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

2016
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Bull. Iranian Math. Soc
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unpublished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

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