On a question of Igusa III: A generalized Poisson formula for pairs of polynomials

This paper is the first response to a question of Igusa that asked for a multivariable extension of his adelic Poisson formula in one variable. We prove a Poisson summation formula over the adeles between two distributions, depending upon two variables. Each distribution is defined by a mapping p:Q(n)-> Q(2) whose components are homogeneous polynomials, and whose critical locus is a union of lines.