Related papers: Character sums for elliptic curve densities

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when…

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.

We study the character sums \[\phi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } \psi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left((x^{m}+a)(x^{n}+b)\right)\] where $\phi$ is the quadratic…

Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the existence of supersingular primes. This…

Let $p \in \{3, 5\}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $ E $ defined over $ \mathbb{Q} $, ordered by height, that are diophantine stable in $ L $.

The article presents a generalization of the classical Hardy-Littlewood conjecture concerning the density of prime tuples to the case of tuples consisting of almost-prime numbers (numbers with a specified quantity of prime divisors). The…

Let $E_1$ and $E_2$ be $\overline{\mathbb{Q}}$-nonisogenous, semistable elliptic curves over $\mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by…

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ be an odd prime number at which $E$ has good ordinary reduction. Let $Sel_{p^\infty}(\mathbb{Q}_\infty, E)$ denote the $p$-primary Selmer group of $E$ considered over the cyclotomic…

If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting $M_E(N):=…

Let E be an elliptic curve with additive reduction over the p-adic numbers, and let G be the group of p-adic points on E that have good reduction. This paper gives necessary and sufficient conditions for G to contain non-trivial p-torsion.

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over…

Let $E$ be an elliptic curve over $\Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let…

Extending Gross's result, we prove that a certain factorizaton of measures holds for all $p$ and any finite even Dirichlet character $\chi$ of any conductor, rather than only for split $p$ and $\chi$ with conductor a power of $p$. Using…

Let $E$ be an elliptic curve over the rationals that does not have complex multiplication. For each prime $\ell$, the action of the absolute Galois group on the $\ell$-torsion points of $E$ can be given in terms of a Galois representation…

We consider the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic…

Let $p\ge5$ be a prime number. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an imaginary quadratic field $K$ such that $p$ is inert in $K$ and that $E$ has good reduction at $p$. Let $K_\infty$ be the…

For E/k an elliptic curve with CM by O, we determine a formula for (a generalization of) the arithmetic local constant of [4] at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to…

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\geq 3$ and $a_p=0$. We generalise the definition of Kobayashi's plus/minus Selmer groups over $\mathbb{Q}(\mu_{p^\infty})$ to $p$-adic Lie…

In this paper, we study certain Kummer characters, which we call the elliptic Soul\'e characters, arising from Galois actions on the pro-$p$ fundamental groups of once-punctured elliptic curves with complex multiplication. In particular, we…