Schläfli modular equations for generalized Weber functions

Abstract

Abstract. Sets of appropriately normalized eta quotients, that we call level n Weber functions, are defined, and certain identities generalizing Weber function identities are proved for these functions. Schläfli type modular equations are explicitly obtained for Generalized Weber Functions associated with a Fricke group Γ 0 (n) + , for n = 2, 3, 5, 7, 11, 13 and 17.

Class invariants from a new kind of Weber-like modular equation

“…This was essentially the program of [8] (though the results are somewhat different in character to the current ones). For now however, we move on to higher class numbers.…”

“…which has roots f(ω) 8 , −f 1 (ω) 8 and −f 2 (ω) 8 , to calculate minimal polynomials for f(ω) at certain values of ω. However this method only works for him in situations where he can explicitly calculate the value γ 2 (ω) and where it has a 'nice' value, such as a rational integer.…”

Modularity of Galois traces of class invariants

“…The Siegel function has a long history [9,18] and the study on its 24-th root of unity for arbitrary n can be found in [12,15,16]. The functions in (3.7) are modular of level 72 and the action of S and T again permutes them up to multiplication by some roots of unity:…”

Evaluation of the Dedekind Eta Function

“…In particular, in this section we consider a case with fundamental discriminant d = −56, but where the class number is four. In fact there are four reduced binary quadratic forms for this discriminant, (1,0,14), (2,0,7), (3,2,5) and (3,-2,5).…”

“…In §3 we report on a solution for the case of fundamental discriminant and class number five (d = −47), using a known evaluation of a Weber function. In §4 we use the results of [14] on modular equations for new Weber-like functions to provide an evaluation where the class number is seven.…”