Greenberg's $\mu =0$ conjecture for lisse sheaves over global function fields

Let $K$ be a global function field of characteristic $p>0$ and $\mathrm{\ell }\mathrm{\ne }p$ be a prime number. We study Selmer groups over a ${\mathbb{Z}}_{\mathrm{\ell }}$-extension $K_{\mathrm{\infty }}\mathrm{/}K$. For a lisse ${\mathbb{Z}}_{\mathrm{\ell }}$-sheaf we prove that the Pontryagin dual of the associated Selmer group is a finitely generated torsion module over the Iwasawa algebra and has $\mu$-invariant equal to zero. This gives a positive-characteristic, prime to $p$, analogue of Greenberg’s $\mu =0$ conjecture. Our result applies in particular to abelian varieties, fine Selmer groups, and adjoint representations. We also prove an analogue of the weak Leopoldt conjecture in this context over $K_{\mathrm{\infty }}$, and deduce that the framed deformation ring of a residual representation is a formal power series ring. The same conclusion holds for the unframed deformation ring if the residual representation has no non-scalar endomorphisms.