Abstract
Given an imaginary quadratic field k of discriminant d , let E be an elliptic curve defined over Q, the algebraic closure of Q in C, which admits complex multiplication by some order in k. Let ~ be a non-zero differential of the first kind on E, defined over Q. We may consider ~o E as a holomorphic 1-form whose rational period lattice has dimension 1 over k; it is therefore reasonable to ask for the "transcendental factor" of this lattice. Since all curves with non-trivial multipliers in k are isogenous over Q, this factor will depend, up to an algebraic number, only on the field of multiplication. This problem was solved by Chowla and Selberg in 1949 [C-S]. Let
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