Class invariants by Shimura's reciprocity law

Alice Gee

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Let $K$ be a quadratic imaginary number field of discriminant $d$ . For $t \in ℕ$ let $𝔒_{t}$ denote the order of conductor $t$ in $K$ and $j (𝔒_{t})$ its modular invariant which is known to generate the ring class field modulo $t$ over $K$ . The coefficients of the minimal equation of $j (𝔒_{t})$ being quite large Weber considered in [We] the functions $f, f_{1}, f_{2}, γ_{2}, γ_{3}$ defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of...