Drinfeld modules with maximal Galois action
David Zywina
TL;DR
The work investigates rank-2 Drinfeld A-modules over F=_q(t) and their Galois actions on torsion, constructing the adelic representation ho_{} and proving that its image is as large as possible for many choices of a=(a_1,a_2) in A^2. Using a combination of λ-adic and adelic group-theoretic criteria, inertia and Frobenius data, determinant analysis, irreducibility results, a Hilbert irreducibility framework, and sieving, the authors show that for a density-1 set of a, ρ_{(a)}(Gal_F) contains SL_2(A) with index dividing q-1 (or 4 when q=2), and in particular many a yield ρ_{(a)}(Gal_F)=GL_2(A). They also provide explicit Drinfeld module examples realizing maximal image, and treat the delicate q=2 case with wild ramification at infinity. The results parallel Serre-style open-image phenomena for elliptic curves, but in the Drinfeld-module setting, yielding strong uniformity and density statements for maximal Galois action with concrete arithmetic and geometric consequences. The methods significantly advance understanding of how often rank-2 Drinfeld modules over function fields exhibit maximal adelic Galois images and illuminate the role of inertia, Frobenius polynomials, and Hilbert-type phenomena in this context.
Abstract
With a fixed prime power $q>1$, define the ring of polynomials $A=\mathbb{F}_q[t]$ and its fraction field $F=\mathbb{F}_q(t)$. For each pair $a=(a_1,a_2) \in A^2$ with $a_2$ nonzero, let $φ(a)\colon A\to F\{τ\}$ be the Drinfeld $A$-module of rank $2$ satisfying $t\mapsto t+a_1τ+a_2τ^2$. The Galois action on the torsion of $φ(a)$ gives rise to a Galois representation $ρ_{φ(a)}\colon \operatorname{Gal}(F^{\operatorname{sep}}/F)\to \operatorname{GL}_2(\widehat{A})$, where $\widehat{A}$ is the profinite completion of $A$. We show that the image of $ρ_{φ(a)}$ is large for random $a$. More precisely, for all $a\in A^2$ away from a set of density $0$, we prove that the index $[\operatorname{GL}_2(\widehat{A}):ρ_{φ(a)}(\operatorname{Gal}(F^{\operatorname{sep}}/F))]$ divides $q-1$ when $q>2$ and divides $4$ when $q=2$. We also show that the representation $ρ_{φ(a)}$ is surjective for a positive density set of $a\in A^2$.