Adrian Iovita (Concordia)

p-Adic families of half-integral weight modular forms

Joint work in progress with F. Andreatta and M.-H. Nicole.

Half integral weight modular forms have been defined by Jacobi, Shimura and others as holomorphic functions on the complex upper half plane satisfying certain symmetries with respect to the action of certain congruence subgroups on the upper half plane. The easiest example is the function $\theta$ of Riemann $\theta(\tau):=\sum_{n\in \Z} e^{\pi i n^2\tau}$, which is a modular form of weight $1/2$ for the modular group $\Gamma_{\theta}\subset \SL_2(\Z)$.

In this talk we will explain, following work of Deligne, Candelori, Welters, de Jong and others how half-integral weight modular forms can be seen geometrically as sections of modular line bundles on modular curves, and how these modular sheaves can be interpolated $p$-adically when restricted to strict neighborhoods of the ordinary locus in the modular curve.

In fact, all this works without much change for Hilbert and Siegel modular forms.