78 research outputs found
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
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