Mathematics Colloquium
Date: Wednesday, 14 February 2024
Time: 2.00 - 3.00 PM
Venue: Seminar Hall
Delta geometry and the characteristic polynomial of the Frobenius
Arnab Saha
IIT Gandhinagar.
14-02-24

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Abstract

By a previous construction by Borger and the speaker, one attaches a canonical filtered isocrystal $\mathbf{H}_\delta(G)$ associated to the arithmetic jet spaces of a smooth commutative group scheme $G$. In the case when A is an elliptic curve, $\mathbf{H}_\delta(A)$ is isomorphic to the first crystalline cohomology $\mathbf{H}_{\mathrm{cris}}^1(A)$ when $A$ does not have a canonical lift. The above comparison theorem allows us to give a character theoretic interpretation of the crystalline cohomology.

In this talk, using the characteristic polynomial of the Frobenius, we will show that whenever $A$ is an abelian scheme, the module of primitive characters of $A$ is isomorphic to $H^0(A,\Omega_A)$. We will also comment on the relation of $\mathbf{H}_\delta(A)$ with the first crystalline cohomology $\mathbf{H}_{\mathrm{cris}}^1(A)$. This is a joint work with L. Gurney and S. Pandit.