Hankel determinants of Eisenstein series

Abstract

In this paper we prove Garvan's conjectured formula for the square of the modular discriminant $\Delta$ as a 3 by 3 Hankel determinant of classical Eisenstein series $E_{2n}$. We then obtain similar formulas involving minors of Hankel determinants for $E_{2r}\Delta^m$, for $m=1,2,3$ and $r=2,3,4,5,7$, and $E_{14}\Delta^4$. We next use Mathematica to discover, and then the standard structure theory of the ring of modular forms, to derive the general form of our infinite family of formulas extending the classical formula for $\Delta$ and Garvan's formula for $\Delta^2$. This general formula expresses the $n\times n$ Hankel determinant $\det(E_{2(i+j)}(q))_{1\leq i,j\leq n}$ as the product of $\Delta^{n-1}(\tau)$, a homogeneous polynomial in $E_4^3$ and $E_6^2$, and if needed, $E_4$. We also include a simple verification proof of the classical 2 by 2 Hankel determinant formula for $\Delta$. This proof depends upon polynomial properties of elliptic function parameters from Jacobi's Fundamenta Nova. The modular forms approach provides a convenient explanation for the determinant identities in this paper.