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Exceptional cases of adelic surjectivity for Drinfeld modules of rank 2. (English) Zbl 1501.11064

Let \(E\) be an elliptic curve over \({\mathbb Q}\) without complex multiplication. J.-P. Serre [Invent. Math. 15, 259–331 (1972; Zbl 0235.14012)] proved that the image of the Galois representation \[ \rho_E\colon \mathrm{Gal}(\bar{\mathbb Q}/{\mathbb Q})\longrightarrow \lim_{\substack{\longleftarrow\\ n}}\mathrm{Aut} (E[n])\cong \mathrm{GL}_2(\hat{\mathbb Z}) \] is an open subgroup of \(\mathrm{GL}_2(\hat{\mathbb Z})\) and that the adelic surjectivity is never attained for elliptic curves over \({\mathbb Q}\).
The author considers the analogue problem for global function fields. Let \(A={\mathbb F}_q[T]\) and \(F={\mathbb F}_q(T)\). The natural question is to ask whether there is a Drinfeld \(A\)-module \(\phi\) over \(F\) of rank \(r\) with generic characteristic whose adelic Galois representation \[ \rho_{\phi}\colon G_K\longrightarrow \lim_{\substack{\longleftarrow\\ {\mathfrak a}}} \mathrm{Aut}(\phi[{\mathfrak a}])\cong \mathrm{GL}_r(\hat{A}) \] is surjective. D. Zywina [“Drinfeld modules with maximal Galois action on their torsion points”, Preprint, arXiv:1110.4365] proved the adelic surjectivity for a Drinfeld \(A\)-module over \(F\) of rank \(2\) for an odd prime power \(q\geq 5\). The author [J. Number Theory 237, 99–123 (2022; Zbl 1502.11065)] proved the adelic surjectivity for a Drinfeld \(A\)-module over \(F\) of rank \(3\) for \(q=p^e\) a prime power with \(p\geq 5\) and \(q \equiv 1\bmod 3\).
In this paper, the author considers the cases \(q=2, 3\) and \(2^e \geq 4\). The results are the following:
\(\bullet\) Theorem 4.4. If \(q=2\) and \(\phi_T=T+g_1\tau +g_2\tau^2\) with \(g_1\in F\), \(g_2\in F^*\), then \(\rho_{\phi}\) is not surjective.
\(\bullet\) Theorem 5.4. If \(q=3\) and \(\phi_T=T+g_1\tau+g_2\tau^2\) where \(g_1\in F\), \(g_2\in F^*\) and \(-g_2\) is a nonsquare, then \(\rho_{\phi}\) is not surjective.
\(\bullet\) Theorem 5.7. If \(q=3\) and \(\phi_T=T+\tau-T^2\tau^2\), then \(\rho_{\phi}\) is surjective.
\(\bullet\) Theorem 6.2. If \(q=2^e\geq 4\) and \(\phi_T=T+\tau-T^{q-1} \tau^2\), then \(\rho_{\phi}\) is surjective.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R32 Galois theory
11R37 Class field theory
11R58 Arithmetic theory of algebraic function fields
11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)

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References:

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[10] Chien-Hua Chen Department of Mathematics Pennsylvania State University University Park, PA 16802, U.S.A. E-mail: cxc813@psu.edu
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