# London Number Theory Seminar Spring 2019

## Schedule

##### Parity of Selmer ranks in quadratic twist families

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we prove results about the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square.

##### Vanishing theorems for étale sheaves Room 500

The talk is based on two results: Scholze's Artin type vanishing theorem for the projective space, which I proved without perfectoid geometry (which implies in particular that it holds in positive characteristic), and a rigidity theorem for subloci of the $$\ell$$-adic character variety stable under the Galois group over a number field (joint work in progress with Moritz Kerz).

##### Automorphism groups of K3 surfaces over nonclosed fields

Using the Torelli theorem for K3 surfaces of Pyatetskii-Shapiro and Shafarevich one can describe the automorphism group of a K3 surface over $${\mathbb C}$$ up to finite error as the quotient of the orthogonal group of its Picard lattice by the subgroup generated by reflections in classes of square $$-2$$. We will give a similar description valid over an arbitrary field in which the reflection group is replaced by a certain subgroup. We will then illustrate this description by giving several examples of interesting behaviour of the automorphism group, and by showing that the automorphism groups of two families of K3 surfaces that arise from Diophantine problems are finite. This is joint work with Martin Bright and Ronald van Luijk (University of Leiden).

##### Fourier analysis on universal formal covers

The p-adic Fourier transform of Schneider and Teitelbaum has complicated integrality properties which have not yet been fully understood. I will report on an approach to this problem relying on the universal formal cover of a p-divisible group as introduced by Scholze and Weinstein. This has applications to the representation theory of p-adic division algebras.

##### p-adic L-functions of Hilbert cusp forms and the trivial zero conjecture Room 500

In a joint work with Daniel Barrera and Andrei Jorza, we prove a strong form of the trivial zero conjecture at the central point for the p-adic L-function of a non-critically refined cohomological cuspidal automorphic representation of GL(2) over a totally real field, which is Iwahori spherical at places above p. We will focus on the novelty of our approach in the case of a multiple trivial zero, where in order to compute higher order derivatives of the p-adic L-function, we study the variation of the root number in partial finite slope families and establish the vanishing of many Taylor coefficients of the p-adic L-function of the family.

##### Symmetries and spaces

It is a long established idea in mathematics that in order to understand space we need to study its symmetries. This is the centrepoint of the Erlangen program, which, published by Felix Klein in 1872 in Vergleichende Betrachtungen über neuere geometrische Forschungen, is a method of characterizing geometries based on group theory.

In a group we can multiply, while on a space we can integrate. I will explore the link between the two starting with the mathematics of the seventeenth century and leading to the arithmetic of elliptic curves.

##### Rational points over global fields and applications

We present analytic methods for counting rational points on varieties defined over global fields. The main ingredient is obtaining a version of Hardy-Littlewood circle method which incorporates elements of Kloosterman refinement in new settings.

##### How many real Artin-Tate motives are there?

The goals of my talk are 1) to place this question within the framework of tensor-triangular geometry, and 2) to report on joint work with Paul Balmer (UCLA) which provides an answer to the question in this framework.

##### The sup-norm problem over number fields Room 500

In this talk we study the sup-norm of automorphic forms over number fields. This topic sits on the intersection of Quantum chaos, harmonic analysis and number theory and has seen a lot progress lately. We will discuss some of the recent result in the rank one setting.

##### Singular moduli for real quadratic fields and p-adic mock modular forms

The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles. I will discuss p-adic counterparts for our proposed RM invariants of classical relations between singular moduli and the theory of weak harmonic Maass forms.

##### The eigencurve at Eisenstein weight one points

Coleman and Mazur constructed the eigencurve, a rigid analytic space classifying p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is better understood at points corresponding to cuspforms of weight greater than 1, while the weight one case is far more intricate. In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. We focus on the unusual phenomenon of cuspidal Hida families specializing to Eisenstein series at weight one. We discuss the relation between the geometry of the eigencurve and the Gross-Stark Conjecture.

## Directions

##### Finding the seminar room

The UCL Mathematics department is situated in the upper floors of the 25 Gordon Street building of UCL's main campus. From the street the building is most identifiable by its UCL student union branding. To reach the upper floors, entering through the main entrance, go to the right to find a corridor with lifts. The maths department has use of the second set of two lifts but not the first. For Room 505, exit at the 5th floor and go as far to the right as possible, then turn right again. For the common room exit on the 6th floor, turn right and go straight.

##### Getting to UCL

UCL is well-connected by all forms of public transport. The nearest tube stations are Euston (Victoria and Northern Lines) and Euston Square (Circle, Hammersmith & City, Metropolitan Lines). It is also easy walking distance from Euston, King's Cross and St. Pancras (Eurostar) train stations. Of the London airports, Heathrow, Luton and Stansted all work well for UCL.