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

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 \operatorname{GU}(2,1, \mathbb{Q}(\sqrt{-3})) . The formula is based on counting points over finite fields on curves of genu...

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Bibliographic Details
Published inCommentarii mathematici Helvetici Vol. 97; no. 2; pp. 305 - 381
Main Authors Bergström, Jonas, van der Geer, Gerard
Format Journal Article
LanguageEnglish
Published 01.01.2022
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Summary: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 \operatorname{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.
ISSN:0010-2571
1420-8946
1420-8946
DOI:10.4171/cmh/532