Alternatively have a look at the program.

## Critical values, congruences, Selmer groups

Since the seminar I am moderating is called congruences, I'll start off by explaining the possibilities for the composition factors of the reduction mod p of the type of 4-dimensional Galois representation coming from a CY 3-fold, and comparing these with the kinds of congruences observed by Anton Mellit, calling on him to report on observations about pairs of congruences and common values of $p$. This would set the scene.

## Formal groups associated to pencils of Calabi-Yau varieties.

## A characterization of toric varieties in characteristic $p$

## Introduction to $F$-isocrystals on the line

## Introductions to log-growth and Frobenius slope filtrations.

## Galois theory of differential equations with an action of an endomorphisms.

## Congruence sheaves via Hecke kernels.

We will introduce Hecke kernels according to Kontsevich and show how to construct congruence $D2$ differential equations via Hecke correspondences in practice.

## Modular D3 equations and spectral elliptic curves

Determinantal differential equations were introduced by Vasily Golyshev and Jan Stienstra around 2005. The motivation comes from mirror symmetry for Fano varieties. I will talk about our recent work with Vasily on such equations of orders 2 and 3, that is D2 and D3. We show that the expansion of the analytic solution of a non-degenerate modular equation of type D3 over the rational numbers with respect to the natural parameter coincides, under certain assumptions, with the $q$-expansion of thenewform of its spectral elliptic curve and therefore possesses a multiplicativity property.