A Simple Trace Formula for Algebraic Modular Forms

Abstract

We derive an elementary formula for the trace of a Hecke operator acting on a space of algebraic modular forms, as a sum of character values. We describe explicit computations in the case of the unitary group that make possible the determination of the eigenvalues of a certain Hecke operator. This produces numerical evidence for a analogue of Harder’s conjecture, on congruences between Hecke eigenvalues modulo divisors of critical L-values.

2000 AMS Subject Classification:

• 11F33
• 11F55

Keywords:

• algebraic modular forms
• Harder’s conjecture
• trace formula

Acknowledgments

I thank G. Harder, D. Loeffler, and J. Tilouine, whose helpful contributions have been noted at various points, and H. Katsurada for directing me to his preprint [Katsurada 10].

Notes

¹For this fact, see the L-functions and modular forms database, available at http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3/9/1/ .