arXiv subjects
Publications and source records attributed to Dwipanjana Shit.
Let $\mathbb{F}_{q}$ be a finite field, and $A:=\mathbb{F}_{q}[T]$. In this article, we give explicit criteria, involving concrete valuations, on the coefficients of the Drinfeld $A$-modules of rank $r$ for $r=2,3$, which ensure the surjectivity of the associated $(T)$-adic Galois representation. As a result, we shall calculate the densities of such Drinfeld $A$-modules.
Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is a prime power and let $A:= \mathbb{F}_{q}[T]$. By~\cite{PR09}, the adelic image of the Galois representation attached to a rank $2$ Drinfeld $A$-module $φ$ is open, and determining when it is surjective remains a subtle problem. To resolve this question, in this article, we study the $\mathfrak{p}$-adic surjectivity of the Galois representations attached to $φ$, where $\mathfrak{p} \in Ω_A:= \mathrm{Spec}(A) \setminus \{ (0) \}$. There are two directions to investigate this problem: one by fixing the prime $\mathfrak{p}$, and the other by fixing $φ$. In the horizontal direction, for a fixed prime $\mathfrak{p} \in Ω_A$, we give explicit and easily verifiable conditions on Drinfeld $ A$-modules $φ$ of rank $2$ which ensure the surjectivity of the $\mathfrak{p}$-adic Galois representation $ρ_{φ,\mathfrak{p}}$. This work not only extends the work of~\cite{Ray24} for $\mathfrak{p}=(T)$, but also obtains a variant of~\cite{Ray24} under comparatively simpler conditions in the case $\mathfrak{p}=(T)$. In the vertical direction, we show that for a fixed rank $2$ Drinfeld $A$-module $φ$, whose coefficients satisfy certain congruence and valuation conditions, the $\mathfrak{p}$-adic Galois representation $ρ_{φ,\mathfrak{p}}$ is surjective for all primes $\mathfrak{p} \in Ω_A$. This recovers the example of \cite{Zyw11} and yields new examples beyond those considered in \cite{Zyw25}. As a consequence, we obtain the surjectivity of the associated adelic Galois representation.
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}.