A Fricke transformation for cubic-residue eta products of level 13
Abstract
We prove an explicit Fricke transformation for the two-dimensional space generated by normalized reciprocal products on the cubic-residue cosets modulo 13. The transformation matrix is a scalar multiple of a matrix of differences of cubic Gaussian periods. Its projective action is defined over the cyclic cubic field, whereas the normalized matrix is defined over the real cyclotomic field. The proof uses generalized eta functions, a modular unit of degree two on X1(13), and a five-coefficient identity. A separate divisor argument lifts the projective transformation to the asserted linear transformation. We also record the Fricke images at an arbitrary prime p≡1pmod3 and prove that their normalized sine constants belong to the associated cyclic cubic subfield.
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BibTeX
@misc{hakimoglubrown2026fricke,
title = {{A Fricke transformation for cubic-residue eta products of level 13}},
author = {Cetin Hakimoglu-Brown},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06804},
eprint = {2609.06804},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06804},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Fricke transformation for cubic-residue eta products of level 13
AU - Cetin Hakimoglu-Brown
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06804
UR - https://arxiv.org/abs/2609.06804
AB - We prove an explicit Fricke transformation for the two-dimensional space generated by normalized reciprocal products on the cubic-residue cosets modulo 13. The transformation matrix is a scalar multiple of a matrix of differences of cubic Gaussian periods. Its projective action is defined over the cyclic cubic field, whereas the normalized matrix is defined over the real cyclotomic field. The proof uses generalized eta functions, a modular unit of degree two on X_1(13), and a five-coefficient identity. A separate divisor argument lifts the projective transformation to the asserted linear transformation. We also record the Fricke images at an arbitrary prime p≡1pmod3 and prove that their normalized sine constants belong to the associated cyclic cubic subfield.
ER -