Rational torsion on optimal curves and rank-one quadratic twists
Let \(E/\mathbb Q\) be an elliptic curve of conductor \(N\). By the modularity of elliptic curves over \(\mathbb Q\), given an elliptic curve \(E/\mathbb Q\) there is a homomorphism \(\pi : J_0(N) \to E\) where \(J_0(N)\) is the Jacobian variety of the modular curve \(X_0(N)\) of level \(N\). An elliptic curve \(E/\mathbb Q\) is said to be \textit{optimal} if \(\ker(\pi)\) is connected, and there is a unique optimal elliptic curve \(E\) in each isogeny class of elliptic curves defined over \(\mathbb Q\) with conductor \(N\). Given an elliptic curve \(E/\mathbb Q\) of conductor \(N\), and a positive divisor \(d\) of \(N\), let \(w_d(E)\) be the number \(\pm1\) such that \(W_d(f) = w_d(E)\, f\) where \(W_d\) is the Atkin-Lehner involution, and \(f\) is the newform associated with \(E\). Proved in [\textit{N. Dummigan}, Int. J. Number Theory 1, No. 4, 513--531 (2005; Zbl 1158.11321)] is as follows: Let \(E'/\mathbb Q\) be an elliptic curve of square-free conductor \(N\) with a rational point of odd prime order \(\ell \nmid N\), and \(E\) be the optimal elliptic curve \(\mathbb Q\)-isogenous to \(E'\). If (a) \(w_p(E')=-1\) for at least one prime divisor \(p\) of \(N\), and (b) \(\ell\) satisfies a certain divisibility condition, then \(E\) has a \(\mathbb Q\)-rational \(\ell\)-torsion point. The authors of the paper under review prove that the conditions (a) and (b) in Dummigan's result are always satisfied, and use it to find the following result on quadratic twists of elliptic curves: Let \(E'/\mathbb Q\) be an elliptic curve of square-free conductor \(N\) with a rational point of order \(3 \nmid N\). If there is only one prime \(p \mid N\) such that \(w_p=-1\), then \(\text{ord}_{s=1} L(s,E'_D)=1\) for a positive proportion of fundamental discriminants \(D\) where \(E'_D\) is the quadratic twist of \(E'\) by \(D\). The authors note that the family of elliptic curves in this result includes the elliptic curves considered in [\textit{D. Byeon, D. Jeon} and \textit{C. H. Kim}, ``Rank-one quadratic twists of an infinite family of elliptic curves'', J. Reine Angew. Math. 633, 67--76 (2009; Zbl 1257.11062)] where conditions on the optimal elliptic curve in an isogeny class were imposed.
scientific article; zbMATH DE number 5853446
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