发表机构
Technische Universität Darmstadt; University of Wisconsin Madison; Renmin University of China（达姆施塔特工业大学; 威斯康星大学麦迪逊分校; 中国人民大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究具有双二次CM域复乘的亏格二曲线，通过正交Shimura簇上的CM环显式描述其雅可比簇，推导稳定坏约化素数的上界并证明其存在性。
AI 中文摘要
我们研究其雅可比簇具有双二次CM域复乘的亏格二曲线。在此情形下，雅可比簇分解为两个椭圆曲线的乘积，这两个椭圆曲线具有同一虚二次域中序的复乘。我们根据CM序的判别式推导了此类曲线的稳定坏约化素数的上界。为此，我们通过签名(3,2)的正交Shimura簇上的小CM环给出了具有双二次域复乘的主极化阿贝尔曲面的显式描述。我们利用该描述证明了此类CM环与Humbert曲面之间的算术相交公式，该公式以不相干Eisenstein级数的系数和三元正定二次型的表示数表示。最后，利用高阶Green函数的算术性质，我们证明了稳定坏约化素数总是存在的。
英文摘要
We study genus two curves whose Jacobians have complex multiplication by a biquadratic CM field. In this setting the Jacobian decomposes as a product of two elliptic curves with complex multiplication by orders in the same imaginary quadratic field. We derive upper bounds for the primes of stable bad reduction of such curves in terms of the discriminants of the CM orders. To do so, we give an explicit description of principally polarized abelian surfaces with CM by a biquadratic field via small CM cycles on an orthogonal Shimura variety of signature (3,2). We use it to prove an arithmetic intersection formula between such CM cycles with Humbert surfaces in terms of coefficients of incoherent Eisenstein series and representation numbers of ternary positive definite quadratic forms. Finally, employing the arithmetic properties of higher Green functions, we show that primes of stable bad reduction always exist.
Comments43 pages
