{"id":"hexagon/2610.00365","pdf":"https://hexagonmath.org/pdf/2610.00365v1","url":"https://hexagonmath.org/abs/2610.00365v1","tags":["math.AG","msc:11G09","Drinfeld modules","Galois representations"],"score":0,"title":"Adelic Galois representations are maximal for almost all Drinfeld modules","domain":"math.NT","source":"hexagon","authors":["Anwesh Ray"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-18","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}