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Twists of elliptic curves. (English) Zbl 0887.11025

Let \(E\) be an elliptic curve over \(\mathbb{Q}\), and for each integer \(D\) denote by \(E(D)\) the quadratic twist of \(E\) by \(D\). It is conjectured that there exist infinitely many prime numbers \(p\) such that \(E(p)\) has rank zero and there exist infinitely many prime numbers \(\ell \) such that \(E(\ell )\) has rank greater than zero. In this paper the author considers certain special families of elliptic curves and shows that there exists a finite set of prime numbers \(S\) of density \(1/3\) satisfying the following property:
If \( D=\prod_{j}p_{j}\) is a squarefree integer with \(p_{j}\in S\) then \(E(D)\) has rank zero.
In particular, for \(p\in S\), \(E(p)\) has rank zero. As an example he considers the elliptic curve \(E_{1}\) given by the Weierstrass equation \(y^{2}=x^{3}+44 x^{2}-19360 x+1682384\). The set \(S_{1} \) of prime numbers associated with \(E_{1}\) consists of \(11\) and the prime numbers such that the number of \(\mathbb{F}_{p}\)-rational points of the reduction \(\widetilde{X_{0}(11)}\) of \(X_{0}(11)\) modulo \(p\) is an odd number. In order to generalize this result for any elliptic curve \(E\) in the family he considers, the author shows that for the prime numbers \(p\in S\) such that the rational factor of the special value \(L(E(p),1)\) of the \(L\)-function of \(E(p)\) at \(1\) is nonzero, the rank of \(E(p)\) is zero. These special values of \(L\)-functions are related to surjective \(\mathbb{Z}/2 \mathbb{Z}\)-Galois representations that are attached to modular forms. The paper is concluded proving the following result related to cubic twists of certain elliptic curves. Let \(c\) be an odd integer which is not a perfect square. Then there exist infinitely many integers \(D\) that are at most the product of two primes for which the cubic twist \(E_{cD^{2}}':y^{2}=x^{3}+cD^{2}\) of \(E_{c}':y^{2}=x^{3}+c\) has positive rank.

MSC:

11G05 Elliptic curves over global fields
14H52 Elliptic curves
14G05 Rational points
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