2015/01/17 by Izadi, Farzali, Nabardi, Kamran
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1501.04179
openalex publication_date 2015/01/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let E be an elliptic curve over ℚ with the given Weierstrass equation y2=x3+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): y2=x3+aD2x+bD3. Let E(D)(ℚ) be the group of ℚ-rational points of E(D). It is conjectured by J. Silverman that there are infinitely many primes p for which E(p)(ℚ) has positive rank, and there are infinitely many primes q for which E(q)(ℚ) has rank 0. In this paper, assuming the parity conjecture, we show that for infinitely many primes p, the elliptic curve En(p): y2=x3-np2x has odd rank and for infinitely many primes p, En(p)(ℚ) has even rank, where n is a positive integer that can be written as biquadrates sums in two different ways, i.e., n=u4+v4=r4+s4, 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 En(p)(ℚ) has odd rank (and so a positive rank) as long as n is odd and p≡5, 7\pmod8 or n is even and p≡1\pmod4. In the end, we also compute the ranks of some specific values of n and p explicitly.