On Colmez’s product formula for periods of CM-abelian varieties
On Colmez’s product formula for periods of CM-abelian varieties
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论科尔梅斯CM-阿贝尔品种时期的产品配方
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发表时间：
2011
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通讯作者：
Andrew Obus
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作者：
Andrew Obus
Colmez通过一个域$$K$$的复乘法，推测出了阿贝尔变周期的乘积公式，类似于代数数论中的标准乘积公式。他证明了这个猜想直到$$K/mathbb{Q }$$阿贝尔的有理2次方。在本文中，我们通过消去2的这个幂，完成了$$K/mathbb{Q }$$阿贝尔的Colmez证明。我们的证明依赖于分析混合特征$$(0,2)$$中费马曲线的De Rham上同调上的伽罗瓦作用，而这又依赖于理解在三个点分叉的投影线的$$mathbb Z /2^n$$ -覆盖的稳定约简。
Colmez conjectured a product formula for periods of abelian varieties with complex multiplication by a field $$K$$, analogous to the standard product formula in algebraic number theory. He proved this conjecture up to a rational power of 2 for $$K/mathbb{Q }$$ abelian. In this paper, we complete the proof of Colmez for $$K/mathbb{Q }$$ abelian by eliminating this power of 2. Our proof relies on analyzing the Galois action on the De Rham cohomology of Fermat curves in mixed characteristic $$(0,2)$$, which in turn relies on understanding the stable reduction of $$mathbb Z /2^n$$-covers of the projective line, branched at three points.