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)
Date: |
2022 |
Abstract: |
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. |
Abstract: |
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. |
Rights: |
Tots els drets reservats. |
Language: |
Anglès |
Document: |
Article ; recerca ; Versió publicada |
Subject: |
Iwasawa theory ;
Rankin-selberg convolution ;
Elliptic modular forms ;
Mazur-tate conjectures |
Published in: |
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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Record created 2022-07-27, last modified 2023-11-29