相关论文: On several families of elliptic curves with arbitr…

In this paper, we are going to compute the average size of 2-Selmer groups of two families of hyperelliptic curves with marked points over function fields. The result will be obtained by a geometric method which could be considered as a…

We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from…

This paper studies fine Selmer groups of elliptic curves in abelian $p$-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic…

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E in…

In this paper the family of elliptic curves over \Q given by the equation E_{p}: Y^2=(X-p)^3+X^3+(X+p)^3 where p is a prime number, is studied. It is shown that the maximal rank of the elliptic curves is at most 3 and some conditions under…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…

This is the first in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we describe are practical in the case n=3 for…

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K)_{tors}$, where $K$ is the compositum of all $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve…

Let $p$ be an odd prime number. In this article, we study the variation of Iwasawa invariants among $p$-congruent elliptic curves over certain $p$-adic Lie extensions. We investigate both the classical Selmer group as well as the fine…

Let $M$ and $N$ be positive integers for which the modular curve $X_1(M,MN)$ has genus $0$, and let $p$ be a prime divisor of $MN$. This article gives asymptotic lower bounds for the average size of the $p$-Selmer group of elliptic curves…

Let $\mathcal{C}$ be a hyperelliptic curve $y^2 = p(x)$ defined over a number field $K$ with $p(x)$ integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the $2$-Selmer group of the Jacobian of…

In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals…

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…

We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.

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…

It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is…

For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*} E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ as a function of $t$ and assuming some standard…

Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $\mu_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin,…

We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group…

Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $\lambda$-invariants of these signed Selmer…