Difference of modular functions and their CM value factorization

Yang, Tonghai; Yin, Hongbo

doi:10.1090/tran/7479

Difference of modular functions and their CM value factorization
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by Tonghai Yang and Hongbo Yin PDF

Trans. Amer. Math. Soc. 371 (2019), 3451-3482 Request permission

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 $\Psi (\frac {d_1+\sqrt {d_1}}2) -\Psi (\frac {d_2+\sqrt {d_2}}2)$, where $\Psi$ is the $j$-invariant or the Weber invariant $\omega _2$. 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 $\omega _2$. The method used here could be extended to deal with other modular functions on a genus zero modular curve.

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