Quadratic Q-curves, units and Hecke L-values

Abstract We show that if K is a quadratic field, and if there exists a quadratic Q-curve E/K of prime degree N, satisfying weak conditions, then any unit u of OK satisfies a congruence ur ≡ 1 (mod N), where r = 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 ψ˜ and, roughly speaking, the elliptic curve. We show that this divisor then occurs in a critical value L(ψ , ˜ 2), by constructing a non-zero element in a Selmer group and applying a theorem of Kato.