Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

Hatley, Jeffrey; Kundu, Debanjana; Lei, Antonio; Ray, Jishnu

doi:10.1007/s11139-022-00560-w

Cited by 2 publications

(4 citation statements)

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“…Note that in this modular form context, there is an extra finiteness hypothesis on H 0 (F v , A f ), and this arises due to a lack of an analogue of Mattuck's theorem [35] for a modular form. We do however remark that although a recent work of Hatley-Kundu-Lei-Ray [14] has provided some sufficient conditions for this finiteness hypothesis to hold, it would seem that the general situation seems out of reach at the moment. We also note that in the event that the level N is not divisible by p, then the finiteness is valid for all primes v above p (cf.…”

Section: Introductionmentioning

confidence: 77%

“…; also see [12,23,31]). Subsequently, there have been much interest on the fine Selmer group of a modular form (for instance, see [14,16,17]) or even more general classes of Galois representations (see [21,26,27,32]). A common feature in these cited works is that they are mainly concerned with working over the cyclotomic Z p -extension.…”

Section: Introductionmentioning

confidence: 99%

“…Hence the above theorem applies in these cases, where Conjecture Y is valid (also see Section 6 for some examples, where F is not necessarily abelian over Q and the theorem applies). This therefore provides a strong evidence to Conjecture Y, and at the same time, a weak partial support towards the pseudo-nullity prediction of Coates-Sujatha. As mentioned in the opening paragraph, there has been much interest in the study of the fine Selmer group of a modular form (for instance, see [14,16,17]). This will be the next theme of the paper which we describe briefly here.…”

Section: Introductionmentioning

confidence: 99%

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Structure of fine Selmer groups over -extensions

LIM

2023

Math. Proc. Camb. Phil. Soc.

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This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$ -extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ , where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$ -extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$ -extension, then it holds for almost all $\mathbb{Z}_{p}$ -extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

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Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Structure of fine Selmer groups over -extensions

LIM

2023

Math. Proc. Camb. Phil. Soc.

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“…To this purpose, we need to assume that a certain finiteness condition (⋆) holds. This is an assumption the occurrence which is quite natural in this context (see for example Hypothesis H cyc in [9], or Assumption (H0) in [16]). We prove the following Theorem 1.2.…”

Section: Introductionmentioning

confidence: 94%

Fine Selmer groups of modular forms

Kleine¹,

Müller²

2022

Preprint

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We compare the Iwasawa invariants of fine Selmer groups of p-adic Galois representations over admissible p-adic Lie extensions of a number field K to the Iwasawa invariants of ideal class groups along these Lie extensions.More precisely, let K be a number field, let V be a p-adic representation of the absolute Galois group G K of K, and choose a G K -invariant lattice T ⊆ V . We study the fine Selmer groups of A = V /T over suitable p-adic Lie extensions K∞/K, comparing their corank and µ-invariant to the corank and the µ-invariant of the Iwasawa module of ideal class groups in K∞/K.In the second part of the article, we compare the Iwasawa µ-and l 0invariants of the fine Selmer groups of CM modular forms on the one hand and the Iwasawa invariants of ideal class groups on the other hand over trivialising multiple Zp-extensions of K.

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Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

Cited by 2 publications

References 12 publications

Structure of fine Selmer groups over -extensions

Structure of fine Selmer groups over -extensions

Fine Selmer groups of modular forms

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