Functional equations of algebraic Rankin–Selberg p-adic L-functions

Abstract

Abstract This article presents an approach to the functional equation for Selmer complexes, which in turn have applications in the Iwasawa theoretic study of Rankin–Selberg products of the Hida and Coleman families. Our treatment establishes the functional equation for algebraic p -adic L -functions (that are given in terms of characteristic ideals of Selmer groups arising as the cohomology of appropriately defined Selmer complexes in degree 2). This is achieved by recovering the characteristic ideal as the determinant of the said Selmer complex, once we prove (under suitable but rather mild hypotheses) that the Selmer complex in question is perfect with amplitude [1, 2], and its cohomology is concentrated in degree-2. The perfectness of these Selmer complexes turns out to be a delicate problem, and the required properties require a study of Tamagawa numbers in families, which may be of independent interest.