We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight $2$ and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than $3$ the image is as large as possible. As a consequence, we prove that the groups $\text{PGL}(2,{𝔽}_{{\ell }^2})$ for every prime $(\ell \equiv 3,5(\text{mod}8),\ell \>3)$, and $\text{PGL}(2,{𝔽}_{{\ell }^5})$ for every prime $\ell \neg \equiv 0\pm 1\left(\text{mod}11\right);\ell \>3)$, are Galois groups...

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].