17 research outputs found
Applications of Commutator-Type Operators to -Groups
For a p-group G admitting an automorphism of order with exactly
fixed points such that has exactly fixed points,
we prove that G has a fully-invariant subgroup of m-bounded nilpotency class
with -bounded index in G. We also establish its analogue for Lie
p-rings. The proofs make use of the theory of commutator-type operators.Comment: 11 page
On certain finiteness questions in the arithmetic of modular forms
We investigate certain finiteness questions that arise naturally when
studying approximations modulo prime powers of p-adic Galois representations
coming from modular forms. We link these finiteness statements with a question
by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms.
Specifically, we conjecture that for fixed N, m, and prime p with p not
dividing N, there is only a finite number of reductions modulo p^m of
normalized eigenforms on \Gamma_1(N). We consider various variants of our basic
finiteness conjecture, prove a weak version of it, and give some numerical
evidence.Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3:
restructered parts of the article; v4: minor corrections and change
Topics on modular Galois representations modulo prime powers
This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences