Number theory seminar, Lent 2017

The seminar will be at Malet Place Engineering, Room 1.02 (except on February 15 and 22, when it will be in Archaeology, Room G6), from 4:30-5:30pm. Here is a map of UCL campus. The seminar will be preceded by study groups, from 1-2:15pm (p-adic integration, organized by Jack Lamplugh and Joe Kramer-Miller) and 2:30-4pm (p-adic Hodge theory over Lubin-Tate extensions, organized by Joaquin Rodrigues Jacinto), in the same room. Here is the programme of the second study group. There will be tea and coffee in room 606 from 4pm onwards.

January 11
Chris Skinner (Princeton University)
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.

January 18
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 Zp^3-extension of F.

January 25
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* and K* as subgroups of J_L and J_K respectively. The norm map N: J_L--> J_K restricts to the usual field norm N: L*--> K* on L*. Thus, if an element of K* is a norm from L*, 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* which is a norm from J_L is in fact a norm from L*. 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.

February 1
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.

February 8
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.

February 15
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 Q, and the points over fields cut out by Artin representations attached to modular forms of weight one.

February 22
Erick Knight (Harvard/Bonn)

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.

March 1
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.

March 8
Céline Maistret (University of 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.

March 15
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.

March 22
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.