Related papers: From potential modularity to modularity for integr…

In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre…

In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text{SL}_2(\mathbb Z)$. When the underlying modular…

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…

We show that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic p>0, exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds…

We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a…

We prove that every odd semisimple reducible (2-dimensional) mod l Galois representation arises from a cuspidal eigenform. In addition, we investigate the possible different types (level, weight, character) of such a modular form. When the…

Given a holomorphic newform $f$ of weight $k$ and with rational coefficients, a question of Mazur and van Straten asks if there is an associated Calabi-Yau variety $X$ over ${\mathbb Q}$ of dimension $k-1$ such that the $\ell$-adic Galois…

We classify 3-dimensional semi-stable representations of the Galois group of Q_p with coefficients and regular Hodge--Tate weights, by determining the isomorphism classes of admissible filtered (phi,N)-modules of Hodge type (0,r,s) with 0 <…

A lot of work has gone into computing images of Galois representations coming from elliptic curves. This article presents an algorithm to determine the image of the mod-$3$ Galois representation associated to a principally polarized abelian…

In their paper Livn\'e and Yui (math.AG/0304497) discuss several examples of non-rigid Calabi-Yau varieties which admit semi-stable K3-fibrations with 6 singular fibres over a base which is a rational modular curve. They also establish the…

We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the…

In this article, we prove the remaining open cases of the Fontaine-Mazur conjecture on two-dimensional regular Galois representations over $\Gal(\overline{\Q}/\Q)$ when $p=3$, hence concluding the conjecture in the regular case for all odd…

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such…

We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_K$ for a finite extension $K/\mathbb{Q}_p$. This is done by considering a moduli space of Breuil--Kisin modules, satisfying an additional Galois…

We review some recent results on the modularity of non-rigid Calabi-Yau threefolds.

We give a direct approach to recover some of the results of Wiles and Tayor on modularity of certain 2-dimensional p-adic representations of the absolute Galois group of Q.

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1…

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well…

We use the theory of trianguline $(\varphi,\Gamma)$-modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at $p$, including those with characteristic $p$ coefficients.…

We prove congruences of Hecke eigenvalues between cuspidal Hilbert newforms $f_{79}$ and $h_{79}$ over $F=\mathbb Q(\sqrt{5})$, of weights (2,2) and (2,4) respectively, level of norm 79. In the main example, the modulus is a divisor of 5 in…