相关论文: Determination of elliptic curves by their adjoint …

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results…

We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb Q$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise…

Let A be a modular elliptic curve over a totally real field F, and let E/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic…

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is…

Let $E$ be an elliptic curve with good reduction at a fixed odd prime $p$ and $K$ an imaginary quadratic field where $p$ splits. We give a growth estimate for the Mordell-Weil rank of $E$ over finite extensions inside the…

We show how non-vanishing of p-adic L functions controls the dimensions of Selmer varieties associated to the complement of the origin in an elliptic curve with CM. As a corollary, one obtains a \pi_1-proof of the theorem of Siegel for such…

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see \cite{schoof_1985}). If $E/K$ is an…

An elliptic curve E defined over \Q is an algebraic variety which forms a finitely generated abelian group, and the structure theorem then implies that E = \Z^r + \Z_{tors} for some r \geq 0; this value r is called the rank of E. It is a…

Let $p$ be a prime and let $K$ be a finite extension of $\mathbb{Q}_p$. Let $E/K$ be an elliptic curve with additive reduction. In this paper, we study the topological group structure of the set of points of good reduction of $E(K)$. In…

Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and $p\nmid 2N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of $p$-adic zeta element for $E$ over $L$, encoding two…

In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding…

Let $q > 5$ be a prime and $K$ a quadratic number field. In this article we extend a previous result of Najman and the author and prove that if $E/K$ is an elliptic curve with potentially multiplicative reduction at all primes $\mathfrak q…

We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree p. For elliptic curves over l-adic fields, the…

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit…

For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. For a prime $p$ of good reduction for $E$, we write $\#E_p(\mathbb{F}_p) = p + 1 - a_p(E)$ for the number of $\mathbb{F}_p$-rational points of the…

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$…

Let $p>3$ be a prime and $E$ be a supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. Let $c$ be a prime with $c < 3p/16$ and $G$ be a subgroup of $E[c]$ of order $c$. The pair $(E,G)$ is called a supersingular elliptic curve with…

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…