Cubic structures, equivariant Euler characteristics and lattices of modular forms

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 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 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.Comment: 40pp, Final version, to appear in the Annals of Mathematic

Exterior powers in Iwasawa theory

The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller Iwasawa modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of p-adic L-functions under the relevant CM main conjectures.Comment: 41 pages, to appear in J. Eur. Math. So