# Abstracts for Conference on Arithmetic Algebraic Geometry on the occasion of Gerd Faltings' 60th birthday

Alternatively have a look at the program.

## Some equidistribution conjectures on curves

We will formulate two sorts of equidistribution conjectures on curves, and discuss the numerical evidence for their truth in the case of elliptic curves.

## On Schmidt's Subspace Theorem

Schmidt's Subspace Theorem is a higher dimensional generalization of Roth's Theorem on the approximation of algebraic numbers by elements from a given number field. It asserts that the set of solutions in $P^n(K)$ ($K$ number field) of a particular Diophantine inequality is contained in finitely many proper linear subspaces of $P^n(K)$. I would like to discuss quantitative versions giving an explicit upper bound for the number of subspaces, and go into ideas of Faltings that went into the proof.

## Supersingular K3 surfaces are unirational

We show that supersingular K3 surfaces are related by purely inseparable isogenies. As an application, we deduce that they are unirational, which confirms conjectures of Artin, Rudakov, Shafarevich, and Shioda. The main ingredient in the proof is to use the formal Brauer group of a Jacobian elliptically fibered K3 surface to construct a family of “moving torsors” under this fibration that eventually relates supersingular K3 surfaces of different Artin invariants by purely inseparable isogenies.

## Class groups of imaginary quadratic fields

## Galois representations attached to torsion in the cohomology of automorphic vector bundles, after Boxer

I will report on work of George Boxer in which he has constructed Hasse like invariants on the closure of Ekedahl-Ooort strata, which cut out the open stratum. This answers in the affirmative a question of Oort as to whether the open Ekedahl-Oort strata (in the minimal compactification) are affine. It also allows Boxer to attach Galois representations to torsion in the (Zariski) cohomology of automorphic vector bundles on certain Shimura varieties.

## Boundedness for representations of the fundamental group

Deligne's finiteness theorem for lisse $\bar{\mathbb{Q}}_\ell$-sheaves ver finite fields for bounded rank and bounded ramification suggests the existence of boundedness theorems for other categories as well, notably for algebraic flat connections over the complex numbers and stratifications in characteristic $p>0$.

We will report on what we understand so far (work in progress).

## Torsion theories for Iwahori-Hecke modules in characteristic $p$

By work of Bernstein and Borel the smooth representation theory of a $p$-adic reductive group $G$ with characteristic zero coefficients is very closely related to the module theory of the corresponding Iwahori-Hecke algebra $H$. With characteristic $p$ coefficients this relation is much more complicated and still mysterious. I will introduce a canonical torsion pair in the module category of $H$ (but which depends on $G$) whose associated class of torsion free modules allows a natural fully faithful embedding into the category of smooth mod $p$ representations of $G$.

## The hyperbolic Ax-Lindemann conjecture

The hyperbolic Ax Lindemann conjecture is a functional transcendental statement which describes the Zariski closure of "algebraic flows" on Shimura varieties. We will describe the proof of this conjecture and its consequences for the André-Oort conjecture. This is a joint work with Bruno Klingler and Andrei Yafaev.

## Some Variants of Jordan's Theorem

Jordan's theorem on finite subgroups of $GL_n(C)$ asserts that each such group is contained in an algebraic subgroup of $GL_n$ which is toral-by-finite and of bounded complexity. As such, it exemplifies the idea of trying to relate various kinds of linear groups to linear algebraic "envelopes" of some kind. I will discuss some variations on this theme.

## Gross-Zagier formula: why is it right?

In his eccentrically written paper in 1952, Heegner showed that the 1000 years old congruent number problem is soluble for all prime $p$ (or $2p$) congruent to 5, 7 (or 6) mod 8, by constructing some rational points on certain elliptic curves of infinite order. In 1986, Gross and Zagier proved a formula which relates the heights of Heegner points and derivatives of $L$-series. At the ICM in 1986, Faltings asked: "Alles in allem handelt es sich um eine schöne Entdeckung, welche wir aber leider noch nicht "erklären" können: Warum ist sie richtig?"