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PreprintCC BY 4.0arXiv

Geometric Proofs of Riemann-Type Theta Relations

  • Sander Zwegers
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Abstract

We present a geometric method for deriving theta function identities from automorphisms of positive definite quadratic forms. Given an automorphism of a quadratic form with matrix A, we construct a rational isometry of In⊗ A that fixes the diagonal subspace pointwise. Elementary lattice decompositions and finite character sums then yield explicit transformation formulas for the associated theta functions. As applications, we give a geometric proof of Schultz's generalization of the Borweins' cubic theta identity and recover two related quadratic theta identities.

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Cite

BibTeXRIS
BibTeX
@misc{zwegers2026geometric,
  title = {{Geometric Proofs of Riemann-Type Theta Relations}},
  author = {Sander Zwegers},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40408},
  eprint = {2610.40408},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40408},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Geometric Proofs of Riemann-Type Theta Relations
AU  - Sander Zwegers
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40408
UR  - https://arxiv.org/abs/2610.40408
AB  - We present a geometric method for deriving theta function identities from automorphisms of positive definite quadratic forms. Given an automorphism of a quadratic form with matrix A, we construct a rational isometry of I_n⊗ A that fixes the diagonal subspace pointwise. Elementary lattice decompositions and finite character sums then yield explicit transformation formulas for the associated theta functions. As applications, we give a geometric proof of Schultz's generalization of the Borweins' cubic theta identity and recover two related quadratic theta identities.
ER  -