On analogues of Mazur-Tate type conjectures in the Rankin-Selberg setting

TL;DR

This paper extends Mazur-Tate conjectures to the Rankin-Selberg setting, defining new Theta elements and proving results close to the weak main conjecture, with applications to elliptic curves and Artin representations.

Contribution

It introduces new Theta elements for Rankin--Selberg convolutions and generalizes recent work to approach the weak main conjecture in this context.

Findings

Proves a result near the weak main conjecture for Rankin--Selberg convolutions.

Defines Theta elements using Loeffler--Zerbes' p-adic L-functions.

Provides bounds on Mordell-Weil group components related to Theta elements.

Abstract

We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic curves to prove a result very close to the \emph{weak main conjecture} of Mazur and Tate for Rankin--Selberg convolutions. Special emphasis is given to the case where $f$ corresponds to an elliptic curve $E$ and $g$ to a two dimensional odd irreducible Artin representation $\rho$ with splitting field $F$. As an application, we give an upper bound of the dimension of the $\rho$-isotypic component of the Mordell-Weil group of $E$ over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $F$ in terms of the order of vanishing of our Theta elements.