Iwasawa theory for Rankin-Selberg convolution at an Eisenstein prime

TL;DR

This paper proves a mod p version of the Iwasawa main conjecture for the Rankin-Selberg convolution of two modular forms when the prime p is an Eisenstein prime for one of them, linking p-adic L-functions and Selmer groups.

Contribution

It establishes a mod p Iwasawa main conjecture for Rankin-Selberg convolutions under Eisenstein prime conditions, providing explicit examples.

Findings

The p-adic L-function and Selmer group characteristic ideal generate the same ideal mod p.

The main conjecture holds modulo p for the Rankin-Selberg product under Eisenstein prime assumptions.

Explicit examples illustrating the congruence are provided.

Abstract

Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and $ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l < k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty} $-Selmer group of the Rankin-Selberg product of $f$ and $h$ under the assumption that $ p $ is an Eisenstein prime for $ h $ i.e. the residual Galois representation of $ h $ at $ p $ is reducible. We show that the $ p $-adic $ L $-function and the characteristic ideal of the $p^\infty$-Selmer group of the Rankin-Selberg product of $f, h$ generate the same ideal modulo $ p $ in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$ holds mod $p$. As an application to our results, we explicitly describe a few examples where the above congruence holds.