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

Abstract

Farkas, Kra and Kopeliovich (Commun. Anal. Geom. 4(2):207–259, 1996) showed that the quotients F1 and F2 of modified theta functions generate the function field \(\mathcal{K}(X(p))\) of the modular curve X(p) for a principal congruence subgroup Γ(p) with prime p≥7. For such primes p we first find affine models of X(p) over ℚ represented by Φp(X,Y)=0, from which we are able to obtain the algebraic relations Ψp(X,Y)=0 of F1 and F2 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.