Explicit Drinfeld moduli schemes and Abhyankar's Generalized Iteration Conjecture

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• 作者：Florian Breuer ^{fbreuer@sun.ac.za" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Drinfeld modules ; Drinfeld moduli schemes ; Galois groups
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：432-450
• 全文大小：382 K

文摘

Let k   be a field containing F_q${\mathbb{F}}_q$. Let ψ be a rank r   Drinfeld F_q[t]${\mathbb{F}}_q[t]$-module determined by ψ_t(X)=tX+a₁X^q+⋯+a_r−1X^qr−1+X^qr${\psi }_t(X)=tX+a_1{X}^q+\cdots +a_{r-1}{X}^{q^{r-1}}+X^{q^r}$, where t,a₁,…,a_r−1$t,a_1,\dots ,a_{r-1}$ are algebraically independent over k  . Let n∈F_q[t]$n\in {\mathbb{F}}_q[t]$ be a monic polynomial. We show that the Galois group of 145b119009754e0bf00767718f0" title="Click to view the MathML source">ψ_n(X)${\psi }_n(X)$ over k(t,a₁,…,a_r−1)$k(t,a_1,\dots ,a_{r-1})$ is isomorphic to GL_r(F_q[t]/nF_q[t])${\mathrm{GL}}_r({\mathbb{F}}_q[t]/n{\mathbb{F}}_q[t])$, settling a conjecture of Abhyankar. Along the way we obtain an explicit construction of Drinfeld moduli schemes of level tn.

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