Related papers: Artin's Conjecture and Elliptic Curves

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture…

We give a classification of all possible $2$-adic images of Galois representations associated to elliptic curves over $\mathbb{Q}$. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop…

In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new…

Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a…

We study the weight part of Serre's conjecture for generic $n$-dimensional mod $p$ Galois representations. We first generalize Herzig's conjecture to the case where the field is ramified at $p$ and prove the weight elimination direction of…

Our main result in this article is a proof (under mild technical assumptions) of an analogue for $p$-adic Galois representations attached to a newform $f$ of even weight $k\geq4$ of Kolyvagin's conjecture on the $p$-indivisibility of…

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (non-monomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to S5 and A5…

We prove some new cases of weight part of Serre's conjectures for mod $p$ Galois representations associated to automorphic representations on unitary groups $U(d)$. The approach is a generalization of the work of Gee-Liu-Savitt, namely, we…

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers…

In a previous paper, the potential automorphy of certain Galois representations to GL_n for n even was established, following work of Harris, Shepherd-Barron and Taylor and using the lifting theorems of Clozel, Harris and Taylor. In this…

In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive…

In this note we show that if an abelian variety possesses a Galois embedding into some projective space, then it must be isogenous to the self product of an elliptic curve. We prove moreover that the self product of an elliptic curve always…

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular…

We give a new local proof of Breuil-M\'ezard conjecture for two dimensional representations of the absolute Galois group of $\mathbb{Q}_p$, when $p\ge 5$ and the representation has scalar endomorphisms.

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a…

The modularity of an elliptic curve $E/\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve…

In 1979 Goldfeld conjectured: 50\% of the quadratic twists of an elliptic curve defined over the rationals have analytic rank zero. In this expository article we present a few recent developments towards the conjecture, especially its first…

We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to $p$-adic $L$-functions under a conjectural formula for the…

We describe the set of points of the trianguline variety over a given local Galois representation. Global analogues describing companion points in eigenvariety by [Bre14] and [HN17], can be thought of as a rational analogue to the weight…

We prove the Breuil-M\'ezard conjecture for 2-dimensional potentially Barsotti-Tate representations of the absolute Galois group G_K, K a finite extension of Q_p, for any p>2 (up to the question of determining precise values for the…