Iranian Mathematical Society Title:. On Silverman's conjecture for a family of elliptic curves ON SILVERMAN'S CONJECTURE FOR A FAMILY OF ELLIPTIC CURVES

Author, Izadi, F Izadi, K Nabardi
2016 Bull. Iranian Math. Soc   unpublished
Let E be an elliptic curve over 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 2 x + bD 3. 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
more » ... suming the parity conjecture, we show that for infinitely many primes p, the elliptic curve E (p) n : y 2 = x 3 − np 2 x has odd rank and for infinitely many primes p, E (p) n (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 (p) n (Q) has odd rank (and so a positive rank) as long as n is odd and p ≡ 5, 7 (mod 8) or n is even and p ≡ 1 (mod 4). In the end, we also compute the ranks of some specific values of n and p explicitly.