Adelic Galois representations are maximal for almost all Drinfeld modules
Let $r\geq 3$ be an integer, let $q$ be an odd prime power, and set $A=\mathbb{F}_q[T]$ and $F=\mathbb{F}_q(T)$. We study the Galois representations associated with rank $r$ Drinfeld modules of the form $\phi_T=T+g_1\tau+\cdots+g_r\tau^r$, where $g_i\in A$ and $g_r\neq 0$. Ordering the coefficient tuples $\vec{g}=(g_1,\ldots,g_r)$ by their maximum degree, we prove that, for a density-one set of tuples, the $\lambda$-adic Galois representation is surjective for every prime $\lambda$ of $A$, and the adelic Galois image contains $\mathrm{SL}_r(\widehat A)$. Moreover, for every tuple in this…
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