Details
| Original language | English |
|---|---|
| Pages (from-to) | 2149-2166 |
| Number of pages | 18 |
| Journal | International Journal of Number Theory |
| Volume | 16 |
| Issue number | 10 |
| Publication status | Published - 28 Nov 2020 |
| Externally published | Yes |
Abstract
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.
Keywords
- André-Oort conjecture, Singular modulus, complex multiplication
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: International Journal of Number Theory, Vol. 16, No. 10, 28.11.2020, p. 2149-2166.
Research output: Contribution to journal › Article › Research › peer review
}
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 -