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)

Centre de conférences Marilyn et James Simons (IHES)

Centre de conférences Marilyn et James Simons


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

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