$p$-adic $L$-functions for elliptic curves over global function fields

Abstract

We introduce a $p$-adic $L$-function $\mathscr L_{A/L}$ associated to each ordinary elliptic curve $A$ over a global function field $K$ of characteristic $p$ together with a $\mathbb{Z}_{p}^{d}$-extension $L/K$, $d=0$ allowed, unramified outside a finite set of places where $A$ has ordinary (good ordinary or multiplicative) reductions. This $\mathscr L_{A/L}$ is characterized by its interpolation of the special values of twisted Hasse-Weil $L$-functions. We show that it satisfies the desired functional equation, specialization formula, and restriction formula in connection with the characteristic ideal of the dual $p^\infty$-Selmer group of $A/L$. The Iwasawa main conjecture having $\mathscr{L}_{A / L}$ as the analytic side is proven in several cases. In the $d\geq 3$ case, the conjecture holds for $A/L$ if and only if it holds for all intermediate $\mathbb{Z}_p^2$-extensions $L'/K$ belonging to a given non-empty Zariski open subset of the Grassmannian $\mathrm{Gr}(d-2,d)(\mathbb{Z}_p)$. Recently, subject to a technical $\mu$-invariant hypothesis, if $A/K$ has semistable reduction everywhere, the Iwasawa main conjecture is proven for $A$ over $L$ \cite{ttt26}.