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arXiv · 1711.02983

Difference of modular functions and their CM value factorization

Tonghai Yang·Hongbo Yin
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Abstract

In this paper, we use Borcherds lifting and the big CM value formula of Bruinier, Kudla, and Yang to give an explicit factorization formula for the norm of $Ψ(\frac{d_1+\sqrt{d_1}}2) -Ψ(\frac{d_2+\sqrt{d_2}}2)$, where $Ψ$ is the $j$-invariant or the Weber invariant $ω_2$. The $j$-invariant case gives another proof of the well-known Gross-Zagier factorization formula of singular moduli, while the Weber invariant case gives a proof of the Yui-Zagier conjecture for $ω_2$. The method used here could be extended to deal with other modular functions on a genus zero modular curve.

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Tonghai Yang, Hongbo Yin. 2017-11-08. Difference of modular functions and their CM value factorization. https://arxiv.org/abs/1711.02983

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