Some Properties of Elliptic Modular Forms at the Supercuspidal Primes

Abstract

The Brauer class of the endomorphism algebra attached to a primitive non-CM cusp form of weight two or more is a two torsion element in the Brauer group of some number field. We give a formula for the ramification of that algebra locally for all places lying above \textbf{all} supercuspidal primes. For $p=2$, we also treat the interesting case where the image of the local Weil-Deligne representation attached to that modular form is an exceptional group. We have completed the programme initiated by Eknath Ghate to give a satisfactory answer to a question asked by Ken Ribet.

In a different project, we studied the variance of the local epsilon factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. As an application, we detect the nature of the supercuspidal representation from that information, similar results are proved by Pacetti for modular forms with trivial nebentypus. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $p \ \Vert \ N$ and $p$ odd, $N$ denote the level of modular forms) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-adic Gamma function.