On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$

[2502.19084] On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2502.19084 (math) [Submitted on 26 Feb 2025 ( v1 ), last revised 10 Apr 2026 (this version, v2)] Title: On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$ Authors: Narasimha Kumar , Dwipanjana Shit View a PDF of the paper titled On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$, by Narasimha Kumar and 1 other authors View PDF HTML (experimental) Abstract: 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 $\varphi$ 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 $\varphi$, where $\mathfrak{p} \in \Omega_A:= \mathrm{Spec}(A) \setminus \{ (0) \}$. There are two directions to investigate this problem: one by fixing the prime $\mathfrak{p}$, and the other by fixing $\varphi$. In the horizontal direction, for a fixed prime $\mathfrak{p} \in \Omega_A$, we give explicit and easily verifiable conditions on Drinfeld $ A$-modules $\varphi$ of rank $2$ which ensure the surjectivity of the $\mathfrak{p}$-adic Galois representation $\rho_{\varphi,\mathfrak{p}}$. 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 $\varphi$, whose coefficients satisfy certain congruence and valuation conditions, the $\mathfrak{p}$-adic Galois representation $\rho_{\varphi,\mathfrak{p}}$ is surjective for all primes $\mathfrak{p} \in \Omega_A$. 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. Comments: The title of the article has been slightly revised, while the results have been significantly generalized to a broader framework. Additionally, the overall presentation has been substantially enhanced for greater clarity and impact Subjects: Number Theory (math.NT) MSC classes: Primary 11G09, 11F52, Secondary 11F80 Cite as: arXiv:2502.19084 [math.NT] (or arXiv:2502.19084v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2502.19084 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Narasimha Kumar Dr. [ view email ] [v1] Wed, 26 Feb 2025 12:23:20 UTC (18 KB) [v2] Fri, 10 Apr 2026 04:09:34 UTC (22 KB) Full-text links: Access Paper: View a PDF of the paper titled On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$, by Narasimha Kumar and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-02 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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