* Thursday, April 19th, 14:00

Differential equations in characteristic 0 and p

Marius van der Put (Groningen University)

Linear differential equations over the field of complex numbers are largely governed by differential Galois theory, monodromy and Stokes matrices. The good notion of linear differential equation over a field of positive characteristic is called “iterative differential module” or ”stratified bundle”. Differential Galois theory exists in this case. The Tannaka group of the stratified bundles on a variety is the algebraic fundamental group in characteristic p. The main problem is producing stratified bundles and the inverse problem for differential Galois group. We will compare characteristic 0 and p. Further methods for the construction of stratified bundles on curves are presented.

* Thursday, April 19th, 11:00

Cohomology of arithmetic groups, periods and special values II

Günter Harder (MPIM)

* Wednesday, April 18th, 14:30 (Venue: MPIM)

First order differential equations

Marius van der Put (Groningen University)

We consider a first order differential equation of the form f(y′;y)=0 with f∈K[S;T] and K a differential field either complex or of positive characteristic. We investigate several properties of f, namely the ‘Painlevé property’ (PP), solvability and stratification. A modern proof of the classication of first order equations with PP is presented for all characteristics. A version of the Grothendieck-Katz conjecture for first order equations is proposed and proven for special cases. Finally the relation with Malgrange’s Galois groupoids and model theory is discussed.

* Tuesday, April 17th, 14:30

Modular forms in Pari/GP

Henri Cohen (Université de Bordeaux)

The aim of this talk is to describe the new modular forms package available in Pari/GP which has a number of features not available in other packages, in particular expansion at all cusps and computation of arbitrary Petersson products.

* Tuesday, April 17th, 11:00

Cohomology of arithmetic groups, periods and special values I

Günter Harder (MPIM)

* Monday, April 16th, 16:30

Serre-Tate theory for Calaby-Yau varieties

Piotr Achinger (Instytut Matematyczny PAN)

Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero. Variants of this theory have been obtained for ordinary K3 surfaces by Nyygard and Ogus, and for ordinary curves by Mochizuki. I will report on a joint project in progress with Maciej Zdanowicz (EPFL), in which we construct canonical liftings of ordinary Calabi–Yau varieties modulo p^2. Consequently, we obtain a Frobenius

lifting on the moduli space of such varieties, and a weak form of canonical coordinates. The construction uses Frobenius splittings and a ‘relative’ version of Witt vectors of length two, and seems closely related to results of Deligne-Illusie on the decomposition of the de Rham complex.