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

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 [32], 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 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.