2007.83: Cubic structures, equivariant Euler characteristics and lattices of modular form

2007.83: Ted Chinburg, Georgios Pappas and Martin J. Taylor (2007) Cubic structures, equivariant Euler characteristics and lattices of modular form. Annals of Mathematics. ISSN 0003-486x

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Abstract

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective at schemes over $\Z$ with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coeficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves.

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