London Number Theory Seminar
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London Number Theory Seminar |
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The London Number Theory Seminar is held weekly during term times. The site rotates between KCL, Imperial College and UCL.
This term, the seminar will be hosted by Imperial College and will be held on Wednesdays from 4:00 pm to 5:00 pm in room 658 of the Huxley Building. This term's organiser is Alexei Skorobogatov.
The seminar is preceded by the "Informal graduate student meeting" from 12:00 to 13:30 (starting on 14th October) and the Study Group on "Deformations of Galois representations" from 14:00 to 15:30 (everything in room 658). Tea is served in the Common Room on the 5th floor at 15:30.
| 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 In unpublished work, Wetherell suggested that Chabauty's method should still be applicable provided the weaker bound $r \leq d(g-1)$ is satisfied.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 $K=\Q(\sqrt[3]{2})$. |
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| 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. |
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| 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). |
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| 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. |
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| 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. |
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| 16 November |
The London-Paris Number Theory Seminar speakers: M. Emerton, A. Skorobogatov, S. David |
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| 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.) |
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| 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). |
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| 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. |
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| 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 $\mathbb{F}_1$. |
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| 16 December | tba tba |
A list of previous seminar talks is here.
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