Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo
# Duality of Drinfeld Modules and P-adic Properties of Drinfeld Modular Forms

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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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