On theta functions and Weil’s generalized Poisson summation formula.

Abstract

In his paper, G. Shimura quotes a proposition [2, Proposition 2.5] which is derived directly from a transformation formula of theta functions due to KrazerPrym and suggests proving this proposition by means of a generalized Poisson summation formula obtained by A. Weil [4, Theoreme 4]. The purpose of this paper is to execute his idea. Let us explain briefly the result. Let V be a 2n dimensional real vector space and let D be a discrete subgroup of rank 2n in V. Suppose that we have a nondegenerate alternate bilinear form A on V x V which assumes integral values on D x D. The form A represents an integral cohomology class on the torus T= VID. A complex vector space structure J on V induces a complex structure on the torus, which may be denoted by the same notation. Such a complex structure J is said to be admissible if there is a positive divisor on the complex torus (T, J) whose cohomology class is A. Making use of the theory of theta functions on the complex vector space (V, J), we assign to each admissible complex structure J on T a holomorphic map 0(J) of the complex torus (T, J) into a complex projective space P of a certain dimension depending only on the form A, in such a way that the cohomology class A is a rational multiple of the image of the cohomology class of hyperplane sections on P under the cohomology homomorphism 0(J)*. Consider the group S(D) of real linear transformations of V which leave the bilinear form A and the subgroup D invariant. If a E S(D), then a induces an isomorphism a of T onto itself. If J is an admissible complex structure on T, there is a unique admissible complex structure Ja on T such that a: (T, Ja) -(T, J) is a holomorphic isomorphism of these two complex tori. Now, the result asserts that if we make a suitable choice of the assignment 0(J) to each admissible complex structure J and if a belongs to a congruence subgroup in S(D) of sufficiently high level with respect to an appropriate base of D, then the following diagram is commutative: