Uniformization of Some Quotients of Modular Curves

In the paper 10] (see also 2]), the authors construct a holomorphic map from the modular curve X(k) = H 2 =?(k), where ?(k) is the congruence subgroup of ? = PSL(2; Z), into the projective space C P (k?3 2) , via theta functions. This holomorphic map is equivariant with respect to the action of PSL(2; Z=k) on both factors, and extends to the conformal compactiication X(k). Thus, X(k) can be exhibited as an algebraic curve in C P (k?3 2). One can produce from the theta functions explicit algebraic identities which are satissed by the image, and in favorable circumstances one can show that this map is an embedding. The object of this note is to extend the discussion to some quotients of the curves X(k). Indeed, let X 1 (k) be the quotient of X(k) by the group generated by the matrix B = 1 1 0 1 ! ; and for k = p a prime congruent to 1 (mod 4), let X 2 (k) be the quotient of X 1 (k) by the involution A p = x 1 ?cp ? x ! ; where x is an integer such that x 2 + 1 = cp for some c. Our main result is the construction of a holomorphic map from X 2 (p) to C P (p?5 4). For some low values of p (p = 13 and 17), we explicitly calculate the equations satissed by the image. In these examples, it follows from the discussion that the holomorphic map is an embedding.