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 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.