The <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-fine Selmer group of an elliptic curve E over a number field F is a subgroup of the classical <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-Selmer group of E over F. Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group of E over a p-adic Lie extension of a number field is intricately related to some deep questions in classical Iwasawa theory; for example, Iwasawa’s classical <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-invariant vanishing conjecture. In this article, we study the properties of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-fine Selmer group of an elliptic curve over certain p-adic Lie extensions of a number field. We also define and discuss <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-fine Selmer group of an elliptic curve over function fields of characteristic p and also of characteristic <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ell \ne p.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≠</mo> <mi>p</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We relate our study with a conjecture of Jannsen.

Iwasawa theory of fine Selmer groups over global fields

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发表日期：

2025年10月8日

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Sohan Ghosh,
Somnath Jha,
Sudhanshu Shekhar

摘要

The $p^\infty $ $p^{\infty}$ -fine Selmer group of an elliptic curve E over a number field F is a subgroup of the classical $p^\infty $ $p^{\infty}$ -Selmer group of E over F. Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group of E over a p-adic Lie extension of a number field is intricately related to some deep questions in classical Iwasawa theory; for example, Iwasawa’s classical $\mu $ $μ$ -invariant vanishing conjecture. In this article, we study the properties of the $p^\infty $ $p^{\infty}$ -fine Selmer group of an elliptic curve over certain p-adic Lie extensions of a number field. We also define and discuss $p^\infty $ $p^{\infty}$ -fine Selmer group of an elliptic curve over function fields of characteristic p and also of characteristic $\ell \ne p.$ $ℓ \neq p .$ We relate our study with a conjecture of Jannsen.