Affine models of the modular curves X(p) and its application

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Farkas, Kra and Kopeliovich (Commun. Anal. Geom. 4(2):207-259, 1996) showed that the quotients F(1) and F(2) of modified theta functions generate the function field K(X(p)) of the modular curve X(p) for a principal congruence subgroup Gamma(p) with prime p >= 7. For such primes p we first find affine models of X(p) over Q represented by Phi(p) (X,Y) = 0, from which we are able to obtain the algebraic relations Psi(p) (X,Y) = 0 of F(1) and F(2) presented by Farkas et al. As its application we construct the ray class field K((p)) modulo p over an imaginary quadratic field K and then explicitly calculate its class polynomial by using the Shimura reciprocity law.
Publisher
SPRINGER
Issue Date
2011-02
Language
English
Article Type
Article
Keywords

CLASS FIELDS

Citation

RAMANUJAN JOURNAL, v.24, no.2, pp.235 - 257

ISSN
1382-4090
DOI
10.1007/s11139-010-9240-7
URI
http://hdl.handle.net/10203/95208
Appears in Collection
MA-Journal Papers(저널논문)
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