Structure of fine Selmer groups in abelian p-adic Lie extensions

Debanjana Kundu (1) , Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio (2, 3, 4) , Sujatha Ramdorai (1)

1 UBC - University of British Columbia
2 CTN - Combinatoire, théorie des nombres
3 ICJ - Institut Camille Jordan
4 UJM - Université Jean Monnet - Saint-Étienne

This paper studies fine Selmer groups of elliptic curves in abelian $p$-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic $\mathbb{Z}_p$-extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.

Mots clés: Elliptic Curve; Selmer group; Generalized Greenberg's Conjecture

Domaines: Théorie des nombres [math.NT]

Debanjana Kundu, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio, Sujatha Ramdorai. Structure of fine Selmer groups in abelian p-adic Lie extensions. Osaka Journal of Mathematics, 2024, 61 (1). ⟨hal-03769801v5⟩