Ramanujan and the Modular j-Invariant

Abstract

Abstract.A new infinite product t n was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about t n by establishing new connections between the modular jinvariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, t n generates the Hilbert class field of Q( √ −n). This shows that t n is a new class invariant according to H. Weber's definition of class invariants.

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

“…The first representation of λ n in (1.4) suggests connections between λ n and Ramanujan's alternative cubic theory. In fact, Berndt and Chan [5] have recently found such a relationship. In Section 8, applying one of their results, we establish an explicit formula for the j-invariant in terms of λ n , and evaluate several values of the j-invariant.…”

“…From the relation between the j-invariant and the modulus √ α in the cubic theory [5] (8.5) j(3τ ) = 27 (1 + 8(1 − α)) 3 (1 − α)α 3 , q = e 2πiτ , and (8.3), we deduce that (8.6) j(3τ ) = 27(9s + 1) 3 s + 1 s .…”

“…The function λ n had been briefly introduced earlier in his third notebook [25, p. 393], where Ramanujan offered a formula for λ n in terms of Klein's j-invariant, first proved by Berndt and Chan [5], [3,p. 318,Entry 11.21] by using Ramanujan's cubic theory of elliptic functions to alternative bases.…”

ON EVALUATIONS OF THE MODULAR j-INVARIANT BY MODULAR EQUATIONS OF DEGREE 2

“…These polynomials are extremely simple, whereas the corresponding polynomials of the same degrees satisfied by J n are more complicated. Refer [4] to see that if n is square free, n ≡ 11 (mod 24), and the class number of the Hilbert class field is odd, then t n and J n satisfy irreducible polynomials of the same degree. Since modular equations are crucial in this study on evaluations of the modular j-invariant in terms of J n , we now give a definition of a modular equation.…”

Some Eisenstein Series Identities

“…Identities (1.3) (1.6) have important applications. First, they are used to derive the interesting formulas [3] j({)= 27a 3 (q)(a 3 (q)+8c 3 (q))…”