177 research outputs found
Iwasawa invariants of Galois deformations
We study the behavior of Iwasawa invariants among ordinary deformations of a
fixed residual Galois representation taking values in a reductive algebraic
group G. In particular, under the assumption that these Selmer groups are
cotorsion modules over the Iwasawa algebra, we show that the vanishing of the
mu invariant is independent of the deformation, while the lambda invariant
depends only on the ramification. This generalizes work of Greenberg-Vatsal and
Emerton-Pollack and the author in the case G = GL(2)
Local torsion on elliptic curves and the deformation theory of Galois representations
We prove that, on average, elliptic curves over Q have finitely many primes p
for which they possess a p-adic point of order p. We include a discussion of
applications to companion forms and the deformation theory of Galois
representations
Kida's formula and congruences
We prove a formula (analogous to that of Kida in classical Iwasawa theory and
generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic
and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian
p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side
our methods, which make use of congruences between modular forms, yield a
Kida-type formula for a very general class of ordinary Galois representations.
We are further able to deduce a Kida-type formula for elliptic curves at
supersingular primes
Mazur-Tate elements of non-ordinary modular forms
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of
cuspidal eigenforms, generalizing known results in weight 2. Our first theorem
deals with forms of "medium" weight, and our second deals with forms of small
slope . We give examples illustrating the strange behavior which can occur in
the high weight, high slope case
Variation of Iwasawa invariants in Hida families
Let r : G_Q -> GL_2(Fpbar) be a p-ordinary and p-distinguished irreducible
residual modular Galois representation. We show that the vanishing of the
algebraic or analytic Iwasawa mu-invariant of a single modular form lifting r
implies the vanishing of the corresponding mu-invariant for all such forms.
Assuming that the mu-invariant vanishes, we also give explicit formulas for the
difference in the algebraic or analytic lambda-invariants of modular forms
lifting r. In particular, our formula shows that the lambda-invariant is
constant on branches of the Hida family of r. We further show that our formulas
are identical for the algebraic and analytic invariants, so that the truth of
the main conjecture of Iwasawa theory for one form in the Hida family of r
implies it for the entire Hida family
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