Algebraic Hecke Characters
An algebraic Hecke character is a Hecke character taking algebraic values: they were introduced by Weil in 1947 under the name type A0. Such characters occur in class field theory and the theory of complex multiplication.
If E is an elliptic curve defined over a number field F with complex multiplication by the imaginary quadratic field K, then there is an algebraic Hecke character χ for K, with exceptional set S the set of primes of bad reduction of E together with the infinite places. This character has the property that for a prime ideal p of good reduction, the value χ(p) is a root of the characteristic polynomial of the Frobenius endomorphism. As a consequence, the Hasse–Weil zeta function for E is a product of two Dirichlet series, for χ and its complex conjugate.
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