Explicit Drinfeld moduli schemes and Abhyankar's Generalized Iteration Conjecture

TextLet k   be a field containing FqFq. Let ψ be a rank r   Drinfeld Fq[t]Fq[t]-module determined by ψt(X)=tX+a1Xq+⋯+ar−1Xqr−1+Xqrψt(X)=tX+a1Xq+⋯+ar−1Xqr−1+Xqr, where t,a1,…,ar−1t,a1,…,ar−1 are algebraically independent over k  . Let n∈Fq[t]n∈Fq[t] be a monic polynomial. We show that the Galois group of ψn(X)ψn(X) over k(t,a1,…,ar−1)k(t,a1,…,ar−1) is isomorphic to GLr(Fq[t]/nFq[t])GLr(Fq[t]/nFq[t]), settling a conjecture of Abhyankar. Along the way we obtain an explicit construction of Drinfeld moduli schemes of level tn.VideoFor a video summary of this paper, please visit https://youtu.be/TInrNq02-UA.