89 research outputs found
Computing generators of free modules over orders in group algebras
Let E be a number field and G be a finite group. Let A be any O_E-order of
full rank in the group algebra E[G] and X be a (left) A-lattice. We give a
necessary and sufficient condition for X to be free of given rank d over A. In
the case that the Wedderburn decomposition of E[G] is explicitly computable and
each component is in fact a matrix ring over a field, this leads to an
algorithm that either gives an A-basis for X or determines that no such basis
exists.
Let L/K be a finite Galois extension of number fields with Galois group G
such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to
certain Galois modules that arise naturally in this situation. For example, one
can take X to be O_L, the ring of algebraic integers of L, and A to be the
associated order A of O_L in E[G]. The application of the algorithm to this
special situation is implemented in Magma under certain extra hypotheses when
K=E=Q.Comment: 17 pages, latex, minor revision
Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions
Let A be an abelian variety over a number field k and F a finite cyclic
extension of k of p-power degree for an odd prime p. Under certain technical
hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number
conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru-
ences involving values of derivatives of the Hasse-Weil L-functions of twists
of A, normalised by completely explicit twisted regulators. This
reinterpretation makes the eTNC amenable to numerical verification and
furthermore leads to explicit predictions which refine well-known conjectures
of Mazur and Tate
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