[Paper Review] A uniform open image theorem for l-adic representations in positive characteristic

Emiliano Ambrosi|arXiv (Cornell University)|Nov 16, 2017

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes a uniform open image theorem for $α$-adic representations in positive characteristic, proving that under the geometric Lie perfect condition, the exceptional locus of $k$-rational points with non-open local image is finite, and the index $[\Pi : \Pi_x]$ is uniformly bounded. The result extends Cadoret-Tamagawa's characteristic 0 result to positive characteristic using generalized modular curves and wild ramification control via Riemann-Hurwitz with $\ell$-adic cohomology.

ABSTRACT

Let $k$ be a finitely generated field of characteristic $p > 0$ and $\\ell$ a prime. Let $X$ be a smooth, separated, geometrically connected curve of finite type over $k$ and $\ ho: \\pi_1(X)\ ightarrow GL_r(\\mathbb Z_{\\ell})$ a continuous representation of the \\etale fundamental group of $X$ with image $G$. Any $k$-rational point $x:Spec(k)\ ightarrow X$ induces a local representation $\ ho_x: \\pi_1(Spec(k)) \ ightarrow \\pi_1(X) \ ightarrow GL_r(\\mathbb Z_{\\ell})$ with image $G_x$. The goal of this paper is to study how $G_x$ varies with $x\\in X(k)$. In particular we prove that if $\\ell\ eq p$ and every open subgroup of $\ ho(\\pi_1(X_{\\overline k}))$ has finite abelianization, then the set $X_{\ ho}^{ex}(k)$ of $k$-rational points such that $G_x$ is not open in $G$ is finite and there exists a constant $C\\geq 0$ such that $[G:G_x]\\leq C$ for all $x\\in X(k)-X_{\ ho}^{ex}(k)$. This result can be applied to obtain uniform bounds for the $\\ell$-primary torsion of groups theoretic invariants in one dimensional families of varieties. For example, torsion of abelian varieties and the Galois invariants of the geometric Brauer group. This extends to positive characteristic previous results of Anna Cadoret and Akio Tamagawa in characteristic 0.

Motivation & Objective

• To extend Cadoret-Tamagawa's uniform open image theorem from characteristic 0 to positive characteristic.
• To establish finiteness and uniform boundedness of index for $k$-rational points whose local Galois image is not open in the global image.
• To control wild inertia terms in the Riemann-Hurwitz formula for abstract modular curves in positive characteristic.
• To generalize the use of gonality and introduce isogonality as a tool for bounding points of bounded degree in positive characteristic.

Proposed method

• Construct connected étale covers $X_U \to X$ associated to open subgroups $U \subseteq \Pi$, called abstract modular schemes.
• Use the anabelian dictionary to relate $k$-rational points lifting to $X_U$ with $\Pi_x \subseteq U$, enabling diophantine control.
• Apply the Riemann-Hurwitz formula to compute genus of abstract modular curves, with careful treatment of wild inertia terms in positive characteristic.
• Generalize techniques from [CT12a] to control wild ramification in the genus computation.
• Use the growth of gonality and isogonality of compactified covers to bound the number of points of bounded degree.
• Leverage the Lie perfect condition on $\Pi_{\overline{k}}$ to ensure uniform bounds on index $[\Pi : \Pi_x]$.

Experimental results

Research questions

• RQ1 Can the uniform open image theorem for $\ell$-adic representations be extended from characteristic 0 to positive characteristic?
• RQ2 Under what conditions is the set of $k$-rational points with non-open local image finite in positive characteristic?
• RQ3 How can wild ramification in the Riemann-Hurwitz formula be controlled in positive characteristic modular curve constructions?
• RQ4 Can isogonality replace gonality as a tool to bound points of bounded degree in positive characteristic?
• RQ5 Is the isogonality of the compactified covers $X_{C\Pi_{\overline{k}}(n)}$ unbounded as $n \to \infty$ under the geometric Lie perfect condition?

Key findings

• The exceptional locus $X_{\rho}^{ex}(k)$ is finite when $X$ is a curve, $k$ is finitely generated, and $\rho$ is geometrically Lie perfect.
• There exists an integer $N \geq 1$, depending only on $\rho$, such that $[\Pi : \Pi_x] \leq N$ for all $x \in X_{\rho}^{gen}(k)$.
• The genus of the abstract modular curves $X_U$ grows with the level, enabling control over rational points via diophantine finiteness.
• Wild inertia terms in the Riemann-Hurwitz formula are controlled via generalizations of [CT12a] techniques.
• The isogonality $\gamma^{\text{iso}}_{X_{C\Pi_{\overline{k}}(n)}}$ tends to infinity as $n \to \infty$, suggesting a potential path to extending the result to points of bounded degree.
• The result implies uniform bounds on $\ell$-primary torsion in invariants like the geometric Brauer group and abelian varieties in one-dimensional families.