On the periods of abelian integrals and a formula of Chowla and Selberg
On the periods of abelian integrals and a formula of Chowla and Selberg
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关于阿贝尔积分的周期以及 Chowla 和 Selberg 公式
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发表时间：
1978
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通讯作者：
B. Gross
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作者：
B. Gross
给定一个判别式d的虚二次域k，设E是定义在Q上的椭圆曲线，Q是C中Q的代数闭包，它允许k中某阶的复数乘法。设~是定义在Q上的E上的第一类非零微分。我们可以把~o E看作是一个全纯1-形式，它的有理周期格的维数为1/k，因此要求这个格的“超越因子”是合理的。由于所有在k中具有非平凡乘子的曲线在Q上是同构的，所以这个因子将只依赖于乘法域，直到一个代数数。这个问题是解决了Chowla和Selberg在1949年[C-S]。让
Given an imaginary quadratic field k of discriminant d , let E be an elliptic curve defined over Q, the algebraic closure of Q in C, which admits complex multiplication by some order in k. Let ~ be a non-zero differential of the first kind on E, defined over Q. We may consider ~o E as a holomorphic 1-form whose rational period lattice has dimension 1 over k; it is therefore reasonable to ask for the "transcendental factor" of this lattice. Since all curves with non-trivial multipliers in k are isogenous over Q, this factor will depend, up to an algebraic number, only on the field of multiplication. This problem was solved by Chowla and Selberg in 1949 [C-S]. Let