Generators and equations for modular function fields of principal congruence subgroups

Defining equations of modular curves

“…Thus, the relation between these two functions defines the curve X 1 (N ). A similar method is also used to obtain defining equations of X(N) by Ishida [11]. In general, though, the degree of the equations obtained in this fashion is not optimal.…”

“…×(6Y 4 + 13Y 3 + 12Y 2 + 5Y + 1)X 11 12 X = 5/1, Y = 4 · 6/1 2 Y 12 = X(X − 1) 2 (X + 1) 6 (X 2 + 1) 4 (X 2 − X + 1) 3…”

Class invariants and cyclotomic unit groups from special values of modular units

“…In [11], the author gives a pair of generators X 2 and X 3 for F , for prime ≥ 5, (see also [9,10] for the more general setting) with q−product expansions…”

Remarks for Basic Appell Series

“…where K u,v (τ ) are Klein forms of level N ( [3], [4], [5], [6] and [11]). In a neighborhood of the cusp i∞ of Γ(2N 2 ), the function X r (τ ) has an infinite product expansion:…”