Gene Kopp

Indefinite Theta Functions and Zeta Functions

Hilbert's 12th problem for a base field K asks for a class of analytic functions with special values generating the maximal abelian extension of K. One approach to this problem is given by the Stark conjectures, which express special values of (derivatives of) Hecke L-functions in terms of algebraic units in abelian extensions of K. For a real quadratic base field, we present a new analytic formula for "Stark units" inspired by classical Kronecker limit formulas and by Sander Zwegers's theory of indefinite theta functions. We define a generalized indefinite theta function in dimension g and index 1 whose modular parameter transforms by a symplectic group. We introduce the indefinite zeta function, defined from the indefinite theta function using a Mellin transform, and show that it interpolates between certain zeta functions attached to ray ideal classes of real quadratic fields in a manner that preserves the functional equation. A formula for the indefinite zeta function at s=1 and by corollary, for the conjectural Stark units is obtained.