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ON FINE SELMER GROUPS AND SIGNED SELMER GROUPS OF ELLIPTIC MODULAR FORMS

Published online by Cambridge University Press:  19 April 2022

ANTONIO LEI*
Affiliation:
Département de Mathématiques et de Statistique, Université Laval, Pavillion Alexandre-Vachon, 1045 Avenue de la Médecine, Québec, QCG1V 0A6, Canada
MENG FAI LIM
Affiliation:
School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan430079, PR China e-mail: limmf@mail.ccnu.edu.cn

Abstract

Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic $\mathbb {Z}_p$ extension of $\mathbb {Q}$ and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The authors’ research is partially supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096 (Lei), and the National Natural Science Foundation of China under Grant No. 11771164 and the Fundamental Research Funds for the Central Universities of CCNU under Grant CCNU20TD002 (Lim).

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ON FINE SELMER GROUPS AND SIGNED SELMER GROUPS OF ELLIPTIC MODULAR FORMS
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