An effective result of André–Oort type II
Volume 161 / 2013
Acta Arithmetica 161 (2013), 1-19 MSC: Primary 14G35; Secondary 11G18, 14G05. DOI: 10.4064/aa161-1-1
We prove some new effective results of André–Oort type. In particular, we state certain uniform improvements of the main result in [L. Kühne, Ann. of Math. 176 (2012), 651–671]. We also show that the equation $X+Y=1$ has no solution in singular moduli. As a by-product, we indicate a simple trick rendering André's proof of the André–Oort conjecture effective. A significantly new aspect is the usage of both the Siegel–Tatuzawa theorem and the weak effective lower bound on the class number of an imaginary quadratic field given by Gross and Zagier. The results of this article were partially announced in the above-cited paper.