Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

# Duality of Drinfeld Modules and P-adic Properties of Drinfeld Modular Forms

## by Shin Hattori (Tokyo City University)

Europe/Paris
Centre de conférences Marilyn et James Simons (IHES)

### Centre de conférences Marilyn et James Simons

#### IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description

Let p be a rational prime, $q>1$ a p-power and P a non-constant irreducible polynomial in $F_q[t]$. The notion of Drinfeld modular form is an analogue over $F_q(t)$ of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite $\nu$-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

Organized by

Ahmed Abbes

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