TY - JOUR

T1 - Triples of singular moduli with rational product

AU - Fowler, Guy

N1 - Funding information: I would like to thank Jonathan Pila and Yuri Bilu for helpful comments and advice. This work was supported by an EPSRC doctoral scholarship.

PY - 2020/11/28

Y1 - 2020/11/28

N2 - We show that all triples (x1,x2,x3) of singular moduli satisfying x1x2x3 as× are "trivial". That is, either x1,x2,x3 as; some xi as and the remaining xj,xk are distinct, of degree 2, and conjugate over as; or x1,x2,x3 are pairwise distinct, of degree 3, and conjugate over as. This theorem is best possible and is the natural three-dimensional analogue of a result of Bilu, Luca and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the André-Oort conjecture for the family of subvarieties Vα as' a'3 defined by an equation x1x2x3 = α as.

AB - We show that all triples (x1,x2,x3) of singular moduli satisfying x1x2x3 as× are "trivial". That is, either x1,x2,x3 as; some xi as and the remaining xj,xk are distinct, of degree 2, and conjugate over as; or x1,x2,x3 are pairwise distinct, of degree 3, and conjugate over as. This theorem is best possible and is the natural three-dimensional analogue of a result of Bilu, Luca and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the André-Oort conjecture for the family of subvarieties Vα as' a'3 defined by an equation x1x2x3 = α as.

KW - André-Oort conjecture

KW - Singular modulus

KW - complex multiplication

UR - http://www.scopus.com/inward/record.url?scp=85094565462&partnerID=8YFLogxK

U2 - 10.1142/S1793042120501110

DO - 10.1142/S1793042120501110

M3 - Article

VL - 16

SP - 2149

EP - 2166

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 10

ER -