Quadratic Reciprocity , after Weil

The character associated to a quadratic extension field K of Q, χ : Z −→ C, χ(n) = (disc(K)/n) (Jacobi symbol), is in fact a Dirichlet character; specifically its conductor is |disc(K)|. This fact encodes basic quadratic reciprocity from elementary number theory, phrasing it in terms that presage class field theory. This writeup discusses Hilbert quadratic reciprocity in the same spirit. Let k be a number field, and let K be a quadratic extension field of k. We show that a global quadratic norm residue character, νK/k : Ak −→ {±1}, is a Hecke character, i.e., it is trivial on k×. This fact encodes the Hilbert quadratic reciprocity rule, i.e., the product formula for quadratic Hilbert symbols. The reciprocity rule in turn encodes elementary quadratic reciprocity statements, including basic quadratic reciprocity. The ideas here work in the geometric (function field) case as well, but for simplicity we discuss only number fields. This writeup is modeled on a writeup by Paul Garrett, http://www.math.umn.edu/∼garrett/m/v/quad rec 02.pdf . The Weil representation appears in A. Weil, Sur certaines d’opérateurs unitaires, Acta Math. 111 (1964), 143–211, and Hilbert quadratic reciprocity in D. Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresber. Deutsch. Math. Verein. 4 (1897), 175–546.