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NONCOMMUTATIVE IWASAWA THEORY OF ABELIAN VARIETIES OVER GLOBAL FUNCTION FIELDS

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  01 December 2025

LI-TONG DENG Open the ORCID record for LI-TONG DENG [Opens in a new window] ,
YUKAKO KEZUKA Open the ORCID record for YUKAKO KEZUKA [Opens in a new window] ,
YONG-XIONG LI Open the ORCID record for YONG-XIONG LI [Opens in a new window]  and
MENG FAI LIM Open the ORCID record for MENG FAI LIM [Opens in a new window]
LI-TONG DENG
Affiliation:
Tsinghua University , Beijing, 100084, China e-mail: dlt23@mails.tsinghua.edu.cn
YUKAKO KEZUKA
Affiliation:
Kanazawa University , Ishikawa 920-1192, Japan e-mail: kezuka@se.kanazawa-u.ac.jp
YONG-XIONG LI
Affiliation:
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications , Beijing, China e-mail: yongxiongli@gmail.com
MENG FAI LIM*
Affiliation:
Central China Normal University , Hubei Province, 430079, China
*
e-mail: limmf@ccnu.edu.cn
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Abstract

Let A be an abelian variety defined over a global function field F and let p be a prime distinct from the characteristic of F. Let $F_\infty $ be a p-adic Lie extension of F that contains the cyclotomic $\mathbb {Z}_p$-extension $F^{\mathrm {cyc}}$ of F. In this paper, we investigate the structure of the p-primary Selmer group $\mathrm {Sel}(A/F_\infty )$ of A over $F_\infty $. We prove the $\mathfrak {M}_H(G)$-conjecture for $A/F_\infty $. Furthermore, we show that both the $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F^{\mathrm {cyc}})$ and the generalized $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F_\infty )$ are zero, thereby proving Mazur’s conjecture for $A/F$. We then relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of A over the base field F. Assuming the finiteness of the Tate–Shafarevich group, we establish that this corank equals the order of vanishing of the L-function of $A/F$ at $s=1$. Finally, we extend a theorem of Sechi—originally proved for elliptic curves without complex multiplication—to abelian varieties over global function fields. This is achieved by adapting the notion of generalized Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalized Euler characteristic of $\mathrm {Sel}(A/F_\infty )$ to the Euler characteristic of $\mathrm {Sel}(A/F^{\mathrm {cyc}})$.

Keywords

Abelian varietiesIwasawa theoryglobal function fieldsgeneralized Euler characteristics

MSC classification

Primary: 11R23: Iwasawa theory
Secondary: 11R34: Galois cohomology

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 35
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Daniel Chan

The second author was partially supported by the ANR project Coloss, the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 101026826, and JSPS KAKENHI Grant JP25K17227.

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