Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 256-297 |
| Seitenumfang | 42 |
| Fachzeitschrift | Journal of number theory |
| Jahrgang | 243 |
| Frühes Online-Datum | 20 Okt. 2022 |
| Publikationsstatus | Veröffentlicht - Feb. 2023 |
Abstract
Let a∈Z>0 and ϵ1,ϵ2,ϵ3∈{±1}. We classify explicitly all singular moduli x1,x2,x3 satisfying either ϵ1x1a+ϵ2x2a+ϵ3x3a∈Q or (x1ϵ1x2ϵ2x3ϵ3)a∈Q×. In particular, we show that all the solutions in singular moduli x1,x2,x3 to the Fermat equations x1a+x2a+x3a=0 and x1a+x2a−x3a=0 satisfy x1x2x3=0. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establish.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
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in: Journal of number theory, Jahrgang 243, 02.2023, S. 256-297.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Equations in three singular moduli
T2 - The equal exponent case
AU - Fowler, Guy
N1 - Funding Information: Acknowledgements: The author would like to thank Jonathan Pila for helpful comments and the referee for a careful reading of this paper. The author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 945714 ).
PY - 2023/2
Y1 - 2023/2
N2 - Let a∈Z>0 and ϵ1,ϵ2,ϵ3∈{±1}. We classify explicitly all singular moduli x1,x2,x3 satisfying either ϵ1x1a+ϵ2x2a+ϵ3x3a∈Q or (x1ϵ1x2ϵ2x3ϵ3)a∈Q×. In particular, we show that all the solutions in singular moduli x1,x2,x3 to the Fermat equations x1a+x2a+x3a=0 and x1a+x2a−x3a=0 satisfy x1x2x3=0. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establish.
AB - Let a∈Z>0 and ϵ1,ϵ2,ϵ3∈{±1}. We classify explicitly all singular moduli x1,x2,x3 satisfying either ϵ1x1a+ϵ2x2a+ϵ3x3a∈Q or (x1ϵ1x2ϵ2x3ϵ3)a∈Q×. In particular, we show that all the solutions in singular moduli x1,x2,x3 to the Fermat equations x1a+x2a+x3a=0 and x1a+x2a−x3a=0 satisfy x1x2x3=0. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establish.
KW - Andre–Oort conjecture
KW - Singular moduli
UR - http://www.scopus.com/inward/record.url?scp=85140235166&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2105.12696
DO - 10.48550/arXiv.2105.12696
M3 - Article
AN - SCOPUS:85140235166
VL - 243
SP - 256
EP - 297
JO - Journal of number theory
JF - Journal of number theory
SN - 0022-314X
ER -