Picard modular forms and the cohomology of local systems on a Picard modular surface

TL;DR

This paper proposes a conjectural formula linking the cohomology of local systems on Picard modular surfaces to point counting over finite fields, enabling calculations of Hecke operator traces and revealing new structural results.

Contribution

It formulates a conjectural Eichler-Shimura type formula for Picard modular surfaces and proves new results on Frobenius actions, modular form spaces, and Euler characteristics.

Findings

Derived formulas for Frobenius characteristic polynomials

Calculated dimensions of Picard modular form spaces

Provided evidence supporting the conjectural cohomology formula

Abstract

We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems.