Related papers: Galois representations modulo $p$ that do not lift…

Let $p$ be an odd prime. Let $\rho: G_F \to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a Galois representation of a totally real field $F$. For a small partial weight one weight $(k,0)$, we prove that modularity of $\rho$ can be…

For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside 2.

Let $X$ be a variety (possibly non-complete or singular) over a finitely generated field $k$ of characteristic $0$. For a prime number $\ell$, let $\rho_\ell$ be the Galois representation on the first $\ell$-adic cohomology of $X$. We show…

We give an effective version of a result reported by Serre asserting that the images of the Galois representations attached to an abelian surface with $\End(A)= \mathbb{Z}$ are as large as possible for almost every prime. Our algorithm…

Let $p$ be a prime number and let $k\geq 2$ be an integer. In this article we study the semi-simple reductions modulo $p$ of two-dimensional irreducible crystalline $p$-adic Galois representations with Hodge-Tate weights $0$ and $k-1$ and…

Fix a prime $p > 2$. Let $\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida's $p$-ordinary Hecke algebra. Assume the…

Let $F$ be a number field, let $N\geq 3$ be an integer, and let $k$ be a finite field of characteristic $\ell$. We show that if $\rb:G_F\longrightarrow GL_N(k)$ is a continuous representation with image of $\rb$ containing $SL_N(k)$ then,…

This brief note only contains a modest contribution: we just fix some inaccuracies in the proof of the prime level weight 2 case of Serre's conjecture given in Khare's preprint "On Serre's modularity conjecture for 2-dimensional mod p…

Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G…

In this article we consider mod p modular Galois representations which are unramified at p such that the Frobenius element at p acts through a scalar matrix. The principal result states that the multiplicity of any such representation is…

For each of the groups PSL2(F25), PSL2(F32), PSL2(F49), PGL2(F25), and PGL2(F27), we display the first explicitly known polynomials over Q having that group as Galois group. Each polynomial is related to a Galois representation associated…

The Elementary Type Conjecture in Galois theory provides a concrete inductive description of the finitely generated maximal pro-$p$ Galois groups $G_F(p)$ of fields $F$ containing a root of unity of order $p$. We describe several variants…

We construct various explicit Herr complexes that compute the Galois cohomology of a $p$-adic representation of the absolute Galois group of a complete discrete valuation field of characteristic $0$ with a perfect residue field of…

Let $f(x)$ be a monic polynomial in $\dZ[x]$ with no rational roots but with roots in $\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more…

Let G be the absolute Galois group of a global field. Let r1 and r2 be two p-adic, finite dimensional representations of G. Then there exists a finite number of primes q such that if the characteristic polynomials of r1(Frob_q) and…

Let F be a number field with adele ring A_F, and \pi an isobaric, algebraic automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional…

Let K be a totally real Galois number field and let A be a set of elliptic curves over K. We give sufficient conditions for the existence of a finite computable set of rational primes P such that for p not in P and E in A, the…

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the…

Let F be the function field of a curve over totally imaginary number field. Let p be a prime. If F contains a primitive p th root of unity, then every element in the third Galois cohomology group of F with values in the group of p th roots…

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $\rho_f$ and $\mod{p^n}$ reductions $\rho_{f,n}$. Under suitable hypotheses on the…