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

TL;DR

This paper establishes the algebraic functional equation for $p$-adic $L$-functions associated with Rankin-Selberg products, using Selmer complexes and their properties in Iwasawa theory.

Contribution

It proves the perfectness of Selmer complexes in degree 2 and derives the functional equation for algebraic $p$-adic $L$-functions under mild hypotheses.

Findings

Functional equation for algebraic $p$-adic $L$-functions established.

Selmer complexes are shown to be perfect with amplitude [1,2].

Study of Tamagawa factors in families is developed.

Abstract

This article presents an approach to the algebraic 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 (which are given in terms of characteristic ideals of Selmer groups, which arise 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 factors in families, which may be of independent interest.