London Number Theory Seminar

UCL King's Imperial College

The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL.

This term (Spring 2017), the seminar will be hosted by UCL, and will be on Wednesdays at 4:30pm-5:30pm (note non-standard time) in UCL in Malet Place Engineering, Room 1.02 (except on February 15 and 22, when it will be in Archaeology, Room G6) (note non-standard place), starting on 11th January and finishing on March 22nd. The seminar is organised by Sarah Zerbes. The only hint I have for finding the seminar this term is this generic map link for UCL. Good luck.

The seminar will be preceded by tea and biscuits at 4pm in room 606.

The most up-to-date list of speakers/titles is at Sarah Zerbes' website here. The last time I looked it said this:

11/01/17 Chris Skinner (Princeton)
Title: Recent progress on the Iwasawa theory of elliptic curves and modular forms.
Abstract: This talk will describe some of the recent work on the Iwasawa theory of modular forms (at both ordinary and non-ordinary primes) with an emphasize on the strategy of proof, which involves two different main conjectures.

18/01/17 Jack Lamplugh (UCL)
Title: An Euler system for a pair of CM modular forms.
Abstract: Given a pair of modular forms and a prime p, Lei-Loeffler-Zerbes have constructed an Euler system for the tensor product of the p-adic Galois representations attached to each of the forms. When the forms have CM by distinct imaginary quadratic fields, this representation is induced from a character $\chi$ over an imaginary biquadratic field F. I will explain how one can use this Euler system to obtain upper bounds for Selmer groups associated to $\chi$ over the $\mathbf{Z}_p^3$-extension of F.

25/01/17 Rachel Newton (Reading University)
Title: The Hasse norm principle for abelian extensions
Abstract: Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^\times$ and $K^\times$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to J_K$ restricts to the usual field norm $N: L^\times\to K^\times$ on $L^\times$. Thus, if an element of $K^\times$ is a norm from $L^\times$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^\times$ which is a norm from $J_L$ is in fact a norm from $L^\times$. The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis. This is joint work with Christopher Frei and Daniel Loughran.

1/2/17 Atsuhira Nagano (KCL/Waseda University)
Title: K3 surfaces and a construction of a Shimura variety
Abstract: In old times, elliptic modular functions appeared in the study of elliptic curves. They are applied to the construction of class fields. (This is classically called Kronecker's Jugendtraum.) K3 surfaces are 2-dimensional analogy of elliptic curves. In this talk, the speaker will present an extension of the classical result by using K3 surfaces. Namely, we will obtain Hilbert modular functions via the periods of K3 surfaces and construct a certain model of a Shimura variety explicitly.

8/2/17 Christian Johansson (University of Cambridge)
Title: Integral models for eigenvarieties
Abstract: I will discuss a construction of integral models of eigenvarieties using a generalization of the overconvergent distribution modules of Ash and Stevens, and their relation to recent work of Andreatta-Iovita-Pilloni and Liu-Wan-Xiao on the geometry of the Coleman-Mazur eigencurve near the boundary of weight space. This is joint work with James Newton.

15/2/17 (note room change: Archaeology, Room G6)
Alan Lauder (Oxford)
Title: Stark points on elliptic curves and modular forms of weight one
Abstract: I shall discuss some work with Henri Darmon and Victor Rotger on the explicit construction of points on elliptic curves. The elliptic curves are defined over $\mathbb{Q}$, and the points over fields cut out by Artin representations attached to modular forms of weight one.

22/2/17 (note room change: Archaeology, Room G6)
Erick Knight (Harvard/Bonn)
Title: A p-adic Jacquet-Langlands correspondence
Abstract: In this talk, I will construct a p-adic Jacquet-Langlands correspondence, which is a correspondence between Banach space representations of GL2(Qp) and Banach space representations of the unit group of the quaternion algebra D over Qp. The correspondence satisfies local-global compatibility with the completed cohomology of Shimura curves, as well as a compatibility with the classical Jacquet-Langlands correspondence, in the sense that the $D^\times$ representations can often be shown to have the expected locally algebraic vectors.

1/3/17 Giuseppe Ancona (Université de Strasbourg)
Title: Standard conjectures for abelian fourfolds
Abstract: Let X be a smooth projective variety and V be the finite dimensional Q vector space of algebraic cycles on X modulo numerical equivalence. Grothendieck defined a quadratic form on V (basically using the intersection product) and conjectured that it is positive definite. This conjecture is a formal consequence of Hodge Theory in characteristic zero, but almost nothing is known in positive characteristic. Instead of studying this quadratic form at the non archimedean place (the signature) we will study it at the p-adic places. It turns out that this question is more treatable. Moreover, using a product formula formula, the p-adic information will give us non trivial informations on the non archimedean place. For instance we will show the original conjecture when X is an abelian variety of dimension 4.

8/3/17 CĂ©line Maistret (Warwick)
Title: Parity of ranks of abelian surfaces
Abstract: Let K be a number field and A/K an abelian surface (dimension 2 analogue of an elliptic curve). By the Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.

15/3/17 Aurel Page (Warwick)
Title: Computing the homotopy type of compact arithmetic manifolds
Abstract: Cohomology of arithmetic manifolds equipped with the action of Hecke operators provides concrete realisations of automorphic representations. I will present joint work with Michael Lipnowski where we describe and analyse an algorithm to compute such objects in the compact case. I will give a gentle introduction to the known case of dimension 0, sketch ideas and limitations of previous algorithms in small dimensions, and then explain some details and ideas from the new algorithm.

22/3/17 Otto Overkamp (Imperial)
Title: Finite descent obstruction and non-abelian reciprocity
Abstract: For a nice algebraic variety X over a number field F, one of the central problems of Diophantine Geometry is to locate precisely the set X(F) inside X(A_F), where A_F denotes the ring of adeles of F. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for finite étale group schemes over F on X. More recently, Kim proposed an iterative construction of another subset of X(A_F) which contains the set of rational points. In this paper, we compare the two constructions. Our main result shows that the two approaches are equivalent.

The seminar will be preceded by the Study Groups, and this term there are two of them, one on p-adic integration running 1300-1415 and one on p-adic Hodge theory running 1430-1600.

A list of previous seminar talks is here.

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This page is maintained by Kevin Buzzard.