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.
Read more about this topic: Hecke Character
Famous quotes containing the words algebraic and/or characters:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“For the most part, only the light characters travel. Who are you that have no task to keep you at home?”
—Ralph Waldo Emerson (1803–1882)