Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields

Abstract

For an elliptic curve E over Q, and a real quadratic extension F of Q, satisfying suitable hypotheses, we study the algebraic part of certain twisted L-values for E/F . The Birch and SwinnertonDyer conjecture predicts that these L-values are squares of rational numbers. We show that this question is related to the ratio of Petersson inner products of a quaternionic form on a definite quaternion algebra over Q and its base change to F .

Heegner points and $p$-adic $L$-functions for elliptic curves over certain totally real fields

“…This would be consistent with the Birch and Swinnerton-Dyer conjecture. For more on this, see [27]. We remark that for such ı, the restriction ıj A Q is not trivial, so this is not a situation where Waldspurger type results [35] can be directly applied.…”