Arithmetic Geometry and Number Theory RTG Seminar: Dwork Crystals and Related Congruences

Seminar | April 1 | 3-5 p.m. | 891 Evans Hall | Note change in location

Masha Vlasenko, IMPAN Warsaw

Department of Mathematics

I will show a new, simple construction of crystals associated with toric hypersurfaces and exploit it to prove p-adic congruences for expansion coefficients of rational functions. This is joint work with Frits Beukers.

The exposition will be self-contained, but I shall explain that our ideas evolve from those of Bernard Dwork. Since he constructed an explicit Frobenius operator which does point counting for hypersurfaces, attempts to give a cohomological interpretaion of Dwork's work resulted in the Monsky–Washnitzer theory. Leaving out the $p$-adic counterpart, in 1990s Batyrev used solely the de Rham aspect of Dwork's theory to study mixed Hodge structure on the middle cohomology of toric hypersurfaces. Our construction basically adds the Frobenius structure back to this picture. As one of the applications, we will do a version of Katz's internal reconstruction of unit-root crystals via expansion coefficients of differential forms.

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