# Abstracts for Galois representations and pencils of Calabi-Yau motives

Alternatively have a look at the program.

## Critical values, congruences, Selmer groups

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Speaker:
Neil Dummigan
Date:
Sat, 2012-12-15 12:00 - 13:00
Location:
MPIM Lecture Hall

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.

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Speaker:
Jan Stienstra
Date:
Sat, 2012-12-15 15:00 - 16:00
Location:
MPIM Lecture Hall
The Cartier-Dieudonne module of the Artin-Mazur formal group (AMFG) equals the the unit root crystal in crystalline cohomology. A Laurent polynomial (LP) with reflexive Newton polytope defines Calabi-Yau hypersurfaces in toric varieties. There is a very concrete formula for a logarithm for a group law for the AMFG in terms of the constant terms in powers of the LP. The AMFG is a formal group over the ring of coefficients of the LP. This ring has a natural structure of a $\lambda$-ring.

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

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Speaker:
Piotr Achinger
Date:
Sat, 2012-12-15 17:00 - 18:00
Location:
MPIM Lecture Hall
A theorem of J. F. Thomsen states that Frobenius push-forwards of line bundles on smooth toric varieties are direct sums of line bundles. Using characterization of toric varieties in terms of their Cox rings, we show that this property in fact characterizes smooth projective toric varieties.

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

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Speaker:
Richard Crew
Date:
Sun, 2012-12-16 12:00 - 13:00
Location:
MPIM Lecture Hall
We will review basic properties of $F$-isocrystals on a smooth variety, with particular attention to the case of an open subset of the projective line. Topics: convergence conditions, Dwork's trick, the slope filtration, and the local monodromy theorem and its applications. If time permits we will discuss Lauder's work on the explicit computation of Frobenius matrices, and Katz's congruence formulas for the Frobenius of a curve.

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

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Speaker:
Francesco Baldassarri
Date:
Sun, 2012-12-16 15:00 - 16:00
Location:
MPIM Lecture Hall
I plan to give an overview of the literature on this topic, which originates from the variation of $p$-adic De Rham cohomology in a family of varieties over a field of positive characteristic $p$. So, I would like to start discussing the two polygons that vary over the moduli space: the geometric Hodge polygon and the Newton polygon of $p$-adic size of Frobenius eigenvalues. The evidence emerging from Dwork's theory, lead to the "Katz conjecture", stating that the Frobenius polygon stays above the Hodge one.

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

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Speaker:
Lucia di Vizio
Date:
Sun, 2012-12-16 17:00 - 18:00
I'll explain how one can construct a Galois theory for differential equations that takes into account the action of a difference operator,i.e., an endomorphisms, on the solutions.The theory attaches a group scheme to a differential equation, which encodes the algebraic difference relations among the solutions of the differential equation.This is typically the case of $p$-adic differential equation with a Frobenius structure. This is a joint work with C. Hardouin and M. Wibmer.

## Congruence sheaves via Hecke kernels.

Posted in
Speaker:
Anton Mellit
Date:
Mon, 2012-12-17 12:00 - 13:00
Location:
MPIM Lecture Hall

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

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Speaker:
Masha Vlasenko
Date:
Mon, 2012-12-17 17:00 - 18:00
Location:
MPIM Lecture Hall

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.

## From motivic L-functions to paramodular forms

Posted in
Speaker:
Anton Mellit
Date:
Wed, 2012-12-19 11:00 - 11:30
Location:
MPIM Lecture Hall

## Paramodular forms: L-functions, algebraic geometry, Kac-Moody algebras

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Speaker:
Valery Gritsenko
Date:
Wed, 2012-12-19 11:30 - 12:00
Location:
MPIM Lecture Hall
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