On Silverman's conjecture for a family of elliptic curves

TL;DR

This paper investigates the ranks of certain quadratic twists of elliptic curves, providing conditional results on their parity and explicit rank computations for specific cases, based on Silverman's conjecture and the parity conjecture.

Contribution

It establishes new conditional results on the parity of ranks for a family of elliptic curves twisted by primes, under specific biquartic sum conditions and congruences.

Findings

Infinitely many primes p yield elliptic curves with odd rank under certain conditions.

Infinitely many primes p yield elliptic curves with even rank under certain conditions.

Explicit rank computations for specific n and p values are provided.

Abstract

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 $E^{(D)}(\Bbb{Q})$ be the group of $\Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $E^{(p)}(\Bbb{Q})$ has positive rank, and there are infinitely many primes $q$ for which $E^{(q)}(\Bbb{Q})$ has rank $0$. In this paper, assuming the parity conjecture, we show that for infinitely many primes $p$, the elliptic curve $E_n^{(p)}: y^2=x^3-np^2x$ has odd rank and for infinitely many primes $p$, $E_n^{(p)}(\Bbb{Q})$ has even rank, where $n$ is a positive integer that can be written as biquadrates sums in two different ways, i.e., $n=u^4+v^4=r^4+s^4$, where $u, v, r, s$ are positive integers such that $\gcd(u,v)=\gcd(r,s)=1$. More precisely, we prove that: if $n$ can be written in two different ways as biquartic sums and $p$ is prime, then under the assumption of the parity conjecture $E_n^{(p)}(\Bbb{Q})$ has odd rank (and so a positive rank) as long as $n$ is odd and $p\equiv5, 7\pmod{8}$ or $n$ is even and $p\equiv1\pmod{4}$. In the end, we also compute the ranks of some specific values of $n$ and $p$ explicitly.