相关论文: Serre's Uniformity Conjecture for Elliptic Curves …

We exhibit two non-isogenous rational elliptic curves with $17$-torsion subgroups isomorphic as Galois modules.

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…

Let $\ell$ and $p \geq 3$ be different primes. Let $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ be elliptic curves with isomorphic $p$-torsion. Assume that $E$ has potentially multiplicative reduction. We classify when all…

In this paper, we study tame Galois coverings of semistable models that arise from torsion points on elliptic curves. These coverings induce Galois morphisms of intersection graphs and we express the decomposition groups of the edges in…

We unconditionally determine $I_\Q(d)$, the set of possible prime degrees of cyclic $K$-isogneies of elliptic curves with $\Q$-rational $j$-invariants and without complex multiplication over number fields $K$ of degree $\leq d$, for $d\leq…

The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We…

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give…

We formulate and prove the weight part of Serre's conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption in \cite{arXiv:1512.06380},…

Let $E$ be an elliptic curve described by either an Edwards model or a twisted Edwards model over $\mathbb{F}_p$, namely, $E$ is defined by one of the following equations $x^2+y^2=a^2(1+x^2y^2),\, a^5-a\not\equiv 0$ mod $p$, or,…

The main goal of this paper is to determine for which prime numbers $r\geq 3$ can an elliptic curve~$E$ defined over $\mathbb Q$ have an $r$-isogeny over $\mathbb Q(\zeta_r)$. We study this question under various assumptions on the…

We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension…

Let G be a finite group and V a finite-dimensional rational G-representation. We ask whether there exists a finite Galois extension L/K of number fields with Galois group G, an elliptic curve E/K, and a G-submodule of E(L) tensor Q…

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the…

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…

An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $\phi:E\rightarrow E^\prime$ comes equipped with an invariant $\alpha_{\phi/K}$ that measures the valuation of the leading term of the formal group homomorphism…

Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$. We compute certain twists of the classical modular curves $X(8)$. Searching for rational points on these twists enables us to find non-trivial pairs of $8$-congruent…

Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular graph whose nodes are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of…

Let $E/\mathbf{Q}$ be an elliptic curve and $p\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is…

Modular Galois representations into GL_2(F_p) with cyclotomic determinant arise from elliptic curves for p = 2,3,5. We show (by constructing explicit examples) that such elliptic curves cannot be chosen to have conductor as small as…

In this paper, we establish the modularity of every elliptic curve $E/F$, where $F$ runs over infinitely many imaginary quadratic fields, including $\mathbb{Q}(\sqrt{-d})$ for $d=1,2,3,5$. More precisely, let $F$ be imaginary quadratic and…