Cubic structures, equivariant Euler characteristics and lattices of modular forms

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 flat schemes over ‫ޚ‬ 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 coefficients 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.

Equivariant Euler characteristics and sheaf resolvents

“…The fibres are of constant dimension and we denote this fibral dimension by d. Let π : X → Y be a G-cover which is generically a G-torsor on Y with X is regular. Note that TOME 62 (2012), FASCICULE 6 it follows from the assumptions that π : X → Y is flat (see Remark 3.1.a in [7]). When R = Z we suppose that the ramification locus of this cover is supported on a finite set of rational primes Σ which is disjoint with the set of prime divisors of the order of G.…”

“…-1. The factor 2 in our formulas derives from the need to consider the determinant of cohomology of bundles of even rank in [7]; note that this factor can be removed in certain situations. (See Section 3 in [7] for further details.)…”

“…Let p ≡ 1 mod 24 be a prime number and let X 1 (p) be the model over Spec(Z) of the modular curve associated to the congruence subgroup Γ 1 (p) as described in Section 4 in [7]. The group Γ = (Z/pZ) * /{±1} acts faithfully on X 1 (p).…”

“…We shall see presently that when the cover π : X → Y is not etale then our sheaves do not necessarily have NIB. A new approach to such questions has been recently developed in [7]. Our results depend strongly on the use of this paper, and illustrate how their new techniques permit the efficient calculation of such Euler characteristics.…”

“…In fact we shall be interested in twice this class and so, with the notation of [7], we set δ(F) = det(Rh(π * (F)) ⊗2 .…”