A class of Drinfeld $A$-modules of rank $3$ with surjective Galois representations
Narasimha Kumar, Dwipanjana Shit
TL;DR
The paper constructs an infinite two-parameter family of rank-3 Drinfeld $A$-modules $\varphi$ over $F=\mathbb{F}_{q}(T)$ with $q\ge 7$ odd, defined by $\varphi_T=T+g_1^{q-1}\tau+g_2^{q-1}\tau^2+T^{q-1}\tau^3$ for $(g_1,g_2)$ in an infinite set $\mathcal{G}$, and proves that for every non-zero prime ideal $\mathfrak{l}$ of $A$, the mod-$\mathfrak{l}$, $\mathfrak{l}$-adic, and adelic Galois representations are surjective. The strategy combines Tate uniformization at $(T)$, irreducibility results for mod-$\mathfrak{l}$ representations (via inertia analysis and permutation-polynomial techniques), explicit control of inertia images to bound the mod-$\mathfrak{l}$-adic image, and Pink–Rütsche criteria to promote to $\mathfrak{l}$-adic surjectivity; together these yield adelic surjectivity. The work extends Che22 by providing an infinite family valid for all $q\ge 7$ and clarifies the mechanisms that force surjectivity, including a detailed comparison to the Carlitz case and a comprehensive case analysis using Aschbacher’s classification. Altogether, the results contribute concrete surjective adelic representations for rank-3 Drinfeld modules in the function-field setting, analogous to Serre’s and Greicius’s results in the number-field setting but with explicit construction and broad generality.
Abstract
Let $q = p^e \geq 7$ be an odd prime power, and set $A := \mathbb{F}_q[T]$. In this article, we construct an infinite two-parameter family of Drinfeld $A$-modules of rank $3$ such that, for every non-zero prime ideal $\mathfrak{l}$ of $A$, the associated mod-$\mathfrak{l}$, $\mathfrak{l}$-adic, and adelic Galois representations are surjective. These results generalise the specific example, constructed only for primes $p\equiv 1\pmod{3}$, in~\cite{Che22}.