An effective result of André–Oort type II

Lars Kühne¹

¹ SNS Pisa Piazza dei Cavalieri 7 56126 Pisa, Italy

Acta Arithmetica, Tome 161 (2013) no. 1, pp. 1-19.

Résumé

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.

Détail

DOI : 10.4064/aa161-1-1

Mots-clés : prove effective results andr oort type particular state certain uniform improvements main result hne ann math equation has solution singular moduli by product indicate simple trick rendering andr proof andr oort conjecture effective significantly aspect usage siegel tatuzawa theorem weak effective lower bound class number imaginary quadratic field given gross zagier results article partially announced nbsp above cited paper