Quadratic $${\mathbb {Q}}$$ Q -curves, units and Hecke $$L$$ L -values

Abstract

We show that if \(K\) is a quadratic field, and if there exists a quadratic \({\mathbb {Q}}\)-curve \(E/K\) of prime degree \(N\), satisfying weak conditions, then any unit \(u\) of \(O_K\) satisfies a congruence \(u^r\equiv 1\pmod {N}\), where \(r={\mathrm {g.c.d.}}(N-1,12)\). If \(K\) is imaginary quadratic, we prove a congruence, modulo a divisor of \(N\), between an algebraic Hecke character \(\tilde{\psi }\) and, roughly speaking, the elliptic curve. We show that this divisor then occurs in a critical value \(L(\tilde{\psi },2)\), by constructing a non-zero element in a Selmer group and applying a theorem of Kato.