Related papers: Experimental Evidence on a Refined Conjecture of t…

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which…

We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of the generic fibre. As a byproduct, we obtain a new proof of a theorem of Geisser…

Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having analytic rank zero, i.e., such that the $L$-function $L_E(s)$ of $E$ does not vanish at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of the same…

Let E be an elliptic curve over Q, and let F be a finite abelian extension of Q. Using Beilinson's theorem on a suitable modular curve, we prove a weak version of Zagier's conjecture for L(E/F,2), where E/F is the base extension of E to F.

The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained…

We consider surfaces with a double elliptic fibration, with two sections. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a…

Let $p$ be a prime. Tate and Voloch proved that a point of finite order in the algebraic torus cannot be $p$-adically too close to a fixed subvariety without lying on it. The current work is motivated by the analogy between torsion points…

This is an expository article, based on a lecture course given at CRM Barcelona in December 2009. The purpose of these notes is to prove, in a reasonably self-contained way, that finiteness of the Tate-Shafarevich group implies the parity…

We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the…

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…

For a given elliptic curve $E$ defined over the rationals, we study the density of primes $p$ satisfying $\mathrm{gcd}(\#E(\mathbb{F}_p),p-1)=1$ and give a conjectural value for this density with strong heuristic evidence for most elliptic…

We formulate a refined version of the Birch and Swinnerton-Dyer conjecture for abelian varieties over global function fields. This refinement incorporates both families of congruences between the leading terms of Artin-Hasse-Weil $L$-series…

In this paper we generalize the Deuring theorem on a reduction of elliptic curve with complex multiplication. More precisely, for an Abelian variety $A$, arising after reduction of an Abelian variety with complex multiplication by a CM…

Let $k$ be a field and $X$ a smooth projective variety over $k$. When $k$ is a number field, the Beilinson-Bloch conjecture relates the ranks of the Chow groups of $X$ to the order of vanishing of certain $L$-functions. We consider the same…

Nagao's conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized for smooth irreducible projective surfaces by M. Hindry and A. Pacheco.…

This paper focuses on the proof of Serge Lang's Heights Conjecture in a form that is completely effective. As a complementary result the author provides a new proof of Mazur-Merel theorem about a bound for the torsion of elliptic curves in…

Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be…

We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with…

The theory of elliptic pairs, as investigated in a paper by Castravet, Laface, Tevelev, and Ugaglia, provides useful conditions to determine polyhedrality of the pseudo-effective cone, which give rise to interesting arithmetic questions…

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our…