On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$
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| Format: | Preprint |
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2025
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| _version_ | 1866915928098209792 |
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| author | Kumar, Narasimha Shit, Dwipanjana |
| author_facet | Kumar, Narasimha Shit, Dwipanjana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$ Kumar, Narasimha Shit, Dwipanjana Number Theory Primary 11G09, 11F52, Secondary 11F80 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. |
| title | On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$ |
| topic | Number Theory Primary 11G09, 11F52, Secondary 11F80 |
| url | https://arxiv.org/abs/2502.19084 |