Bad reduction of genus $2$ curves with CM jacobian varieties
Bad reduction of genus $2$ curves with CM jacobian varieties
复制标题
CM 雅可比品种的属 $2$ 曲线减少不良
作者：
P. Habegger;F. Pazuki
我们证明，在雅可比具有复数乘法的数域上的 genus $2$ 曲线通常会在某个素数处具有稳定的坏约简。我们通过两种不同的方式计算雅可比的法尔廷斯高度来证明这一点。首先，我们使用由 Colmez 和 Obus 提出的 Colmez 猜想的已知情况，当 CM 场是有理数的阿贝尔扩展时，该猜想是有效的。它将 $L$ 函数的高度和对数导数联系起来。第二个公式涉及基于超椭圆模型将高度分解为局部项。我们使用 Igusa、Liu、Saito 和 Ueno 提出的 genus $2$ 曲线约简理论，将有限位置的贡献与曲线的稳定不良约简联系起来。 Michel 和 Venkatesh 的次凸界与 Zhang 的等分布结果一起用于限制无限位置。
We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a known case of the Colmez conjecture, due to Colmez and Obus, that is valid when the CM field is an abelian extension of the rationals. It links the height and the logarithmic derivatives of an $L$ -function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use the reduction theory of genus $2$ curves as developed by Igusa, Liu, Saito, and Ueno to relate the contribution at the finite places with the stable bad reduction of the curve. The subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang are used to bound the infinite places.