Geodesic
Singular Moduli of Shimura Curves
Canadian journal of mathematics, Tome 63 (2011) no. 4, pp. 826-861

Voir la notice de l'article provenant de la source Cambridge

DOI

The $j$ -function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over $\mathbf{Q}$ . This paper shows that the coordinate maps given by N. Elkies for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.
DOI : 10.4153/CJM-2011-023-7
Mots-clés : 11G18, 11F12 
Errthum, Eric. Singular Moduli of Shimura Curves. Canadian journal of mathematics, Tome 63 (2011) no. 4, pp. 826-861. doi: 10.4153/CJM-2011-023-7
@article{10_4153_CJM_2011_023_7,
     author = {Errthum, Eric},
     title = {Singular {Moduli} of {Shimura} {Curves}},
     journal = {Canadian journal of mathematics},
     pages = {826--861},
     year = {2011},
     volume = {63},
     number = {4},
     doi = {10.4153/CJM-2011-023-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-023-7/}
}
TY  - JOUR
AU  - Errthum, Eric
TI  - Singular Moduli of Shimura Curves
JO  - Canadian journal of mathematics
PY  - 2011
SP  - 826
EP  - 861
VL  - 63
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-023-7/
DO  - 10.4153/CJM-2011-023-7
ID  - 10_4153_CJM_2011_023_7
ER  - 
%0 Journal Article
%A Errthum, Eric
%T Singular Moduli of Shimura Curves
%J Canadian journal of mathematics
%D 2011
%P 826-861
%V 63
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-023-7/
%R 10.4153/CJM-2011-023-7
%F 10_4153_CJM_2011_023_7

Cité par Sources :