Year: 2026
Type: preprint
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 $X_1(13)$, 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 $p\equiv1\pmod3$ and prove that their normalized sine constants belong to the associated cyclic cubic subfield.
Source: arXiv (Cornell University)
Author: Hakimoglu-Brown, Cetin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212078987