On Singular Moduli for Arbitrary Discriminants
On Singular Moduli for Arbitrary Discriminants
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DOI：
10.1093/imrn/rnu223
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发表时间：
2012-06
影响因子：
1
通讯作者：
K. Lauter;B. Viray
中科院分区：
文献类型：
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作者：
K. Lauter;B. Viray
设d1和d2是不同二次虚阶Od1和Od2的判别式，设J（d1, d2）表示CM J不变量与判别式d1和d2的差之积。1985年，Gross和Zagier给出了整数J（d1, d2）在d1和d2是最大阶的相对素数和判别式的情况下的一个优雅的分解公式。为了计算这个公式，他们首先将问题简化为计算Od1和Od2同时嵌入到超奇异曲线的自同态环中的次数，然后解决这个计数问题。有趣的是，这个计数问题也出现在计算2属曲线不变量的类多项式时。然而，在这个应用程序中，必须考虑阶Od1和阶Od2是非最大值。将Gross和Zagier的方法推广到2属曲线上，给出了任意一对判判式d1 6= d2和任意素数> 2的v ' （J(d1, d2)）的可计算公式。在d1是无平方的，d2是任意二次虚阶的判判式的情况下，我们的公式可以用一个简单的封闭形式表示。当d1和d2的导体是相对素数时，我们也给出了一个猜想的封闭公式。
Let d1 and d2 be discriminants of distinct quadratic imaginary orders Od1 and Od2 and let J(d1, d2) denote the product of differences of CM j-invariants with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1, d2) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. To compute this formula, they first reduce the problem to counting the number of simultaneous embeddings of Od1 and Od2 into endomorphism rings of supersingular curves, and then solve this counting problem. Interestingly, this counting problem also appears when computing class polynomials for invariants of genus 2 curves. However, in this application, one must consider orders Od1 and Od2 that are non-maximal. Motivated by the application to genus 2 curves, we generalize the methods of Gross and Zagier and give a computable formula for v`(J(d1, d2)) for any pair of discriminants d1 6= d2 and any prime ` > 2. In the case that d1 is squarefree and d2 is the discriminant of any quadratic imaginary order, our formula can be stated in a simple closed form. We also give a conjectural closed formula when the conductors of d1 and d2 are relatively prime.