A \(p\)-adic Criterion for Quadratic Twists of CM Elliptic Curves to Have Rank \(1\)

Abstract

We present a \(p\)-adic criterion for a family of quadratic twists of an elliptic curve \(E/\mathbb{Q}\) with complex multiplication (CM) by the full ring of integers to have analytic rank \(1\). This criterion is obtained by studying the \(p\)-adic \(L\)-value \(\mathscr{L}_p(\psi_E^*)\), where \(\psi_E^*\) is the Hecke character associated \(E\). By relating \(\psi_E^*\) to a congruent Hecke character that lies in the range of classical interpolation, we reduce the required nonvanishing of \(\mathscr{L}_p(\psi_E^*)\) to a calculation of the \(p\)-adic valuation of the central \(L\)-value of a classical modular form. A theorem of Waldspurger then provides a half-integer weight eigenform whose Fourier coefficients control whether a quadratic twist of \(E\) has analytic rank \(1\).