Quadratic Q-curves, units and Hecke L-values

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 O K satisfies a congruence u r ≡ 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.