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

Technical Summary: Greenberg's $\mu = 0$ Conjecture for Lisse Sheaves over Global Function Fields

Problem Statement
The paper addresses the structure of Selmer groups associated with lisse $\mathbb{Z}_\ell$-sheaves over global function fields $K$ of characteristic $p > 0$, where $\ell \neq p$. Specifically, it investigates the Iwasawa theory of these groups over the unique constant $\mathbb{Z}_\ell$-extension $K_\infty/K$. The central problem is to determine whether the Pontryagin dual of the Selmer group is a finitely generated torsion module over the Iwasawa algebra $\Lambda = \mathbb{Z}_\ell[[\Gamma]]$ and, crucially, whether its $\mu$-invariant vanishes. This constitutes a positive-characteristic analogue of Greenberg's $\mu = 0$ conjecture, which predicts the vanishing of the $\mu$-invariant for elliptic curves and other Galois representations over cyclotomic $\mathbb{Z}_\ell$-extensions of number fields.

Methodology
The author employs a combination of étale cohomology, class field theory for function fields, and deformation theory.

1. Geometric Setup: The work is framed within the geometry of a smooth, projective, geometrically integral curve $X$ over a finite field $k = \mathbb{F}_q$. The global function field is $K = k(X)$. The extension $K_\infty$ is defined as the constant extension $K \cdot k_\infty$, where $k_\infty$ is the unique $\mathbb{Z}_\ell$-extension of $k$.
2. Sheaf Theory: The study focuses on lisse $\mathbb{Z}_\ell$-sheaves $T$ on an open affine subscheme $U \subseteq X$. The associated discrete sheaf $W = T \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell/\mathbb{Z}_\ell$ and the residual sheaf $M = T/\ell T$ are used to define Selmer groups via global cohomology groups $H^1_{\text{et}}(U, \cdot)$ and local conditions at places in $S = X \setminus U$.
3. Cartesian Condition: A key technical assumption is that the Selmer structure is "cartesian" with respect to multiplication by $\ell$. This ensures compatibility between the Selmer groups of $W$ and $M$, allowing the finiteness of the residual Selmer group to imply structural properties of the full Selmer group.
4. Cohomological Vanishing: The proof relies heavily on the vanishing of higher cohomology groups for constructible sheaves over affine curves in positive characteristic. Specifically, the author utilizes the fact that the Galois group of the constant extension $G_{k_\infty}$ is a pro-prime-to-$\ell$ group, leading to the vanishing of $H^a(G_{k_\infty}, N)$ for finite $\ell$-primary modules $N$ and $a > 0$.
5. Deformation Theory: The paper applies these cohomological results to the deformation theory of residual Galois representations $\bar{r}$. By analyzing the adjoint representation $\text{Ad}\bar{r}$ as a constructible sheaf, the author relates the vanishing of $H^2$ to the unobstructedness of deformation functors.

Key Contributions and Results

• Theorem 1.1 (Vanishing of $\mu$): The primary result establishes that for a lisse $\mathbb{Z}_\ell$-sheaf $T$ on $U$, the Pontryagin dual of the Selmer group $\text{Sel}_\mathcal{F}(W/K_\infty)^\vee$ is a finitely generated torsion module over the Iwasawa algebra $\Lambda$ with $\mu$-invariant equal to zero. This holds without imposing the irreducibility hypotheses often required in the number-field setting.
• Theorem 1.2 (Weak Leopoldt Conjecture): The paper proves an analogue of the weak Leopoldt conjecture for lisse sheaves, showing that $H^2(G_S(K_\infty), W) = 0$. This vanishing is derived from the finiteness of cohomology groups over the geometric base change $U_{\bar{k}}$ and the vanishing of cohomology over the constant extension.
• Theorem 1.3 (Deformation Rings): The author proves that the framed deformation functor of the residual representation $\bar{r}_\infty$ (restricted to $G_S(K_\infty)$) is formally smooth over the ring of Witt vectors $O = W(\mathbb{F})$. Consequently, the framed deformation ring is isomorphic to a formal power series ring $O[[x_1, \dots, x_m]]$. If the residual representation has no non-scalar endomorphisms ($\text{End}(V_{\bar{r}}) = \mathbb{F}$), the unframed deformation ring is also a formal power series ring.
• Finiteness of Residual Selmer Groups: A crucial intermediate step involves proving that the residual Selmer group $\text{Sel}_\mathcal{F}(M/K_\infty)$ is finite. This is achieved by showing that $H^1_{\text{et}}(U_\infty, M)$ is finite, utilizing the Hochschild-Serre spectral sequence and the vanishing of cohomology for the constant extension.

Significance and Claims
The paper claims to provide a definitive positive-characteristic, prime-to-$p$ analogue of Greenberg's $\mu = 0$ conjecture. Unlike the number-field case, where the irreducibility of the residual representation is often essential to prevent positive $\mu$-invariants (as seen in Mazur's examples), this result holds generally for lisse sheaves over the constant $\mathbb{Z}_\ell$-extension of a global function field.

The author notes that this work extends previous foundational results by Bandini, Longhi, and others regarding Selmer groups for elliptic curves and abelian varieties over function fields. The main contribution is the generalization to arbitrary lisse $\mathbb{Z}_\ell$-sheaves and the explicit proof of the $\mu = 0$ property in this broader context.

Furthermore, the paper highlights that these results yield strong structural information about deformation rings in the function-field setting. Specifically, it proves that the framed deformation ring is a formal power series ring without requiring a Noetherian hypothesis on the deformation ring itself or a separate $\mu = 0$ hypothesis for the adjoint representation; the vanishing of the relevant cohomology group $H^2$ is derived directly from the geometric properties of the curve and the nature of the constant extension. This establishes that ordinary deformations are unobstructed in the cyclotomic limit for these function fields.