Theta Relations
Abstract
Motivated by the classical quartic and cubic theta identities, we establish a product formula for Siegel--Jacobi theta series associated with rationally isometric positive-definite quadratic spaces. Our main theorem expresses a product of theta series of arbitrary Siegel degree as a finite double character sum over quotients of commensurable lattices. For the root lattice A2, it yields a finite Fourier identity for cubes of cubic theta functions, a new proof of the two-variable cubic identity, and seven independent relations in degree two. A second specialization gives a new square analogue of the two-variable cubic identity. At the origin it reduces to a(q)a(q2)=b(q)b(q2)+c(q)c(q2); in degree two, the same seven relation patterns recur with each cube replaced by the corresponding product at τ and 2τ. We further extend the cubic identities on A2 to ℓ-th-power identities on Aℓ−1 for every ℓ≥3, and to 2ℓ-th-power identities on Aℓ−1 for every ℓ≥2, in arbitrary Siegel degree. The doubled-order construction recovers Jacobi's quartic identity and gives its analogues in arbitrary Siegel degree.
Related
- Geometric Proofs of Riemann-Type Theta Relations
- Period Relations for Theta Products and Elliptic Integral Moments
- Modularity of higher theta series I: cohomology of the generic fiber
- Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta-functions
- Local theta correspondence and generalized Shalika models
BibTeX
@misc{chen2026theta,
title = {{Theta Relations}},
author = {Rong Chen and Haigang Zhou and Xiao-Jie Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27959},
eprint = {2610.27959},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27959},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Theta Relations
AU - Rong Chen
AU - Haigang Zhou
AU - Xiao-Jie Zhu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27959
UR - https://arxiv.org/abs/2610.27959
AB - Motivated by the classical quartic and cubic theta identities, we establish a product formula for Siegel--Jacobi theta series associated with rationally isometric positive-definite quadratic spaces. Our main theorem expresses a product of theta series of arbitrary Siegel degree as a finite double character sum over quotients of commensurable lattices. For the root lattice A_2, it yields a finite Fourier identity for cubes of cubic theta functions, a new proof of the two-variable cubic identity, and seven independent relations in degree two. A second specialization gives a new square analogue of the two-variable cubic identity. At the origin it reduces to a(q)a(q^2)=b(q)b(q^2)+c(q)c(q^2); in degree two, the same seven relation patterns recur with each cube replaced by the corresponding product at τ and 2τ. We further extend the cubic identities on A_2 to ℓ-th-power identities on A_ℓ−1 for every ℓ≥3, and to 2ℓ-th-power identities on A_ℓ−1 for every ℓ≥2, in arbitrary Siegel degree. The doubled-order construction recovers Jacobi's quartic identity and gives its analogues in arbitrary Siegel degree.
ER -