Research Seminar Number Theory and Arithmetic Geometry

The Relative Bogomolov Conjecture for Fibered Products of Elliptic Families

I will talk about the deduction of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of elliptic families from my recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin-Mumford type proven by Masser and Zannier to results of Bogomolov type.

Fr 13.5.2022

Uniform potential density for rational points on algebraic groups and elliptic K3 surfaces

A variety satisfies potential density if it contains a dense subset of rational points after extending its ground field by a finite degree. A collection of varieties satisfies uniform potential density if that degree can be uniformly bounded. I will discuss this property for connected algebraic groups of a fixed dimension and elliptic K3 surfaces. This is joint work with Kuan-Wen Lai.

Fr 20.5.2022

Diophantine equations in families

In this talk we consider quantitative aspects of the question of the solvability of diophantine equations in (algebraic) families, namely for "how many" choices of the parameters the corresponding equation admits a solution. I will consider the case of integral solvability for one parameter family of conics: I will explain a recent joint work with Peter Koymans settling a conjecture of Stevenhagen for the, so called, negative Pell equation. I will also discuss a joint work in progress with Efthymios Sofos on the case of adelic/rational solvability (studied by Serre and Laughran-Smeets) and its application to Diophantine stability questions. Finally I will explain the (unexpected) relation(s) both works have with statistics of L-functions and Iwasawa towers.

The Hirzebruch-Mumford volume of unitary groups and its application to the geometry of ball quotients

Mo 30.5.2022 (15:00-17:30, online)

Fr 3.6.2022

Weak weak approximation and conic bundles

In joint work with Sam Streeter, we show that a surface X with two distinct conic bundles and Zariski-dense rational points satisfies, under some mild conditions, the weak weak approximation property. In particular, our result shows that a "generic" del Pezzo surface of degree 1 or 2 with Zariski-dense rational points and (even just) one conic fibration satisfies the weak weak approximation property. Our method is based on an application of a result of Denef on certain auxiliary rational higher-dimensional covers of X.

Fr 17.6.2022

Exact verification of the strong BSD conjecture for some absolutely simple modular abelian surfaces

The strong Birch-Swinnerton-Dyer conjecture and in particular the exact order of the Shafarevich-Tate group for abelian varieties over the rationals has only been known for elliptic curves (dimension 1) or in higher dimension where the conjecture could be reduced to dimension 1. We give the first absolutely simple examples of dimension 2 where the conjecture can be verified:

Let X be (1) a quotient of the modular curve X₀(N) by a subgroup generated by Atkin-Lehner involutions such that its Jacobian J is an absolutely simple modular abelian surface, or, more generally, (2) an absolutely simple factor of J₀(N) isomorphic to the Jacobian J of a genus-2 curve X. We prove that for all such J from (1), the Shafarevich-Tate group of J is trivial and satisfies the strong Birch-Swinnerton-Dyer conjecture. We further indicate how to verify strong BSD in the cases (2) in principle and in many cases in practice.

To achieve this, we compute the image and the cohomology of the mod-p Galois representations of J, show effectively that almost all of them are irreducible and have maximal image, make the Heegner points Euler system of Kolyvagin-Logachev effective, compute the Heegner points and Heegner indices, compute the p-adic L-function, and perform p-descents. Since many ingredients are involved in the proof, we will give an overview of the methods involved and give more details regarding the computation of the Galois representations and Heegner indices.

This is joint work with Michael Stoll.

The Hodge locus

I will report on a joint work with Klingler and Ullmo. Given a polarizable variation of Hodge structures on a smooth quasi-projective variety S (e.g. the one associated to a family of pure motives over S), Cattani, Deligne and Kaplan proved that its Hodge locus (the locus of closed points of S where exceptional Hodge tenors do appear) is a countable union of closed algebraic subvarieties of S. I will explain when this Hodge locus is actually algebraic. Depending on the interest of the audience, I will conclude the talk either by describing how such an algebraicity statement complements the Lawrence-Venkatesh method or by sketching how similar ideas guarantee the existence of infinitely many 4-dimensinal Jacobians (defined over Qbar) of "Mumford’s type".

Around the support problem for Hilbert class polynomials

I will report on joint work in progress with Francesco Campagna (MPIM Bonn). Let H_D(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D. We study the rate of growth of the greatest common divisor of H_D(a) and H_D(b) as |D| → ∞ for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing H_D(a) also divides H_D(b), what can we say about a and b? This is inspired by work of Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and Corrales-Rodrigáñez-Schoof, who studied these questions with Tⁿ-1 in place of H_D(T) in the ring of S-integers in some number field.

Equidistribution and the Bogomolov conjecture for dynamical systems

In recent joint work with Myrto Mavraki we proved a uniform version of the Bogomolov conjecture for curves and split rational maps. I will discuss some of the ingredients of our proof and its relation to a relative version of the Bogomolov conjecture formulated by Lars Kühne. Time permitting I will talk about connections to problems in measure theory as well as related work in progress.