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

Abstract

Let $K$ be a global function field of characteristic $p>0$ and $\ell\neq p$ be a prime number. We study Selmer groups over a $\mathbb{Z}_\ell$-extension $K_\infty/K$. For a lisse $\mathbb Z_\ell$-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 $K_\infty$, 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.