math.NT — Number Theory

Adelic Galois representations are maximal for almost all Drinfeld modules

Contributed by Anwesh Ray

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 set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides $q-1$. These results extend Zywina's theorem in rank $2$ to arbitrary ranks $r\geq 3$. As a further consequence, we establish the existence of rank $r$ Drinfeld modules with surjective adelic Galois representations for every $r\geq 3$, extending previous existence results which were known for prime values of $r$.