On a generating function of the basis of weakly holomorphic functions on an elliptic curve

Abstract

Let $\Gz$ be a cofinite Fuchsian group whose quotient $\Gz\backslash\HH$ has genus one and a single cusp, normalized to be at $\infty$ with width one. Let $\Xg$ be its smooth compactification, which is an elliptic curve over $\CC$. Let $x,y$ be canonical generators of the function field on $\Xg$, which have poles of order $2,3$ respectively at the cusp and which {satisfy} a generalized Weierstrass relation $y^2+(a_{1}x+a_3)y=x^3+a_2{x}^2+a_{4}x+a_6$. Let $f$ be the unique (up to scale) weight two cusp form for $\Gz$, normalized so that the associated holomorphic differential is $\omega =dx/(2y+a_{1}x+a_3)$. With all this, we prove that the generating function identity \[ \frac{\bigl(y(τ)+y(z)+a_1x(z)+a_3\bigr)f(z)}{x(z)-x(τ)}=\sum_{m\ge0}Φ_m(τ)q_z^m \] defines a family $\{{\Phi }_m{\}}_{m\ge 0}$ of weakly holomorphic modular functions on $\Xg$ such that the set $\{{\Phi }_0\}\cup \{{\Phi }_m{\}}_{m\ge 2}$ is a canonical basis of the space $M_{0,\Gz}^{!,\infty}$ of weakly holomorphic modular functions on $\Xg$ with poles supported only at the cusp. As an application, by comparing the generating function of the family $\{{\Phi }_m{\}}_{m\ge 0}$ with the generating function of the Niebur--Poincaré series, we show that the generating function of the special values of the Kloosterman zeta function at $1$ is the holomorphic part of a certain weight two harmonic Maass form up to an additive constant.