Non-commutative Iwasawa theory of abelian varieties over global function fields
Li-Tong Deng, Yukako Kezuka, Yong-Xiong Li, Meng Fai Lim
TL;DR
This work extends non-commutative Iwasawa theory to global function fields by studying the p-primary Selmer groups of abelian varieties over admissible p-adic Lie extensions. It proves the M_H(G) conjecture for the Pontryagin duals, shows vanishing μ-invariants and thereby Mazur's conjecture in this setting, and develops a robust Akashi-series framework to relate global Selmer data to local twists and L-values. By establishing a surjectivity of localisation maps, a control theorem, and a connection to generalized Euler characteristics, the authors connect the order of vanishing of characteristic elements to Selmer twists and BSD-type invariants. The generalised Euler characteristic framework then enables explicit computations, yielding formulas that mirror known results in number fields while highlighting distinctive function-field phenomena.
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 $μ$-invariant of the Pontryagin dual of the Selmer group $\mathrm{Sel}(A/F^\mathrm{cyc})$ and the generalised $μ$-invariant of the Pontryagin dual of the Selmer group $\mathrm{Sel}(A/F_\infty)$ are zero, therby 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 generalised Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalised Euler characteristic of $\mathrm{Sel}(A/F_\infty)$ to the Euler characteristic of $\mathrm{Sel}(A/F^{\mathrm{cyc}})$.