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 GzbackslashHH has genus one and a single cusp, normalized to be at ∞ 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 y2+(a1x+a3)y=x3+a2x2+a4x+a6. Let f be the unique (up to scale) weight two cusp form for Gz, normalized so that the associated holomorphic differential is ω=dx/(2y+a1x+a3). With all this, we prove that the generating function identity ((y(τ)+y(z)+a1x(z)+a3)f(z))/(x(z)−x(τ))=∑m≥0Φm(τ)qzm defines a family {Φm}m≥0 of weakly holomorphic modular functions on Xg such that the set {Φ0}∪{Φm}m≥2 is a canonical basis of the space M0,Gz!,∞ 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 {Φm}m≥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.
Related
- Holomorphic planes in M× 𝕋2 from weakly exact Lagrangians in M
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- K-holomorphic functions with definable real part
- Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces
BibTeX
@misc{friedman2026generating,
title = {{On a generating function of the basis of weakly holomorphic functions on an elliptic curve}},
author = {Joshua S. Friedman and Jay Jorgenson and Lejla Smajlović},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35382},
eprint = {2609.35382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35382},
}
RIS
TY - UNPB
TI - On a generating function of the basis of weakly holomorphic functions on an elliptic curve
AU - Joshua S. Friedman
AU - Jay Jorgenson
AU - Lejla Smajlović
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35382
UR - https://arxiv.org/abs/2609.35382
AB - Let Gz be a cofinite Fuchsian group whose quotient GzbackslashHH has genus one and a single cusp, normalized to be at ∞ 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_1x+a_3)y=x^3+a_2x^2+a_4x+a_6. Let f be the unique (up to scale) weight two cusp form for Gz, normalized so that the associated holomorphic differential is ω=dx/(2y+a_1x+a_3). With all this, we prove that the generating function identity ((y(τ)+y(z)+a_1x(z)+a_3)f(z))/(x(z)−x(τ))=∑_m≥0Φ_m(τ)q_z^m defines a family {Φ_m}_m≥0 of weakly holomorphic modular functions on Xg such that the set {Φ_0}∪{Φ_m}_m≥2 is a canonical basis of the space M_0,Gz^!,∞ 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 {Φ_m}_m≥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.
ER -