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

Cauchi, Antonio ^[1] ; Lei, Antonio ^[2]
1. [1] Universitat Politècnica de Catalunya
  
  Universitat Politècnica de Catalunya
  
  Barcelona, España
2. [2] Laval University
  
  Laval University
  
  Canadá
Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 2, 2022, págs. 571-630
Idioma: inglés
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Resumen
- 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.