24 research outputs found
Ihara's lemma for imaginary quadratic fields
An analogue over imaginary quadratic fields of a result in algebraic number
theory known as Ihara's lemma is established. More precisely, we show that for
a prime ideal P of the ring of integers of an imaginary quadratic field F, the
kernel of the sum of the two standard P-degeneracy maps between the cuspidal
sheaf cohomology H^1_!(X_0, M_0)^2 --> H^1_!(X_1, M_1) is Eisenstein. Here X_0
and X_1 are analogues over F of the modular curves X_0(N) and X_0(Np),
respectively. To prove our theorem we use the method of modular symbols and the
congruence subgroup property for the group SL(2) which is due to Serre.Comment: 10 page
On deformation rings of residually reducible Galois representations and R=T theorems
We study the crystalline universal deformation ring R (and its ideal of
reducibility I) of a mod p Galois representation rho_0 of dimension n whose
semisimplification is the direct sum of two absolutely irreducible mutually
non-isomorphic constituents rho_1 and rho_2. Under some assumptions on Selmer
groups associated with rho_1 and rho_2 we show that R/I is cyclic and often
finite. Using ideas and results of (but somewhat different assumptions from)
Bellaiche and Chenevier we prove that I is principal for essentially self-dual
representations and deduce statements about the structure of R. Using a new
commutative algebra criterion we show that given enough information on the
Hecke side one gets an R=T-theorem. We then apply the technique to modularity
problems for 2-dimensional representations over an imaginary quadratic field
and a 4-dimensional representation over the rationals.Comment: 32 page