On Russell-Type Modular Equations

Abstract

Abstract. In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell's theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature 3.

Schläfli modular equations for generalized Weber functions

“…Note that ν n is a root of unity, and in fact it is, at worst, an s(n)-th root of unity with s(n) defined as above, (5).…”

“…Modular equations of this type have also been found by Watson [16]. Modular equations also remain a topic of active interest; see for example the work of Chan and Liaw [5], [6]. For more information on the literature associated with modular equations and class invariants, the reader can very profitably consult Berndt's book [2].…”

A certain quotient of eta-functions found in Ramanujan’s lost notebook

“…The modular equation of degree 17 with which we start was proved by Chan and W.-C. Liaw [15] and is given by , after a slight rearrangement.…”

“…In Section 5, we employ recent discoveries of Chan and W.-C. Liaw [15], [20] on Russell-type modular equations of degrees 13, 17, and 19 in the theory of signature 3. In these two sections, we also determine the values of λ 5 , λ 7 , λ 11 , and λ 17 .…”

Cubic singular moduli, Ramanujan’s class invariants λnand the explicit Shimura Reciprocity Law

“…They determined α * n for n = 2, 3, 4, 5 and 6 from known values of Ramanujan-Weber class invariants G 3n and g 6n , and deduced three new series for 1/π corresponding to n = 2, 3 and 6. Recently, Chan and Liaw [16] succeeded in evaluating α * n for n = 2, 5, 7, 11, and 23 using cubic Russell-type modular equations. From the values of α * 7 and α * 11 , they discovered that when 3n is an Euler convenient number, α * n can be determined using Kronecker's Limit Formula.…”