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

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