The p-rank strata of the moduli space of hyperelliptic curves
The p-rank strata of the moduli space of hyperelliptic curves
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超椭圆曲线模空间的p阶层
DOI：
10.1016/j.aim.2011.04.004
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发表时间：
2009
影响因子：
1.7
通讯作者：
R. Pries
中科院分区：
文献类型：
--
作者：
Jeff Achter;R. Pries
证明了特征p⩾3中超椭圆曲线的模空间的边界与p阶层的交集的结果.这为我们分析亏格为g且p阶为f的超椭圆曲线的层Hgf提供了一个强有力的技巧.利用这一技巧，我们证明了亏格为g且p阶为f的一般超椭圆曲线的Jacobian的自同构环与Z同构当g⩾4时.此外，我们还证明了Hgf的每个不可约分支的Z/ℓ单列是辛群Sp 2 g(Z/ℓ)当g⩾3，ℓ≠p是奇素数(当f=0时，ℓ上有温和的假设)。这些结果给出了有限域上给定亏格和p-秩超椭圆曲线的一般行为的大量应用，包括牛顿多边形、绝对单雅可比、类群和Zeta函数的应用。
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p⩾ 3. This yields a strong technique that allows us to analyze the stratum H g f of hyperelliptic curves of genus g and p-rank f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and p-rank f is isomorphic to Z if g⩾ 4. Furthermore, we prove that the Z/ℓ-monodromy of every irreducible component of H g f is the symplectic group Sp 2 g (Z/ℓ) if g⩾ 3, and ℓ≠ p is an odd prime (with mild hypotheses on ℓ when f= 0). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.