The theta multiplier for number fields via p-adic planes

Abstract

In [8] Tate showed how zeta functions associated to grossencharacters of number fields and their functional equations could be developed from local considerations. Since that work is really an adaptation to an adelic setting of Hecke’s [1] original approach to proving the validity of the functional equations, a transformation formula (Tate’s Riemann-Roch Theorem) for what is essentially an adelic theta function plays a crucial role. However, the theta function is developed as a function of a single adelic variable since that is all that is needed for the intended application. On the other hand, the classical theta functions (of real variables) which motivated that work, are often extended to functions defined on (products of) upper half-planes, giving modular forms. As in [6] in the rational case, Dirichlet characters can be incorporated into the definition of the theta function, giving forms on congruence subgroups. (In Tate’s work this was not necessary since he integrates the theta functions against the characters to obtain the zeta functions.) A. Schwartz [5], in the rational case, extended the classical theta function with trivial character defined on the upper half-plane to a function defined on a product of the upper half-plane and a p-adic “half-plane.” While the existence of the theta multiplier is shown in that paper through local calculations, Schwartz returned to the classical theta function in order to deduce an explicit formula for it. In this paper we associate to any number field and Dirichlet character a theta function of an appropriate adelic variable and establish its transformation properties under a discrete theta group. The theta multiplier is developed entirely by piecing together local calculations in an adelic argument. We follow the approach taken in [7], [4], and [5], where p-adic planes