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

DOI10.5565/PUBLMAT6622204arXiv2005.12105WikidataQ123334047 ScholiaQ123334047MaRDI QIDQ6341365

Publication date: 25 May 2020

Abstract: We study the Fitting ideals over the finite layers of the cyclotomic

m a t h b b Z_{p}

-extension of

m a t h b b 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 ``Refined conjectures of the Birch and Swinnerton-Dyer type, 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 emph{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 $h o$ with splitting field $F$ . As an application, we give an upper bound of the dimension of the $h o$ -isotypic component of the Mordell-Weil group of $E$ over the finite layers of the cyclotomic $m a t h b b Z_{p}$ -extension of $F$ in terms of the order of vanishing of our Theta elements.

Mathematics Subject Classification ID

Holomorphic modular forms of integral weight (11F11) Iwasawa theory (11R23) Other abelian and metabelian extensions (11R20)

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