5,327 research outputs found
Factor equivalence of Galois modules and regulator constants
We compare two approaches to the study of Galois module structures: on the
one hand factor equivalence, a technique that has been used by Fr\"ohlich and
others to investigate the Galois module structure of rings of integers of
number fields and of their unit groups, and on the other hand regulator
constants, a set of invariants attached to integral group representations by
Dokchitser and Dokchitser, and used by the author, among others, to study
Galois module structures. We show that the two approaches are in fact closely
related, and interpret results arising from these two approaches in terms of
each other. We also use this comparison to derive a factorisability result on
higher -groups of rings of integers, which is a direct analogue of a theorem
of de Smit on -units.Comment: Minor corrections and some more details added in proofs; 11 pages.
Final version to appear in Int. J. Number Theor
Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations
We report on the computation of torsion in certain homology theories of
congruence subgroups of SL(4,Z). Among these are the usual group cohomology,
the Tate-Farrell cohomology, and the homology of the sharbly complex. All of
these theories yield Hecke modules. We conjecture that the Hecke eigenclasses
in these theories have attached Galois representations. The interpretation of
our computations at the torsion primes 2,3,5 is explained. We provide evidence
for our conjecture in the 15 cases of odd torsion that we found in levels up to
31
The Shimura-Taniyama Conjecture and Conformal Field Theory
The Shimura-Taniyama conjecture states that the Mellin transform of the
Hasse-Weil L-function of any elliptic curve defined over the rational numbers
is a modular form. Recent work of Wiles, Taylor-Wiles and
Breuil-Conrad-Diamond-Taylor has provided a proof of this longstanding
conjecture. Elliptic curves provide the simplest framework for a class of
Calabi-Yau manifolds which have been conjectured to be exactly solvable. It is
shown that the Hasse-Weil modular form determined by the arithmetic structure
of the Fermat type elliptic curve is related in a natural way to a modular form
arising from the character of a conformal field theory derived from an affine
Kac-Moody algebra
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