Greenberg studied the variation of Iwasawa invariants across <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-extensions of a number field F. One of his key ideas was to appropriately topologize the set of all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Z}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-extensions of F. Sören Kleine refined Greenberg’s topology to investigate the variation of Iwasawa invariants for Selmer groups of abelian varieties over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {Z}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-extensions of number fields, providing bounds on their <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-invariants. In this article, we adapt Kleine’s techniques to the context of function fields. Specifically, we generalize the application of topology to analyze the variation of Iwasawa invariants associated with Selmer groups over <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Z}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {Z}_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>ℓ</mi> </msub> </math></EquationSource> </InlineEquation>-extensions of function fields.

On Selmer groups over Greenberg neighbourhoods in the function field case

期刊论文
发表日期：

2025年9月16日

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Sohan Ghosh,
Jishnu Ray

摘要

Greenberg studied the variation of Iwasawa invariants across $\mathbb {Z}_p$ $Z_{p}$ -extensions of a number field F. One of his key ideas was to appropriately topologize the set of all $\mathbb {Z}_p$ $Z_{p}$ -extensions of F. Sören Kleine refined Greenberg’s topology to investigate the variation of Iwasawa invariants for Selmer groups of abelian varieties over $\mathbb {Z}_p$ $Z_{p}$ -extensions of number fields, providing bounds on their $\mu $ $μ$ -and $\lambda $ $λ$ -invariants. In this article, we adapt Kleine’s techniques to the context of function fields. Specifically, we generalize the application of topology to analyze the variation of Iwasawa invariants associated with Selmer groups over $\mathbb {Z}_p$ $Z_{p}$ - and $\mathbb {Z}_\ell $ $Z_{ℓ}$ -extensions of function fields.