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

Antonio Cauchi, Antonio Lei

• We study the Fitting ideals over the finite layers of the cyclotomic Zp-extension of 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 andTate 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 ρ with splitting field F. As an application, we give an upper bound of the dimension of the ρ-isotypic component of the Mordell–Weil group of E over the finite layers of the cyclotomic Zp-extension of F in terms of the order of vanishing of our theta elements.