Finiteness of superelliptic curves with CM Jacobians
Finiteness of superelliptic curves with CM Jacobians
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CM Jacobian 超椭圆曲线的有限性
DOI：
10.24033/asens.2489
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Kang Zuo
中科院分区：
文献类型：
--
作者：
Ke Chen;Xin Lu;Kang Zuo
本文证明了超椭圆曲线的科尔曼猜想：在同构之前，最多有1000条超椭圆曲线的Jacobian是CM交换簇，只要这些曲线的亏格至少为8。这里超椭圆曲线是复的光滑射影曲线，它在射影直线上有一个循环分支覆盖。证明归结为Siegel模簇A_g中超椭圆Torelli轨迹TS_g的几何性质：我们建立了A_g中维数> 0的任何特殊子簇（g至少为8）从TS_g中的通有排除，并且与表面纤维化相关的Higgs丛的稳定性在我们的研究中起着至关重要的作用。
This paper proves the Coleman conjecture for superelliptic curves: there are, up to isomorphism, at most finitely many superelliptic curves whose Jacobians are CM abelian varieties, as long as these curves are of genus at least 8. Here superelliptic curves are complex smooth projective curves admitting a cyclic branched cover over the projective line. The proof is reduced to the geometry of superelliptic Torelli locus TS_g in the Siegel modular variety A_g : we establish the generic exclusion from TS_g of any special subvariety of dimension > 0 in A_g for g at least 8, and the stability properties of Higgs bundles associated to surface fibrations play a crucial role in our study.