相关论文: Counterexamples to the local-global divisibility o…

Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers, $\mathbb{Q}$, with good reduction at an odd prime $p$ and equivalent mod $p$ Galois representation, we compare the $p$-Selmer rank, global and local root…

Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\operatorname{Gal}(K(E[p])\cap K(E[q])/K)$. The size of…

We give a complete answer to the local-global divisibility problem for algebraic tori. In particular, we prove that given an odd prime $p$, if $T$ is an algebraic torus of dimension $r< p-1$ defined over a number field $k$, then the…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers…

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to…

Let $K$ be a number field. For which primes $p$ does there exist an elliptic curve $E / K$ admitting a $K$-rational $p$-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a…

We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all…

For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in $\mathrm{GL}_2(\hat{\mathbb{Z}})$. The obstructions to the…

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion…

Let $k$ be a global field and $\mathbb{A}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to $\overline{\mathbb{Q}}_{\ell}$. Let $\Pi_1$ and $\Pi_2$ be cuspidal automorphic…

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…

For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the action on the Tate modules of…

We generalise Coleman's construction of Hecke operators to define an action of GL_2(Q_l) on the space of finite slope overconvergent p-adic modular forms (l not equal p). In this way we associate to any C_p-valued point on the tame level N…

In this article, we introduce a systematic and uniform construction of non-singular plane curves of odd degrees $n \geq 5$ which violate the local-global principle. Our construction works unconditionally for $n$ divisible by $p^2$ for some…

Let $\mathcal{E}$ be an elliptic curve defined over a number field $K$. Let $m$ be a positive integer. We denote by ${\mathcal{E}}[m]$ the $m$-torsion subgroup of $\mathcal{E}$ and by $K_m:=K({\mathcal{E}}[m])$ the number field obtained by…

Using the rank of the Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve $E$ over $\mathbb{Q}$, we give a lower bound of the class number of the number field $\mathbb{Q}(E[p^n])$ generated by $p^n$-division points of $E$ when the curve…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\rho_E\colon {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}(2,\widehat{ \mathbb{Z} })$ be the adelic representation associated to the natural action of Galois on the…

We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic…

Let p be a prime and K be a number field. Let rho_{E,p}:G_K \longrightarrow Aut(T_p E)\cong GL_2(Z_p) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the…

We extend existing results characterizing Weil-Ch\^atelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of Gonz\'alez-Avil\'es and Tan, we characterize when…