Fabien Pazuki (University of Copenhagen)

Regulators of number fields and abelian varieties

In the general study of regulators, we present three inequalities. We first bound from below the regulators of number fields, following previous works of Silverman and Friedman. We then bound from below the regulators of Mordell-Weil groups of abelian varieties defined over a number field, assuming a conjecture of Lang and Silverman. Finally we explain how to prove an unconditional statement for elliptic curves of rank at least 4. This third inequality is joint work with Pascal Autissier and Marc Hindry.

7 May

Tuesday 3pm MS.03

Nicole Looper (University of Cambridge)

Dynamical uniform boundedness and the abc-conjecture

In 1994, Morton and Silverman formulated what is now known as the Uniform Boundedness Conjecture. A major open problem in arithmetic dynamics, the conjecture includes as particular cases the theorems of Mazur and Merel on uniform bounds on the rational torsion points on elliptic curves. In this talk, I will discuss a uniform boundedness theorem for unicritical polynomials. I will also discuss the relevant tools from Diophantine geometry and their relation to the abc-conjecture.

Kevin Ford (University of Illinois at Urbana-Champaign)

Prime number models, large gaps, prime tuples and the square-root sieve

We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve". In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

Kirsti Biggs (University of Bristol)

Heights of algebraic numbers and Lehmer's conjecture

In 1933, D.H. Lehmer asked a question about roots of integer polynomials which can be translated into the language of the Weil height of algebraic numbers. Although his question - now known as Lehmer's conjecture - remains open, it is possible to prove bounds of the required type in certain specific cases. In this talk, I'll discuss a generalisation of a result of Amoroso and Masser to algebraic numbers generating Galois extensions of an arbitrary number field, which is joint work with S. Akhtari, K. Aktas, A. Hamieh, K. Petersen and L. Thompson.

Alice Pozzi (UCL)

The eigencurve at Eisenstein weight one points

In this talk, we discuss the geometry of the Coleman-Mazur eigencurve at weight one Eisenstein points. The local nature of the eigencurve is mostly understood at classical points of weight greater than one, while one observes some unusual behaviours at weight one. In particular, we study cuspidal Hida families specializing to Eisenstein series at weight one. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations.

We discuss the implications of our analysis on the classicality of a certain overconvergent eigenspace. Finally, we explain how this Galois-theoretic method yields some new insight on Gross’s formula relating the leading term of the p-adic L-function to a Stark unit. This is joint work with Adel Betina and Mladen Dimitrov.

Maxim Gerspach (ETH Zürich)

Even moments of random matrix-valued multiplicative functions

Random multiplicative functions have received a lot of attention recently as random models for number-theoretic objects such as Dirichlet characters or the Möbius function, and understanding their behaviour has allowed us to get heuristic, and sometimes even rigorous, insights into the behaviour of their deterministic counter-parts.

I will introduce a generalization of random multiplicative functions which allows matrix values, and I will give some bounds on their moments. These turn out to be related to the so-called generalized joint spectral radii, which have been studied in a wide range of fields such as dynamical systems, wavelets and control theory.

Alessandro Arlandini (Warwick)

Factorisation of the p-adic Rankin-Selberg L-function in the supersingular case

In this talk I will discuss how to express the p-adic Rankin-Selberg L-function as a product of other p-adic L-functions. In order to do this I will rely on the existence of Euler systems, in particular of the Beilinson-Flach one. The crucial point of the strategy is the relation between the base class of an Euler system and some special value of the attached complex and p-adic L-functions. I will focus on the case of a supersingular modular form, which extends results already known in the ordinary case.

Adam Morgan (University of Glasgow)

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.