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Non-commutative Iwasawa theory of abelian varieties over global function fields by Deng, Li-Tong; Kezuka, Yukako; Li, Yong-Xiong; Lim, Meng Fai is a scholarly article available to read on EtoBox.

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Let $A$ be an abelian variety defined over a global function field $F$. We investigate the structure of the $p$-primary Selmer group $\mathrm{Sel}(A/F_\infty)$ for any prime number $p$ distinct from the characteristic of $F$, over $p$-adic Lie extensions $F_\infty$ of $F$ which contain the cyclotomic $\mathbb{Z}_p$-extension $F^{\mathrm{cyc}}$. In particular, we prove that the Pontryagin dual of the Selmer group $\mathrm{Sel}(A/F^\mathrm{cyc})$ is a torsion $\mathbb{Z}_p[[\mathrm{Gal}(F^\mathrm{cyc}/F)]]$-module with trivial $\mu$-invariant, and we establish the $\mathfrak{M}_H(G)$-conjecture of Coates-Fukaya-Kato-Sujatha-Venjakob for $A/F_\infty$. In view of the validity of the $\mathfrak{M}_H(G)$-conjecture, it therefore makes sense to speak of the characteristic element (in the sense of Coates et al.) attached to the Pontryagin dual of $\mathrm{Sel}(A/F_\infty)$. We 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 $F$. Combining this with the deep results of Tate, Milne and Kato-Trihan, we show that the order of vanishing of the characteristic

Author

Deng, Li-Tong; Kezuka, Yukako; Li, Yong-Xiong; Lim, Meng Fai

Published

2024

Language

EN