Chandrashekhar Khare

On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q)

We prove the existence in many cases of minimally ramied p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre’s conjectures in low levels and weights. For instance we prove that there are no irreducible (p;p) type group schemes over Z. We prove that a as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Delta-function. In the last part of the paper we present arguments that reduce Serre’s conjecture to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.

Functoriality and the inverse Galois problem

We prove that, for any primeand any even integer n, there are infinitely many exponents k for which PSp n (Fk) appears as a Galois group over Q. This generalizes a result of Wiese from 2006, which inspired this paper.

Mod

As a sequel to our proof of the analog of Serres conjecture for function fields in Part I of this work, we study in this paper the deformation rings of

\ell

n

representations of arithmetic fundamental groups II: A conjecture of A. J. de Jong

-dimensional mod

Finiteness of Selmer groups and deformation rings

\ell

On isomorphisms between deformation rings and Hecke rings

representations

The density of Ramified primes in semisimple p-adic Galois representations

\rho

Constructing semisimple p-adic Galois representations with prescribed properties

of the arithmetic fundamental group

Reciprocity Law for Compatible Systems of Abelian mod p Galois Representations

\pi_1(X)