December		Shaun Stevens (UEA) "Supercuspidal representations of p-adic classical groups" The Local Langlands Correspondence relates the representations of the Weil-Deligne group of a locally compact non-archimedean local field to the irreducible smooth representations of general linear groups. Mostly conjecturally, there are also such correspondences for other p-adic groups. In this talk I will try to describe what this means, what's known and also some explicit constructions of representations for p-adic symplectic, orthogonal and unitary groups.
8 December		Jens Marklof (Bristol) "Number Theory and Quantum Chaos" I will review recent developments in some fundamental problems in quantum chaos that have attracted the interest of number theorists. No knowledge of quantum mechanics is necessary.
15 December		Denis Charles(Wisconsin) "Computing Modular Polynomials" (joint work with Kristin Lauter) We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our algorithm has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. We avoid computing the exponentially large integral coefficients by working directly modulo a prime and computing isogenies between elliptic curves via Velu's formulas.

Summer 2004

This term, the seminar was held at Imperial College and was organised by Kevin Buzzard.

The seminar programme for Summer 2004

28 April Dan Snaith (Imperial)
"Overconvergent Siegel modular forms"

5 May Luis Dieulefait (Barcelona)
"Existence of compatible families of Galois representations and the Fontaine-Mazur conjecture for elliptic curves"

12 May Bruno Kahn (Paris 7)
"Birational motives"

19 May Fre Vercauteren (Bristol)
"Zeta functions: the p-adic approach"

26 May Sarah Zerbes (Cambridge)
"Selmer groups over p-adic Lie extensions"

2 June Adam Joyce (Imperial)
"The Manin constant of modular abelian varieties"

The Manin constant of a (modular) elliptic curve was introduced by Manin in a paper in the mid-70s, in which he also conjectured that it is always 1. I shall give this definition and discuss the results in the direction of proving the conjecture. I'll then generalise the definition to abelian varieties of arbitrary dimension and discuss the extensions of the above results to the general setting.

9 June Brian Conrey (AIM)
"Random matrix theory and ranks of elliptic curves"

16 June Detlev Hoffman (Nottingham)
"Isotropy of quadratic forms in finite and infinite dimension"

23 June Teruyoshi Yoshida (Harvard, visiting Imperial)
"Non-abelian Lubin-Tate theory and Deligne-Lusztig theory"

4 August Chandrashekhar Khare (Utah/Tata)
"Transcendental Galois representations"

Spring 2004

This term, the seminar was held at Imperial College and was organised by Prof. Alexei Skorobogatov.

The seminar programme for Spring 2004

14 January Kevin Buzzard (Imperial)
"The 2-adic eigencurve at the boundary of weight space"

Eigencurves are geometric objects parameterising certain modular forms. These objects were introduced by Coleman and Mazur in the mid-1990s and at the time very little was known about what they "looked like". Lloyd Kilford and I can write down equations for one of these eigencurves, and these equations are sufficiently explicit to enable us to pin down exactly what this eigencurve looks like near its boundary.

In my talk I will give an introduction to the theory of eigencurves and then will explain a sketch of our result.

21 January Alexei Skorobogatov (Imperial)
"Rational points on Enriques surfaces"

The Enriques surfaces are cohomologically indistinguishable from rational surfaces, however unlike rational surfaces they have a non-trivial though very small fundamental group. For rational surfaces it is conjectured by Colliot-Th e and Sansuc, and is proved in many cases that the failure of the Hasse principle and weak approximation is controlled by the obstruction based on the Brauer group of the surface. In a joint work with David Harari we construct an Enriques surface over Q with an adelic point satisfying all the conditions provided by the Brauer group, but not in the closure of the set of Q-rational points. The proof uses descent to a torsor on this surface; the structure group of this torsor is a form of a (non-abelian) 1-dimensional orthogonal group.

28 January Roger Heath-Brown (Oxford)
"Cayley's cubic surface"

Roughly how many non-trivial primitive integer solutions does the equation

1/X₀+1/X₁+1/X₂+1/X₃=0
have, in a large cube max |X_i| < B ? On clearing the denominators this gives us a Cayley's cubic surface.

Manin's conjecture predicts a growth rate of order B(log B)⁶. This has now been proved, by using the Universal Torsor for the surface. The latter is an affine variety in 16 dimensional space, which encodes all the relevant divisibility information. The proof entails counting points on this affine variety, for which the key tool comes from the geometry of numbers.

4 February Anthony Hayward (King's)
"Congruences satisfied by Stark units"

"Refined abelian Stark conjectures" are special value conjectures for equivariant L-functions associated to Galois extensions of global fields at "s=0". There has been a proliferation of such conjectures since Stark's original work in the 1970s, notably those of Rubin and Popescu, who generalised Stark's abelian conjecture to account for differing orders of vanishing of the L-functions, and Gross, who gave congruences for the value of the L-function. These conjectures admit a natural refinement, due to Burns, which gives Gross-style congruences in the situation of Rubin's conjecture. The formulation of the refinement is inspired by the Equivariant Tamagawa Conjecture, and this over-riding conjecture provides a unifying overview of the field which was previously sorely lacking. After describing this general situation in the first half of the talk, I move on to study those situations in which systems of explicit units provide proofs of the Burns refinement, hopefully giving more of a sense of what these conjectures are "about". The proof involves details on the explicit units, a study of ray-class fields, and some unusual combinatorics and integer identities.

11 February Victor Abrashkin (Durham)
"Galois modules arising from Faltings's strict modules"

A classical analogue of the concept of a finite flat group scheme over a complete discrete valuation ring loses all interesting properties in the equal characteristic case. In the past year Faltings proposed a modified definition of a group scheme with strict action and showed that it works perfectly in some situations. It will be explained in the talk that many known results about classification and arising Galois modules still hold for Faltings's modules.

18 February Kanetomo Sato (Nagoya)
"p-adic ale Tate twists and arithmetic duality"

For an algebraic variety X over a field k and a positive integer n invertible in k, the ale sheaf of n-th roots of unity and its tensor powers are called ( ale) Tate twists, and play fundamental roles both in number theory and in arithmetic geometry. In this talk, I will talk about a construction of p-adic Tate twists on regular arithmetic schemes (=schemes which are regular flat of finite type over Spec Z), and arithmetic duality theorems for p-adic Tate twists, which generalizes the classical Artin-Verdier duality theorem.

25 February Toby Gee (Imperial)
"Companion forms over totally real fields"

The companion forms conjecture was part of Serre's conjecture on the modularity of mod p Galois representations. This was proved in the early 90s by Gross and Coleman & Voloch. More recently Fred Diamond has conjectured extensions of these results to totally real fields; I will describe my recent progress on cases of these conjectures.

3 March Aleksandra Shlapentokh (East Carolina)
"Hilbert's tenth problem and Mazur's conjectures"

We discuss recent results concerning extensions of Hilbert's tenth problem to rings of integers of number fields and the field of rational numbers, and related conjectures of Mazur.

10 March R is de la Bret he (ENS Paris)
"Counting points on varieties using universal torsors"

We study the asymptotic order of the number of points of bounded height on certain varieties. Universal torsors turned out to be a useful tool to attack this kind of problem. We will give few examples to explain how to prove asymptotic estimations using universal torsors and tools of analytic number theory.

17 March Nikolaos Diamantis (Nottingham)
"Second order cusp forms and L-functions"

Second-order modular forms are functions that have recently appeared in several contexts: Eisenstein series formed with modular symbols, converse theorems of L-functions, percolation theory etc. They satisfy a functional equation that extends naturally that of the usual modular forms and their study is important for the topics that have motivated their introduction. We will discuss the ways they arise in various contexts, their classification and their L-functions.

24 March Adam Logan (Liverpool)
"Heegner points on elliptic curves over real quadratic fields"

Henri Darmon formulated a concrete version of a conjecture of Oda according to which it should be possible to construct rational points on elliptic curves over real quadratic fields defined over quadratic extensions of the field by integrating the Hilbert modular form associated to the curve. We present the conjecture together with some numerical evidence for it. This is joint work with Darmon.

Autumn 2003

This term, the seminar was held on Wednesdays, between 4:15 and 5:15 pm in room 423 of KCL,
and was organised by Dr David Solomon.

The seminar programme for Autumn 2003

1/10/03    David Solomon, KCL 

         `Twisted Zeta-Functions, Stark Conjectures and Hilbert Symbols'
   
   
----------------------------------------------------------------
8/10/03   John Cremona, Nottingham: 

         `Explicit Higher Descents on Elliptic Curves'

 
----------------------------------------------------------------
		
15/10/03  Richard Hill, UCL: 

         `Fractional weights, Borcherds products and 
		            the Congruence Subgroup Problem'

----------------------------------------------------------------
22/10/03  Tim Browning, Oxford: 

         `Counting rational points on singular cubic surfaces'   

----------------------------------------------------------------
29/10/03  Christian Elsholtz,  Royal Holloway College: 

         `Additive decompositions of the set of primes' 
     
----------------------------------------------------------------
5/11/03  Daniel Delbourgo, Nottingham 

        "Euler characteristics of elliptic curves via p-adic modular forms"

----------------------------------------------------------------
12/11/03 Neil Dummigan, Sheffield,
  
        `Critical values of tensor-product L-functions'
      
----------------------------------------------------------------
19/11/03 Dan Evans, Nottingham: 

        `Harmonic analysis on higher dimensional local fields'
        
----------------------------------------------------------------
26/11/03 Daniel Barsky, U Paris 13 [NOTE CHANGE OF DATE]

       `Norms of Iwasawa series attached to totally real fields'
     
----------------------------------------------------------------
3/12/03  Nigel Byott, Exeter: 
  
        `Hopf-Galois strucutres of field extensions'
       
----------------------------------------------------------------
10/12/03 Prof. Igor Shparlinski, Macquarrie U. 

        `Euler Function: Smooth and Square'

-------------------------------------------------

Further details will appear when available.

The seminar programme, Summer 2003

This term it was held in the maths department of Imperial College and was organised by
Dr Kevin Buzzard (buzzard@ic.ac.uk).

The seminar programme

----------------------------------------------------------------------

30 April --- TWO TALKS 

14:15   Elmar Grosse-Kloenne (Muenster) "On twisted unit root L-functions
^^^^^                                    of families of varieties over 
                                         finite fields"

15:45   Steve Gelbart (Weizmann)        "On lower bounds for automorphic
^^^^^                                    L-functions".

----------------------------------------------------------------------

7 May   Allan Lauder (Oxford)   "Deformation theory and the computation
                                 of zeta functions"

----------------------------------------------------------------------

14 May  Vic Snaith (Southampton)  "On the Kummer--Vandiver conjecture"

----------------------------------------------------------------------
***Tuesday 20th May*** at 1600
         Victor Rotger (Barcelona)---"Diophantine properties of fake
                                      elliptic curves and their moduli spaces"

----------------------------------------------------------------------
21 May---Ed Nevens (Imperial)---TBA (something about moduli space
                                     of abelian varieties and/or canonical
                                     subgroups, perhaps)

----------------------------------------------------------------------
28 May---Neil Strickland (Sheffield)---"Elliptic cohomology"

----------------------------------------------------------------------

The seminar programme, Spring 2003

This term the seminar was held in the maths department of Imperial College on Wednesdays at 4.15 pm and was organised by Dr Kevin Buzzard (buzzard@ic.ac.uk).

The seminar programme

----------------------------------------------------------------------

15/1  Andrei Yafaev, Imperial
  
 Title: `Descent on certain Shimura curves'

 Abstract:

 This is a joint work with Alexei Skorobogatov.  Applying descent to
 certain unramified coverings of Shimura curves we offer an explicit
 method of constructing Shimura curves that do not satisfy Hasse
 principle; the failure of the Hasse principle is being explained by
 the Manin obstruction.

----------------------------------------------------------------------

22/1  Kevin Buzzard, Imperial

 Title: `Overconvergent 2-adic modular forms'

 Abstract:

 This is joint work with Frank Calegari. Some computations I did (and
 some known conjectures and theorems) led me to believe that in some
 cases there are very precise formulae for the p-adic valuations of
 the eigenvalues of T_p on various spaces of modular forms. Calegari
 and I have made these conjectures completely explicit and precise in
 the case p=2 and N=1 (for any weight k) and can prove them in some
 cases using a combination of deep theorems of Coleman and elementary
 combinatorial results involving hypergeometric function identities.

----------------------------------------------------------------------

29/1  I. Tomasic (Leeds)

         "Weil conjectures--with a DIFFERENCE"

----------------------------------------------------------------------

 5/2  A. Hayward (Kings)

         "A conjectural class-number formula for 
          higher derivatives of abelian L-functions"

----------------------------------------------------------------------

12/2  R. Kucera (Brno)

         "Cyclotomic units"

----------------------------------------------------------------------

19/2 (1430)  J. Nekovar (Jussieu)
^^^^^^^^^^^                      

         "On the parity of ranks of Selmer
          groups associated to Hilbert modular forms"

19/2 (1615)  T. Ochiai (Tokyo)
^^^^^^^^^^^                   

         "Results and examples for Iwasawa
          theory on Hida deformations."

----------------------------------------------------------------------

26/2  H. Narita (Tokyo)

         "Fourier-Jacobi expansion of certain
          automorphic forms on Sp(1,q)"

----------------------------------------------------------------------

 5/3  M. Breuning (Kings)

          TBA (something about local epsilon constants)

----------------------------------------------------------------------

12/3  D. Harari (Strasbourg)

         "Arithmetic duality theorems for 1-motives"

----------------------------------------------------------------------

19/3  N. Broberg (Durham)

         "Counting rational points on finite
          covers of the projective plane"

----------------------------------------------------------------------

 9/4  

1400: A. Yakovlev    
^^^^
         "Multiplicative Galois modules in local fields"

1530: G. Henniart 
^^^^
         "Expliciting the Langlands conjecture:the tame case"

-------------------------------------------------------------------------

The seminar programme, Autumn 2002

This term, the seminar was held on Wednesdays, in room 423 of KCL, and was organised by Dr David Solomon. Also on Wednesdays in King's this term was the London Number Theory Study Group, which met from 2:45 to 3:45 pm.

    The seminar programme

      9/10  FIRST MEETING: EXCEPTIONALLY A DOUBLE-HEADER STARTING AT 2:45 in room 436: 

      2:45 - 3:45  (room 436) Amnon Besser, Ben Gurion University, 

      4:15 - 5:15 (room 423)  Takao Yamazaki, Tsukuba University, 


                   Title:  `On the structure of Chow groups of 
                                 surfaces over local fields' 
                           
      Abstract:   Let X be a surface over a p-adic field with good reduction 
      and let Y be its special fiber. We consider the structure of 
      the Chow group CH0(X) of zero-cycles on X. If we write T(X)
      for the kernel of the Albanese map of X, then the structure of 
      the quotient group CH0(X)/T(X) is well understood. Hence we only
      have to study T(X). Let D(X) be the maximal divisible subgroup
      of T(X). Then, it is conjectured that F(X) = T(X)/D(X) is finite
      and that F(X) is isomorphic to the Albanese kernel T(Y) of Y
      modulo p-primary torsion. On the contrary, we shall show that the 
      p-primary torsion subgroup of F(X) can be arbitrary large 
      even though we fix the special fiber Y.

      ----------------------------------------------------------------------

      16/10  Mohammed Saidi, University of Durham
        
       Title: `On the fundamental group of complete curves in positive 
               characteristics' 

      ----------------------------------------------------------------------

      30/10  Richard Hill, UCL

       Title: `Shintani Cocyles on GL_n'

      ------------------------------------------------------------------------

      6/11  No Seminar (Reading Week)

      ------------------------------------------------------------------------

      13/11 Sey Yoon Kim, KCL 

       Title: `On the Equivariant Tamagawa Number Conjecture 
               for certain Quaternion Fields ' 

       Abstract: 
       
       Let L/K be a finite extension of number fields.  Then the equivariant
       Tamagawa number conjecture relates the values of the Artin
       L-functions of L/K at integers to various algebraic data of L/K, and
       in particular, the conjecture at s=0,1 implies Chinburg's root number
       conjecture for L/K.  We explain the conjecture at s=0 for abelian
       extensions over Q; then prove it for a family of biquadratic abelian
       extensions over Q to lift a 1989 result of Chinburg on his conjecture
       for the case of quaternion extensions over Q.

      -------------------------------------------------------------------------

      20/11  Martin Taylor, U.M.I.S.T.

       Title: `Arithmetic Euler Characteristics'

       Abstract: 

       I shall start by recalling the basic theory and constructions for
       Euler characteristics of varieties which support an action by a
       finite group. These ideas then extend firstly to the construction of
       equivariant Euler characteristics of arithmetic varieties, and then
       more generally to Euler characteristics which take into account
       metrics and signatures.
            
      -------------------------------------------------------------------------

      27/11 Robin Chapman, Exeter

       Title: `Hermitian structures on lattices'

       Abstract: 

       We consider lattices in Euclidean space Rn. If n is even, Rn can be
       given a structure of a complex vector space in many ways. Given a
       lattice L we investigate which C-structures on Rn have a Hermitian
       form compatible with the Euclidean structure on Rn and for which L
       becomes an O-module for some quadratic order O. In some cases we
       determine explicitly the O-module structure of L.

      -------------------------------------------------------------------------

       4/12 David Burns, King's College London

       Title: `Nearly perfect complexes and Weil-etale cohomology'

       Abstract: 

       We describe a more conceptual approach to the construction of Euler
       characteristics of nearly perfect complexes which was recently
       introduced by Chinburg, Kolster, Pappas and Snaith. We then discuss
       certain applications of our approach in the context of Lichtenbaum's
       theory of Weil- ale cohomology.

      -------------------------------------------------------------------------

      11/12 Rob de Jeu, Durham

       Title: `Zagier's conjecture and (p-adic) regulators'

       Abstract: 

       Let k be a number field. There is a classical relation between the
       residue of the zeta function of k, zetak(s), at s=1, and the
       regulator of the group of units of its ring of integers. Borel proved
       a similar relation between zetak(n) and K2n-1(k) for n>=2.  
       The K-groups are difficult to describe explicitly. We
       discuss a conjecture of Zagier on how this could be done, and
       describe Borel's regulator (as well as a p-adic regulator) in this
       context, involving polylogarithms.

The seminar programme, Summer 2002

This term it was held in the Maths department of Imperial College and organised by
Dr Kevin Buzzard (buzzard@ic.ac.uk).

24 April  Frazer Jarvis (Sheffield)
         "Points on Fermat curves over real quadratic fields"

 1 May    Jayanta Manoharmayum (Sheffield)
         "modularity of GL₂(F₇) Galois representations"

*2 May*   Helena Verrill (Hannover)
         "Transportable modular symbols"

 8 May    Alexei Skorobogatov (Imperial)
         "Some new cases of the Hasse principle and weak approximation"

15 May    Dan Jacobs (Imperial)
          Slopes of Compact Operators

22 May    Ben Green (Cambridge)  
         "Counting sumfree sets in abelian groups"

29 May    Lloyd Kilford (Imperial)
         "Slopes of overconvergent 2-adic modular forms"

 5 June   Denis Petrequin (Cambridge)
         "Chern classes and cycle classes in rigid cohomology"

12 June   Oliver Bltel (Heidelberg)
          TBA

*13 June* Oliver Bueltel (Heidelberg)
          TBA (continued).

19 June   Chad Schoen (Duke)
         "Torsion in the Chow group"

The seminar programme, Spring 2002

This term it was held in the Maths department of Imperial College and organised by
Dr Kevin Buzzard (buzzard@ic.ac.uk).

  Jan 16  Andrei Yafaev (Imperial)  
         "Galois orbits of abelian varieties with complex multiplication

  Jan 23  Denis Benois (Bordeaux)
         "On Tamagawa numbers of crystalline representations"

  Jan 30  Tony Scholl (Cambridge)
         "Local epsilon-factors and tensor products of 
          representations of GL(2)"

  Feb  6  Richard Hill (UCL)
         "something to do with metaplectic groups"

  Feb 13  John Coates (Cambridge)
         "Iwasawa algebras and arithmetic"

  Feb 20  Tim Dokchitser (Durham)
         "TBA"

  Feb 27  Susan Howson (Nottingham)
         "Applications of Euler Characteristics to 
          non-Abelian Iwasawa Theory

  Mar  6  Shaun Stevens (Oxford)
         "TBA"

  Mar 13  John Wilson (Oxford)
         "Abelian surfaces with real multiplication"

  Mar 29  David Solomon (Kings)
         "Abelian Stark Conjectures in Z_p-extensions"

The seminar programme, Autumn 2001

This term, the seminar was held in the Maths department of Imperial College and organised by
Dr Alexei Skorobogatov (a.skorobogatov@ic.ac.uk).

  Oct 10 Andrei Yafaev (Imperial)  

        "Special points on Shimura varieties"

         Abstract

         A conjecture of Andre and Oort predicts that irreducible
         components of a Zariski closure of a set of special points in
         a Shimura variety are subvarieties of Hodge type. This talk
         is devoted to a recent result towards this conjecture
         obtained in a joint work with Bas Edixhoven.   

  Oct 17 Sir Peter Swinnerton-Dyer (Cambridge) 

        "Rational points on certain Kummer surfaces"

         Abstract

         Most of this seminar represents joint work with Alexei
         Skorobogatov. Let E_1,E_2 be elliptic curves defined  over
         an algebraic number field k, and let F_i:y_i^2=f_i(x_i)
         with f_i quartic be a 2-covering of E_i. Then
         V:y^2=f_1(x_1)f_2(x_2) is a Kummer surface associated with
         the Abelian surface E_1\times E_2. In the special case when
         E_1 and E_2 have all their 2-division points defined over
         k, I shall show that the Hasse Principle holds for V
         provided that (i) the Tate-Safarevi\v{c} groups of all the
         twists of E_1 and E_2 are finite, (ii) a certain rather
         weak technical condition holds. Here (i) is necessitated by
         the method of proof, but it is generally believed to be true;
         (ii) can be shown to be strictly stronger than the true
         necessary and sufficient condition, which is conjectured zto
         be the absence of a Brauer-Manin obstruction. Note that
         Schinzel's Hypothesis does not appear. The methods used have
         much in common with those used for diagonal cubic surfaces 
         a_0X_0^3+a_1X_1^3+a_2X_2^3+a_3X_3^3=0 but some stages of the
         argument for the latter are much more complicated. 
         Comparisons will be made between the two.  

  
  Oct 24 Michael Spiess (Nottingham) 

        "Monodromy modules and derivatives of p-adic L-functions"


  Oct 31 Jonathan Dee (Imperial) 

        "Phi-Gamma-modules and families of p-adic Galois representations"


  Nov  7 Jean-Louis Colliot-Th  e (Orsay) 

        "Linear algebraic groups over two-dimensional fields"

         Abstract
         Let $k$ be an algebraically closed field of characteristic
         zero. Let $K$ be either a function field in two variables
         over $k$ or the fraction field of a $2$-dimensional,
         excellent, strictly henselian local domain with residue field
         $k$. We show that linear algebraic groups over such a field
         $K$ satisfy properties which are familar in the context of
         number fields: finiteness of $R$-equivalence, Hasse principle
         forprincipal homogeneous spaces of simply connected
         groups,Hasse principle for complete homogeneous spaces.
         This is joint work with P. Gille (Orsay) and R. Parimala
         (Mumbai). 

  Nov 14 Kevin Buzzard (Imperial) 

        "The eigenvariety"

         Abstract:

         For a general reductive group, people are beginning to
	 believe that certain classes of automorphic forms on this
	 group lie naturally in p-adic analytic families, as the
	 weight varies. For GL_1 one can formulate a precise statement
	 and its proof is an easy consequence of global class field
	 theory. For GL_2 over Q, Coleman and Mazur have constructed
	 families interpolating classical holomorphic modular forms,
	 and have gone onto construct a geometric object, the
	 eigencurve, parameterising the forms. I will explain that if
	 one is willing to do a little rigid geometry then one can
	 generalise much of the Coleman-Mazur construction to a much
	 wider setting, and hence construct "the eigenvariety" in much
	 greater generality. 

  Nov 21 Vic Snaith (Southampton) 

        "Relative K_0, Fitting ideals and the Stickelberger phenomenon" 


  Nov 28 Burt Totaro (Cambridge) 

        "Rational points on homogeneous spaces, and the group E_8"

         Abstract: A homogeneous variety over a field need not have a
         rational point over the same field. The simplest example is a
         conic curve, which in general has a rational point only over
         a quadratic extension field. More generally, given a
         semisimple group G, and a homogeneous G-variety over an
         arbitrary field, we can ask what degree of field extension we
         need in order to find a rational point. I will explain what
         is known about this problem, both for the classical groups
         and the exceptional groups such as E_8.

  Dec  5 Otmar Venjakob (Cambridge) 

        "Iwasawa theory of p-adic Lie extensions"

         Abstract: 
         The most prominent example of a (non-abelian) p-adic Lie
         extension K of a number field k arises maybe by adjoining to
         k the p-power division points of an elliptic curve E (over k)
         without complex conjugation. If G denotes the Galois group of
         K/k one can study the (Pontryagin dual of the) Selmer group
         of E over K as a module over the completed group algebra R(G)
         of G. In this situation it is also reasonable - though not at
         all obvious - to speak about pseudonull modules. One basic
         result is that the Selmer group does not contain any nonzero
         pseudo-null submodule (under certain conditions). If there is
         enough time we are going to discuss also some features of the
         general structure theory of torsion R(G)-modules up to
         pseudo-isomorphism, which was proven by Coates, Schneider and
         Sujatha recently.

  Dec 12 No Talk

Summer 2001

This term, the seminar was held on Wednesdays in room 423 of KCL, and was organised by Dr David Solomon.

Jun 27 Stephen Lichtenbaum (Brown University, USA) -
"A Search for a New Cohomology Theory for Schemes over Z"
(Final Seminar)

Jun 26 Frank Calegari
- `unofficial' seminar at 3.30pm in room 413 in Imperial College -
`Semistable Abelian Varieties of small conductor'
Abstract:
Fontaine proved in 1985 that there do not exist any Abelian varieties over Z. Jacobians of the modular curves X_0(p^n) (non-zero for sufficiently large n) provide examples of Abelian varieties over Z[1/p] for each p. If, however, we restrict our attention to semistable Abelian varieties,then combining Fontaine's theorem, recent papers of Brumer-Kramer and of Schoof, and some new ideas and results, we prove the following: There exists a semistable Abelian variety over Z[1/n] with n squarefree if and only if n is not in the set: {1,2,3,5,6,7,10,13}.

Jun 26 William Stein
- `unofficial' seminar at 2pm in the Clore lecture theatre in Imperial College -
`Mordell-Weil groups of elliptic curves over Q are visible'
Abstract:
During the past few years, Barry Mazur, myself, and others have studied visible subgroups of Shafarevich-Tate groups of abelian varieties. Recently, I've been studying visibility of Mordell-Weil groups of abelian varieties. In this talk, I will very briefly review some results about visibility of Shafarevich-Tate groups, then discuss some of what I've been able to prove about their counterparts in the context of visibility of Mordell-Weil groups. In particular, I will show that Mordell-Weil groups of elliptic curves over Q are visible in modular abelian varieties.

Jun 15 Helena Verrill
- `unofficial' seminar at 3pm in room 413 in Imperial College -
`Modular symbols computations and a certain intermediate Jacobian'

Jun 13 Romyar Sharifi -
`Galois Module Structure of Local Unit Groups and Conductors'
Abstract:
We give an explicit description of generators of the ith unit groups of K = Q_pzeta_pⁿ as Galois submodules of the multiplicative group of K. We can use this to determine the ramification groups of degree pⁿ Kummer extensions of K which are Galois over Q_p.

Jun 6 Richard Hill (University College, London) -
"The Kubota Symbol on SL_n"

May 30 Prof. V. Nikulin (Liverpool) -
"Dynamics of the automorphism group of a K3 surface."

May 23 Daniel Delbourgo (Nottingham) -
"Two-variable Euler systems in the non-CM case"

My 16 Rob de Jeu (Durham) -
"Algebraic K-theory and Syntomic Regulators" (provisional title)

Spring 2001

This term it was held in the maths department of Imperial College, and was organised by Dr Kevin Buzzard (k.buzzard@ic.ac.uk).

Jan 17 Kevin Buzzard (Imperial) -
An overview.

The first talk in the study group on "Remarks on a conjecture of Fontaine and Mazur" by R. Taylor

Jan 24 Al Weiss (U Alberta)
"Equivariant Iwasawa Theory"

Jan 31 Andreas Langer (Bielefeld)
"Relative de Rham-Witt cohomology"

Feb 7 Colin Bushnell (Kings)
"Local Langlands correspondences: some explicit problems"

Feb 14 Alexei Skorobogatov (Imperial)
"Rational solutions of certain equations involving norms"

Feb 21 Christophe Cornut (Strasbourg)
"A conjecture of B. Mazur on higher Heegner points"

Feb 28 Keith Ball (UCL)
"There are infinitely many irrational values of zeta at the odd numbers"

Mar 7 Werner Hoffman (Humboldt U)
"On the differential equations satisfied by weighted orbital integrals"

Mar 14 Anupam Saikia (Cambridge)
Title TBA

Mar 21 Victor Flynn (Liverpool)
"Coverings of curves"

Abstract:
We shall discuss the idea of finding all rational points on a curve C by first finding an associated collection of curves whose rational points cover those of C. This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves and Chabauty techniques. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians.

Autumn 2000

This term it was held in the maths department of Imperial College, and was organised by Dr Alexei Skorobogatov (a.skorobogatov@ic.ac.uk).

Oct 11 Kevin Buzzard (IC)
"Questions about slopes of modular forms"

Oct 18 Alexei Skorobogatov (Imperial)
"Introduction to the study group: McCallum's work on the Brauer points on Fermat curves"

Oct 25 David Solomon (KCL)
"A p-adic Stark conjecture in Z_p-extensions"

Nov 1 Kevin Buzzard (IC)
"Continuation of the study group on McCallum's work:

Coleman's integral and applications to rational points"

Nov 8 Jan Nekovar (Cambridge)
"Some consequences of Mazur's conjecture on Heegner points"

Nov 15 Tom Fisher (Cambridge)
"The Cassels-Tate pairing and the icosahedron"

Nov 22 Roger Heath-Brown (Oxford)
Counting rational points on algebraic curves and surfaces;

Nov 29 Jean-Robert Belliard (Nottingham)
"Real analogues of Stickelberger's theorem"

Dec 6 Richard Hill (UCL)
"Syntomic regulators and K-theory of curves"

Summer 2000

This term, the seminar was held in KCL, and was be organised by Dr David Burns.

May 24	Richard Hill (UCL) "Metacyclic Covers of GL_n"
May 31	Frazer Jarvis (University of Sheffield) "Fermat's last theorem over totally real fields?"
Jun 7	Alexei Skorobogatov (ICL) "Swinnerton-Dyer's theorem on diagonal cubic surfaces" n.b. this talk starts at 5pm
Jun 14	Anton Deitmar (Exeter) "Geometric zeta functions and class numbers of cubic orders"
Jun 21	No seminar scheduled
Jun 27	Cornelius Greither (U. der Bundeswehr, Munich) "TBA" n.b. this is a Tuesday, from 4.30 to 5.30 pm

Spring 2000

This semester, the seminar was held on Wednesdays, in room 807 of UCL, and was organised by Dr Richard Hill.
The programme was:

Jan 26 Neil Dummigan (Oxford)
"Some evidence for the Bloch-Kato conjecture at critical points"

Feb 2 Susan Howson (Nottingham)
"non-Abelian Iwasawa theory and applications to elliptic curves"

Feb 9 David Burns (KCL)
"On the conjectures of Gross and Tate concerning values of abelian L-functions at s = 0"
(This will be the first talk in a study group)

Feb 16 Victor Abrashkin
"Ramification Filtrations and a local analogue of the Grothendieck Conjecture"

Feb 23 David Burns (KCL)
"Continuation of the study group on values of Abelian L-functions at s=0"

Mar 1 Richard Hill
Continuation of the study group.

Mar 8 Robert Vaughan
"A Generalized Divisor Problem".

Mar 15 David Solomon
Continuation of the study group.

Mar 22 David Solomon
Continuation of the study group on values of Abelian L-functions at s=0.

Autumn 1999

Weds Oct 13, KCL: Bart de Smit (Leiden, visiting KCL); "On Imin Chen's jacobian relations"; ABSTRACT:; This is joint work with Bas Edixhoven. We generalise a result of Chen concerning an isogeny between products of jacobians of modular curves associated to subgroups of GL(2,p)
Weds Oct 20, KCL: Olaf Neisse (Augsburg, visiting Southampton); "On balanced modules of group algebras"
Weds Oct 27, KCL: David Solomon (KCL); "Cyclotomic p-Units: a Wild Euler System"
Thurs Nov 4, room 6M42, IC: Kevin Buzzard (IC); "Some new cases of a conjecture of Artin."; This talk will be followed by a short meeting to organise the remainder of this term's seminars. Suggestions include research talks, a study group (a recent preprint of Heath-Brown has been suggested), or a mixture of the two.
Thurs Nov 11,: Cambridge-Oxford-Warwick (COW) algebraic geometry seminar, room 642 IC
2.00 Nick Shepherd-Barron (Cambridge CMS) -
Isomorphism classes in local moduli spaces

3.15 Paul Seidel -
Floer cohomology and the Picard-Lefschetz formula; Isomorphism classes in local moduli spaces; Floer cohomology and the Picard-Lefschetz formula
Thurs Nov 11, 4.30 pm, room 642 IC: Alexei Skorobogatov; "Rational points on diagonal cubic surfaces"; This is the first talk on what will be a short study group on a short paper of Heath-Brown.
Thurs Nov 18, 4.00 pm, room 6M42 IC: Kevin Buzzard; "Some new cases of Artin's conjecture"
Thurs Nov 25, 4.00 pm, room 6M42 IC: Alexei Skorobogatov; "more about Heath-Brown's paper on cubic surfaces"
Thurs Dec 2, 4.00 pm, room 6M42 IC: Richard Hill; "The next installment in Heath-Brown's paper on cubic surfaces"

Summer 1999

`The arithmetic of complex manifolds,

May 5 David Solomon (KCL): Introductory talk
May 19 David Solomon (KCL): Second introductory talk
May 19 Richard Hill: `Rubin's proposition 1 and the Tate pairing'
Jun 3 Richard Hill: Second talk
Jun 9 Kevin Buzzard: First talk
Jun 16 Kevin Buzzard: Second talk
Jun 23 David Burns: Equivariant forms of the Birch-Swinnerton Dyer Conjecture

Spring of 1999

buzzard@ic.ac.uk

Mar 22 1999 Ivan Fesenko (Nottingham)

`Local reciprocity cycles'

Mar 15 1999 Bernhard Koeck (Karlsruhe)

Computing the homology of Koszul complexes

Mar 8 1999 Kevin Buzzard

Euler Systems: the main theorems

Mar 1 1999 David Solomon (Kings)

p-adic Stark Conjectures at s=1: theory and computation II

- the computation

Feb 22 1999 Richard Hill

`Examples of Selmer groups' (study group)

(this talk will be inspired by Rubin's forthcoming book)

Feb 15 1999 David Solomon (Kings)

p-adic Stark Conjectures at s=1: theory and computation

Feb 8 1999 Kevin Buzzard (Imperial)

Tate modules and etale cohomology (study group) sheaves will be assumed)

Feb 1 1999 John Dee (Cambridge)

Chow Groups of Self Products of CM Elliptic Curves

Jan 18 1999 Richard Hill

Geometry of Numbers and Class Field Theory

Autumn of 1998

UCL Maths department

rih@math.ucl.ac.uk

Dec 11 1998 David Solomon

Dedekind Sums and Cocycles on GL_n(Q)

Abstract:
In 1989 Glenn Stevens showed how the `periods' of Eisenstein series could be used to define certain families of 1-cocycles on GL₂(Q) whose values can be expressed in terms of Dedekind Sums (for example, those appearing in the transformation formula for Dedekind's eta-function). In fact, the cocycle relation is equivalent to (a generalisation of) the classical Dedekind Reciprocity Law for these sums. Stevens also showed how these cocycles could be used to evaluate the partial zeta-functions of real quadratic fields at non-positive integers. In 1992, Robert Sczech constructed 1-cocycles on GL₂(Q) by a different method, using only real analysis. He too showed how they related to Dedekind Sums and partial zeta-values. (Indeed they are very closely related to Stevens'). In 1993 Sczech extended his construction to GL_n(Q), producing (n-1)-cocycles that are similarly `universal' for the evaluation of partial zeta-values over totally real fields of degree n.

In this talk, I shall present a third construction of cocycles on GL_n(Q) for n=2 and 3, which was inspired by Shintani's formulae for partial zeta-values. They are again closely related to Stevens' and Sczech's but the construction is algebraic and elementary. I shall also explain the connections with Dedekind Sums and mention (time allowing) partial zeta-values, p-adic interpolation, and the `challenge of higher dimensions'.

Dec 4 1998 Alexei Skorobogatov

Second cohomology and the existence of rational points

Abstract:
Let S be an algebraic group over a field k, and X be a k-variety. We denote by X' the same variety considered over the algebraic closure of k. If Y'/X' is a torsor under S (equipped with a suitable Galois action), then the obstruction for Y'/X' to come from some Y/X (defined over k) lies in the second cohomology set of S. If X contains a k-point, then this class is neutral. In some arithmetically meaningful cases, e.g. X a principal homogeneous space of a semi-simple group over a totally imaginary number field k, and Y'/X' is the universal covering, the converse is also true. Using these ideas one can give a short proof of an old theorem of Sansuc that the Manin obstruction is the only obstruction to the Hasse principle for principal homogeneous spaces of semi- simple groups over number fields (here S is Abelian).

Nov 27 1998 No number theory seminar

(UCL was hosting a one-day event: `A tribute to the life and work of Sir James Lighthill')

Nov 20 1998 Jan Nekovar

A skew-symmetric pairing on the infinite part of a Selmer group

Abstract:
The talk will be about a non-classical generalisation of the Cassels-Tate pairing. It is defined, roughly speaking, on the non-generic part of the dual of the Selmer group. The existence of the pairing has non-trivial consequences; for example, one can deduce results about the parity of the (co)-rank of the Selmer group.

Nov 13 1998 Kevin Buzzard

Families of modular forms and the Gouvea-Mazur conjecture

Abstract:
Recent deep work of Coleman has shown what people have suspected now for a long time - namely that many modular forms `come in families'. Coleman's work establishes (a slightly weak form of) a conjecture of Gouvea and Mazur. I will explain the conjecture and show how one can use a completely different (and much simpler method) to attack it. The simpler method does not give results as strong as Coleman's (one only gets `continuity' results rather than `analyticity' results), but has the advantage that it generalises much more easily to other kinds of modular forms.

KCL Maths Dept Home page

UCL Maths Dept Home page

IC Maths Dept Home page

Department of Mathematics - King's College London

28 April		Dan Snaith (Imperial) "Overconvergent Siegel modular forms"
5 May		Luis Dieulefait (Barcelona) "Existence of compatible families of Galois representations and the Fontaine-Mazur conjecture for elliptic curves"
12 May		Bruno Kahn (Paris 7) "Birational motives"
19 May		Fre Vercauteren (Bristol) "Zeta functions: the p-adic approach"
26 May		Sarah Zerbes (Cambridge) "Selmer groups over p-adic Lie extensions"
2 June		Adam Joyce (Imperial) "The Manin constant of modular abelian varieties" The Manin constant of a (modular) elliptic curve was introduced by Manin in a paper in the mid-70s, in which he also conjectured that it is always 1. I shall give this definition and discuss the results in the direction of proving the conjecture. I'll then generalise the definition to abelian varieties of arbitrary dimension and discuss the extensions of the above results to the general setting.
9 June		Brian Conrey (AIM) "Random matrix theory and ranks of elliptic curves"
16 June		Detlev Hoffman (Nottingham) "Isotropy of quadratic forms in finite and infinite dimension"
23 June		Teruyoshi Yoshida (Harvard, visiting Imperial) "Non-abelian Lubin-Tate theory and Deligne-Lusztig theory"
4 August		Chandrashekhar Khare (Utah/Tata) "Transcendental Galois representations"

14 January		Kevin Buzzard (Imperial) "The 2-adic eigencurve at the boundary of weight space" Eigencurves are geometric objects parameterising certain modular forms. These objects were introduced by Coleman and Mazur in the mid-1990s and at the time very little was known about what they "looked like". Lloyd Kilford and I can write down equations for one of these eigencurves, and these equations are sufficiently explicit to enable us to pin down exactly what this eigencurve looks like near its boundary. In my talk I will give an introduction to the theory of eigencurves and then will explain a sketch of our result.
21 January		Alexei Skorobogatov (Imperial) "Rational points on Enriques surfaces" The Enriques surfaces are cohomologically indistinguishable from rational surfaces, however unlike rational surfaces they have a non-trivial though very small fundamental group. For rational surfaces it is conjectured by Colliot-Th e and Sansuc, and is proved in many cases that the failure of the Hasse principle and weak approximation is controlled by the obstruction based on the Brauer group of the surface. In a joint work with David Harari we construct an Enriques surface over Q with an adelic point satisfying all the conditions provided by the Brauer group, but not in the closure of the set of Q-rational points. The proof uses descent to a torsor on this surface; the structure group of this torsor is a form of a (non-abelian) 1-dimensional orthogonal group.
28 January		Roger Heath-Brown (Oxford) "Cayley's cubic surface" Roughly how many non-trivial primitive integer solutions does the equation 1/X₀+1/X₁+1/X₂+1/X₃=0 have, in a large cube max \|X_i\| < B ? On clearing the denominators this gives us a Cayley's cubic surface. Manin's conjecture predicts a growth rate of order B(log B)⁶. This has now been proved, by using the Universal Torsor for the surface. The latter is an affine variety in 16 dimensional space, which encodes all the relevant divisibility information. The proof entails counting points on this affine variety, for which the key tool comes from the geometry of numbers.
4 February		Anthony Hayward (King's) "Congruences satisfied by Stark units" "Refined abelian Stark conjectures" are special value conjectures for equivariant L-functions associated to Galois extensions of global fields at "s=0". There has been a proliferation of such conjectures since Stark's original work in the 1970s, notably those of Rubin and Popescu, who generalised Stark's abelian conjecture to account for differing orders of vanishing of the L-functions, and Gross, who gave congruences for the value of the L-function. These conjectures admit a natural refinement, due to Burns, which gives Gross-style congruences in the situation of Rubin's conjecture. The formulation of the refinement is inspired by the Equivariant Tamagawa Conjecture, and this over-riding conjecture provides a unifying overview of the field which was previously sorely lacking. After describing this general situation in the first half of the talk, I move on to study those situations in which systems of explicit units provide proofs of the Burns refinement, hopefully giving more of a sense of what these conjectures are "about". The proof involves details on the explicit units, a study of ray-class fields, and some unusual combinatorics and integer identities.
11 February		Victor Abrashkin (Durham) "Galois modules arising from Faltings's strict modules" A classical analogue of the concept of a finite flat group scheme over a complete discrete valuation ring loses all interesting properties in the equal characteristic case. In the past year Faltings proposed a modified definition of a group scheme with strict action and showed that it works perfectly in some situations. It will be explained in the talk that many known results about classification and arising Galois modules still hold for Faltings's modules.
18 February		Kanetomo Sato (Nagoya) "p-adic ale Tate twists and arithmetic duality" For an algebraic variety X over a field k and a positive integer n invertible in k, the ale sheaf of n-th roots of unity and its tensor powers are called ( ale) Tate twists, and play fundamental roles both in number theory and in arithmetic geometry. In this talk, I will talk about a construction of p-adic Tate twists on regular arithmetic schemes (=schemes which are regular flat of finite type over Spec Z), and arithmetic duality theorems for p-adic Tate twists, which generalizes the classical Artin-Verdier duality theorem.
25 February		Toby Gee (Imperial) "Companion forms over totally real fields" The companion forms conjecture was part of Serre's conjecture on the modularity of mod p Galois representations. This was proved in the early 90s by Gross and Coleman & Voloch. More recently Fred Diamond has conjectured extensions of these results to totally real fields; I will describe my recent progress on cases of these conjectures.
3 March		Aleksandra Shlapentokh (East Carolina) "Hilbert's tenth problem and Mazur's conjectures" We discuss recent results concerning extensions of Hilbert's tenth problem to rings of integers of number fields and the field of rational numbers, and related conjectures of Mazur.
10 March		R is de la Bret he (ENS Paris) "Counting points on varieties using universal torsors" We study the asymptotic order of the number of points of bounded height on certain varieties. Universal torsors turned out to be a useful tool to attack this kind of problem. We will give few examples to explain how to prove asymptotic estimations using universal torsors and tools of analytic number theory.
17 March		Nikolaos Diamantis (Nottingham) "Second order cusp forms and L-functions" Second-order modular forms are functions that have recently appeared in several contexts: Eisenstein series formed with modular symbols, converse theorems of L-functions, percolation theory etc. They satisfy a functional equation that extends naturally that of the usual modular forms and their study is important for the topics that have motivated their introduction. We will discuss the ways they arise in various contexts, their classification and their L-functions.
24 March		Adam Logan (Liverpool) "Heegner points on elliptic curves over real quadratic fields" Henri Darmon formulated a concrete version of a conjecture of Oda according to which it should be possible to construct rational points on elliptic curves over real quadratic fields defined over quadratic extensions of the field by integrating the Hilbert modular form associated to the curve. We present the conjecture together with some numerical evidence for it. This is joint work with Darmon.

Jun 27	Stephen Lichtenbaum (Brown University, USA) - "A Search for a New Cohomology Theory for Schemes over Z" (Final Seminar)
Jun 26	Frank Calegari - `unofficial' seminar at 3.30pm in room 413 in Imperial College - `Semistable Abelian Varieties of small conductor' Abstract: Fontaine proved in 1985 that there do not exist any Abelian varieties over Z. Jacobians of the modular curves X_0(p^n) (non-zero for sufficiently large n) provide examples of Abelian varieties over Z[1/p] for each p. If, however, we restrict our attention to semistable Abelian varieties,then combining Fontaine's theorem, recent papers of Brumer-Kramer and of Schoof, and some new ideas and results, we prove the following: There exists a semistable Abelian variety over Z[1/n] with n squarefree if and only if n is not in the set: {1,2,3,5,6,7,10,13}.
Jun 26	William Stein - `unofficial' seminar at 2pm in the Clore lecture theatre in Imperial College - `Mordell-Weil groups of elliptic curves over Q are visible' Abstract: During the past few years, Barry Mazur, myself, and others have studied visible subgroups of Shafarevich-Tate groups of abelian varieties. Recently, I've been studying visibility of Mordell-Weil groups of abelian varieties. In this talk, I will very briefly review some results about visibility of Shafarevich-Tate groups, then discuss some of what I've been able to prove about their counterparts in the context of visibility of Mordell-Weil groups. In particular, I will show that Mordell-Weil groups of elliptic curves over Q are visible in modular abelian varieties.
Jun 15	Helena Verrill - `unofficial' seminar at 3pm in room 413 in Imperial College - `Modular symbols computations and a certain intermediate Jacobian'
Jun 13	Romyar Sharifi - `Galois Module Structure of Local Unit Groups and Conductors' Abstract: We give an explicit description of generators of the ith unit groups of K = Q_pzeta_pⁿ as Galois submodules of the multiplicative group of K. We can use this to determine the ramification groups of degree pⁿ Kummer extensions of K which are Galois over Q_p.
Jun 6	Richard Hill (University College, London) - "The Kubota Symbol on SL_n"
May 30	Prof. V. Nikulin (Liverpool) - "Dynamics of the automorphism group of a K3 surface."
May 23	Daniel Delbourgo (Nottingham) - "Two-variable Euler systems in the non-CM case"
My 16	Rob de Jeu (Durham) - "Algebraic K-theory and Syntomic Regulators" (provisional title)

Jan 17	Kevin Buzzard (Imperial) - An overview. The first talk in the study group on "Remarks on a conjecture of Fontaine and Mazur" by R. Taylor
Jan 24	Al Weiss (U Alberta) "Equivariant Iwasawa Theory"
Jan 31	Andreas Langer (Bielefeld) "Relative de Rham-Witt cohomology"
Feb 7	Colin Bushnell (Kings) "Local Langlands correspondences: some explicit problems"
Feb 14	Alexei Skorobogatov (Imperial) "Rational solutions of certain equations involving norms"
Feb 21	Christophe Cornut (Strasbourg) "A conjecture of B. Mazur on higher Heegner points"
Feb 28	Keith Ball (UCL) "There are infinitely many irrational values of zeta at the odd numbers"
Mar 7	Werner Hoffman (Humboldt U) "On the differential equations satisfied by weighted orbital integrals"
Mar 14	Anupam Saikia (Cambridge) Title TBA
Mar 21	Victor Flynn (Liverpool) "Coverings of curves"
	Abstract: We shall discuss the idea of finding all rational points on a curve C by first finding an associated collection of curves whose rational points cover those of C. This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves and Chabauty techniques. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians.

Oct 11	Kevin Buzzard (IC) "Questions about slopes of modular forms"
Oct 18	Alexei Skorobogatov (Imperial) "Introduction to the study group: McCallum's work on the Brauer points on Fermat curves"
Oct 25	David Solomon (KCL) "A p-adic Stark conjecture in Z_p-extensions"
Nov 1	Kevin Buzzard (IC) "Continuation of the study group on McCallum's work: Coleman's integral and applications to rational points"
Nov 8	Jan Nekovar (Cambridge) "Some consequences of Mazur's conjecture on Heegner points"
Nov 15	Tom Fisher (Cambridge) "The Cassels-Tate pairing and the icosahedron"
Nov 22	Roger Heath-Brown (Oxford) Counting rational points on algebraic curves and surfaces;
Nov 29	Jean-Robert Belliard (Nottingham) "Real analogues of Stickelberger's theorem"
Dec 6	Richard Hill (UCL) "Syntomic regulators and K-theory of curves"

Jan 26	Neil Dummigan (Oxford) "Some evidence for the Bloch-Kato conjecture at critical points"
Feb 2	Susan Howson (Nottingham) "non-Abelian Iwasawa theory and applications to elliptic curves"
Feb 9	David Burns (KCL) "On the conjectures of Gross and Tate concerning values of abelian L-functions at s = 0" (This will be the first talk in a study group)
Feb 16	Victor Abrashkin "Ramification Filtrations and a local analogue of the Grothendieck Conjecture"
Feb 23	David Burns (KCL) "Continuation of the study group on values of Abelian L-functions at s=0"
Mar 1	Richard Hill Continuation of the study group.
Mar 8	Robert Vaughan "A Generalized Divisor Problem".
Mar 15	David Solomon Continuation of the study group.
Mar 22	David Solomon Continuation of the study group on values of Abelian L-functions at s=0.

Cambridge-Oxford-Warwick (COW) algebraic geometry seminar, room 642 IC
2.00	Nick Shepherd-Barron (Cambridge CMS) - Isomorphism classes in local moduli spaces
3.15	Paul Seidel - Floer cohomology and the Picard-Lefschetz formula