Related papers: Level raising mod 2 and arbitrary 2-Selmer ranks

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an…

We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group…

For any quadratic extension $L/K$ of number fields, we prove that there are infinitely many elliptic curves $E$ over $K$ so that the abelian groups $E(K)$ and $E(L)$ both have rank $1$. In particular, there are infinitely many elliptic…

In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of…

We consider elliptic curves over global fields of positive characteristic with two distinct marked non-trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of…

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…

We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the…

In this paper, we determine the abelianization of the level d mapping class group for d=2 and odd d. We also extend the homomorphism of the Torelli group defined by Heap to a homomorphism of the level 2 mapping class group.

We investigate the rank growth of elliptic curves from $\mathbb{Q}$ to $S_4$ and $A_4$ quartic extensions $K/\mathbb{Q}$. In particular, we are interested in the quantity $\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q})$ for fixed $E$ and…

Let $k$ be a field of characteristic $q$, $\cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(\cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$…

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a…

We show that there are infinitely many elliptic curves $E/\mathbb{Q}$, up to isomorphism over $\overline{\mathbb{Q}}$, for which the finitely generated group $E(\mathbb{Q})$ has rank exactly $2$. Our elliptic curves are given by explicit…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ up to…

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related…

This paper introduces explicit Galois cohomological methods for determining the ranks of Bloch--Kato Selmer groups associated to the Tate twists of the 2-adic second \'etale cohomology of the Jacobian of a hyperelliptic curve with a…

In this paper, we prove that when elliptic curves over $\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains.

A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which…

Consider elliptic curves $ E=E_\sigma: y^2 = x (x+\sigma p) (x+\sigma q), $ where$ \sigma =\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(\phi)}(E/{\mathbf{Q})}$, and $\…

This paper generalises previous work of the author to the setting of overconvergent $p$-adic automorphic forms for a definite quaternion algebra over a totally real field. We prove results which are analogues of classical `level raising'…

The modular curves in the family $X_1(N)$ for natural numbers $N$ parametrize elliptic curves over the complex numbers with a distinguished point of order $N$. The purpose of this paper is to better understand how to calculate the degrees…