Ternary Quadratic Forms, Modular Equations, and Certain Positivity Conjectures

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 ∈ N, $$\begin{array}{rcl} & & \left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + {y}^{2} + {z}^{2} = n\right \}\right \vert \geq & &\left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + 3{y}^{2} + 3{z}^{2} = n\right \}\right \vert, \end{array}$$ just to name one among many similar “positivity” results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author, stating that for any n ∈ N, $$\begin{array}{rcl} & & \left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + {y}^{2} + {z}^{2} = n\right \}\right \vert \geq & &\left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + 7{y}^{2} + 7{z}^{2} = n\right \}\right \vert.\end{array}$$ We prove a number of identities for certain ternary forms with discriminants 144, 400, 784, or 3, 600 by converting every ternary identity into an identity for the appropriate η-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 16S 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 − 8S to genera of ternary quadratic forms with discriminant 16S 2.