On the torsion of Drinfeld modules of rank two
On the torsion of Drinfeld modules of rank two
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关于二阶 Drinfeld 模的扭转
DOI：
10.1515/crelle.2010.017
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Ambrus Pál
中科院分区：
文献类型：
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作者：
Ambrus Pál
证明了曲线Y0(？)有无？2(T)-有理点，其中？⊲？2[T]是次数至少为3的素数理想，且Y0(？)仿射Drinfeld模曲线是否对具有Hecke型层次？-结构的一般特征的2阶Drinfeld模进行了参数化处理？作为结果，我们得到了Schweizer的一个猜想，它完全描述了？2(T)上的二阶Drinfeld模的挠率，并在这种情况下隐含了一致有界性猜想。我们用形式浸没方法的一种变体得出我们的结果。此外，我们证明了群Aut(X0(？))有第二个订单。作为我们方法的进一步应用，我们还确定了X0(？)在哪里？⊲？Q[T]是次数至少为3的素数理想，Q是素数p的幂。
Abstract We prove that the curve Y 0(?) has no ?2(T)-rational points where ? ⊲ ?2[T] is a prime ideal of degree at least 3 and Y 0(?) is the affine Drinfeld modular curve parameterizing Drinfeld modules of rank two over ?2[T] of generic characteristic with Hecke-type level ?-structure. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over ?2(T) implying the uniform boundedness conjecture in this particular case. We reach our results with a variant of the formal immersion method. Moreover we show that the group Aut(X 0(?)) has order two. As a further application of our methods we also determine the prime-to-p cuspidal torsion packet of X 0(?) where ? ⊲ ? q [T] is a prime ideal of degree at least 3 and q is a power of the prime p.