Ravi Ramakrishna

Finiteness of Selmer groups and deformation rings

We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were proven to exist by the seond named author.

Constructing semisimple p-adic Galois representations with prescribed properties

Consider a characteristic p representation of the absolute Galois group of the rational numbers. In this paper we show how to deform this representation to the p-adics while guaranteeing that the characteristic polynomials of Frobenius at a density one set of primes are algebraic and pure of specified weight. The resulting representation is ramified at an infinite (density zero) set of primes. As a consequence of the technique of proof we show that one can compatibly lift a mod pq representation. Again, the resulting representation is ramified at infinitely many primes.

Deformations of certain reducible Galois representations, II

We examine the deformation theory of two dimensional mod

Infinitely ramified Galois representations.

p

Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur

reducible indecomposable Galois representations, removing or relaxing many of the hypotheses of part I. We are able to prescribe local conditions on our deformations by allowing ramification at a set of primes congruent to 1 mod

Lifting Galois representations

p

Some supercongruences occurring in truncated hypergeometric series

, not 1 mod

Hypergeometric Functions over Finite Fields

p^2

Transcendental

, and satisfying some other splitting conditions. The new hypotheses admit irreducible ordinary characteristic zero lifts of many mod

\ell

p