Cocycle Relations and Elliptic Gamma Values
Abstract
Recent work of Bergeron-Charollois-García proposed a conjectural analytic expression for elliptic units for complex cubic fields as the special values of smoothed quotients of elliptic Gamma functions. These values depend on the conductor ideal, ray class group element, smoothing ideal, and an admissible element h. If this analytic expression is to fit into the framework of explicit class field theory, then, as in the case of torsion points on CM elliptic curves, the resulting complex number should depend only on the congruence class of h modulo a fractional ideal. This conjecture has been supported by much numerical evidence. In this paper, we use the cocycle relations of the smoothed elliptic Gamma function to prove that the construction is indeed independent of h.
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BibTeX
@misc{gray2026cocycle,
title = {{Cocycle Relations and Elliptic Gamma Values}},
author = {Teymour Gray},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10365},
eprint = {2610.10365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10365},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cocycle Relations and Elliptic Gamma Values
AU - Teymour Gray
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.10365
UR - https://arxiv.org/abs/2610.10365
AB - Recent work of Bergeron-Charollois-García proposed a conjectural analytic expression for elliptic units for complex cubic fields as the special values of smoothed quotients of elliptic Gamma functions. These values depend on the conductor ideal, ray class group element, smoothing ideal, and an admissible element h. If this analytic expression is to fit into the framework of explicit class field theory, then, as in the case of torsion points on CM elliptic curves, the resulting complex number should depend only on the congruence class of h modulo a fractional ideal. This conjecture has been supported by much numerical evidence. In this paper, we use the cocycle relations of the smoothed elliptic Gamma function to prove that the construction is indeed independent of h.
ER -