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

Abstract

We introduce a $p$-adic $L$-function ${\mathcal{L}}_{A/L}$ 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 ${\mathcal{L}}_{A/L}$ 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 ${\mathcal{L}}_{A/L}$ as the analytic side is proven in several cases. In the $d\ge 3$ case, the conjecture holds for $A/L$ if and only if it holds for all intermediate ${\mathbb{Z}}_p^2$-extensions $L^{\prime }/K$ 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 $\hat{I}¼$-invariant hypothesis, if $A/K$ has semistable reduction everywhere, the Iwasawa main conjecture is proven for $A$ over $L$ \cite{ttt26}.

47 pages