# Arithmetic of Differential Equations

September 2 – 8, 2018
Łukęcin, Poland

## Part of the program Varieties: Arithmetic and Transformations at IMPAN

#### Lecture courses

• Kiran S. Kedlaya (UC San Diego)

Frobenius structures

A Frobenius structure on a differential equation is an auxiliary structure that occurs when the equation is of "geometric origin" (that is, it occurs as a Picard-Fuchs equation). We'll start with some examples of Dwork, where the Frobenius structure is related to counting points on certain varieties over finite fields. We'll then formulate some general results along these lines, and show how these can be used productively for machine computation of zeta functions.

References:

• Kiran S. Kedlaya $p$-adic Differential Equations
• Alain S. Robert A Course in $p$-adic Analysis

• Christian Liedtke (Technische Universität München)

Models of curves, abelian varieties, and K3 surfaces

Let $R$ be a local and complete DVR with field of fractions $K$ and let $X$ be a smooth and proper variety over $K$. Now, one can ask whether $X$ admits a "reasonable" model over $R$, that is, whether there exists a regular scheme $\mathcal{X}$ that is proper and flat over $R$ with generic fiber $X$. In particular, one can ask whether one can find even a model that is smooth over $R$, that is, whether $X$ has good reduction. In general, the answer to the latter question is negative: a necessary condition is that the monodromy on all cohomology groups of $X$ is trivial. In same cases, these conditions are even sufficient. In my lectures, I will discuss these notions with a special focus on curves, abelian varieties, and K3 surfaces. If time permits, I will also discuss how non-trivial monodromy give rise to ($p$-adic) differential equations.

• Fernando Rodriguez Villegas (ICTP Trieste)

Rigid local systems: from Goursat to Katz

In concrete terms our goal in these lectures is to discuss in elementary terms when is there a unique solution up to simultaneous conjugation of the equation $A_1 \cdot \ldots \cdot A_k = I_n$, where $A_1, \ldots, A_k$ belong to fixed conjugacy classes in ${\rm GL}_n(\mathbf{C})$ and $I_n$ is the identity matrix. As we will see, this question is tied with the classical problem of describing order $n$ differential equations with no "accessory parameters". Roughly speaking this means that the local behavior of solutions around the singularities completely determines the equation; the $A_i$ corresponds to analytic continuation of solutions to the equation around a singularity. Following Katz, we call this situation rigid. Starting from the remarkable paper of Goursat of 1886 we plan to arrive to the equally remarkable result of Katz on how all rigid local systems can be constructed via middle convolution.

Organizers: Piotr Achinger (IMPAN), Adrian Langer (University of Warsaw), Masha Vlasenko (IMPAN)

#### Venue

The school will take place in a University of Warsaw pension in Lukecin, on Western part of Polish Baltic sea shore, see the map. The accommodation (full board, double room) will cost about 200 PLN a day (1 Euro is approx. 4.3 PLN). Due to space constraints, most rooms will be double or triple.

#### Schedule

All inquiries should be directed to Piotr Achinger, pachinger [at] impan [dot] pl.

Sponsors: Simons Foundation, Banach Center, Polish Academy of Sciences, University of Warsaw

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