Nous développons une nouvelle stratégie pour comprendre la nature des obstructions aux déformations d’une représentation galoisienne globale $\overline{\rho }$ réductible, impaire de dimension 2. Ces obstructions s’interprètent en termes de groupe de Šafarevič. D’après [BöMé], elles sont reliées à des conjecture arithmétiques classiques (Conjecture de Vandiver, conjecture de Greenberg). Dans cet article, nous introduisons un autre groupe de Šafarevič associé au corps $L$ fixe par $ker\overline{\rho }$. Nous comparons les deux groupes...

We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes of modules for a profinite group $G$ over a complete local Noetherian ring $A$ of positive residue characteristic.

We give precise estimates for the number of classical weight one specializations of a non-CM family of ordinary cuspidal eigenforms. We also provide examples to show how uniqueness fails with respect to membership of weight one forms in families.

We give a simple proof that critical values of any Artin $L$-function attached to a representation $\rho$ with character ${\chi }_{\rho }$ are stable under twisting by a totally even character $\chi$, up to the $dim\rho$-th power of the Gauss sum related to $\chi$ and an element in the field generated by the values of ${\chi }_{\rho }$ and $\chi$ over $\mathbb{Q}$. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.

Given an odd prime  $p$  and a representation $\varrho$  of the absolute Galois group of a number field $k$ onto ${\mathrm{PGL}}_2\left({𝔽}_p\right)$ with cyclotomic determinant, the moduli space of elliptic curves defined over $k$ with $p$-torsion giving rise to $\varrho$ consists of two twists of the modular curve $X\left(p\right)$. We make here explicit the only genus-zero cases $p=3$ and $p=5$, which are also the only symmetric cases: ${\mathrm{PGL}}_2\left({𝔽}_p\right)\simeq {𝒮}_n$ for $n=4$ or $n=5$, respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which...

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.

This paper focuses on the Diophantine equation $xⁿ+p^{\alpha }yⁿ=Mz³$, with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if $n\ge ℱ(M,p^{\alpha })$, where $ℱ(M,p^{\alpha })$ is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].

In a recent paper, Freitas and Siksek proved an asymptotic version of Fermat’s Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A{x}^p+B{y}^p+C{z}^p=0$, where A, B, C are odd integers belonging to a totally real field.

We explore the question of how big the image of a Galois representation attached to a $\Lambda$-adic modular form with no complex multiplication is and show that for a “generic” set of $\Lambda$-adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.

Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\mathrm{Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type $\mathrm{U}\left(3\right)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb{Q}]$. As important steps, and in any rank, we prove that any first order...

Let $f$ be a primitive cusp form of weight at least 2, and let ${\rho }_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of ${\rho }_f$ to a decomposition group at $p$ is “upper triangular”. If in addition $f$ has CM, then this representation is even “diagonal”. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members of a non-CM...