On analogues of Mazur-Tate type conjectures in the Rankin-Selberg setting
Cauchi, Antonio (Universitat Politècnica de Catalunya. Departament de Matemàtiques)
Lei, Antonio (Université Laval. Département de Mathématiques et de Statistique)

Data:	2022
Resum:	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 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 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.
Resum:	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.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Iwasawa theory ; Rankin-selberg convolution ; Elliptic modular forms ; Mazur-tate conjectures
Publicat a:	Publicacions matemàtiques, Vol. 66 Núm. 2 (2022) , p. 571-630 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/402238
DOI: 10.5565/PUBLMAT6622204

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