Ternary Quadratic Forms, Modular Equations and Certain Positivity Conjectures

TL;DR

This paper links Ramanujan's modular equations to ternary quadratic forms, proving several positivity conjectures and discovering new modular equations through identities involving eta-quotients and the concept of S-genus.

Contribution

It introduces the notion of S-genus of ternary quadratic forms and proves positivity inequalities and new modular equations, advancing understanding of quadratic forms and modular identities.

Findings

Proved positivity inequalities for ternary quadratic forms.

Discovered new modular equations of degrees 5 and 7.

Introduced the concept of S-genus and its relation to binary quadratic forms.

Abstract

We show that many of Ramanujan's modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2: x,y,zin Z}|, just to mention one among many similar positive results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author stating that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 7y^2 + 7z^2: x,y,z in Z}|. We prove a variety of identities for certain ternary forms with discriminants 144,400, 784,3600 by converting these into identities for appropriate eta- quotients. In the process we discover and prove a few new modular equations of degree 5 and 7. For any square free odd integer S with prime factorization p_1.....p_r, we define the S-genus as a union of 2^r specially selected genera of ternary quadratic forms, all with discriminant 16 S^2. This notion of S-genus arises naturally in the course of our investigation. It entails an interesting injection from genera of binary quadratic forms with discriminant -8 S to genera of ternary quadratic forms with discriminant 16 S^2.