PreprintarXiv licencearXiv
Algebraic independence of the exponential and Weierstrass ℘-functions
Abstract
Let Ω be a lattice in ℂ with algebraic invariants and complex multiplication, let ℰ be the elliptic curve associated with Ω, and let ℘ be the Weierstrass function relative to Ω. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that if t1,…,ts are ℚ-linearly independent algebraic numbers and p1,…,pn are k-linearly independent algebraic numbers, then the s+n numbers et1,…,ets, ℘(p1),…,℘(pn) are algebraically independent over ℚ̅. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.
Related
- Integer points for iterated Weierstrass elliptic functions
- Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies
- An Algebraic Proof of Gauge-Fixing Independence
- Algebraic independence of periods of Anderson modules and their hyperderivatives
- An extension of algebraic independence of special values for non-lacunary power series
BibTeX
@misc{bertolin2026algebraic,
title = {{Algebraic independence of the exponential and Weierstrass $\wp$-functions}},
author = {Cristiana Bertolin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15294},
eprint = {2609.15294},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15294},
}
RIS
TY - UNPB
TI - Algebraic independence of the exponential and Weierstrass ℘-functions
AU - Cristiana Bertolin
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15294
UR - https://arxiv.org/abs/2609.15294
AB - Let Ω be a lattice in ℂ with algebraic invariants and complex multiplication, let ℰ be the elliptic curve associated with Ω, and let ℘ be the Weierstrass function relative to Ω. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that if t_1,…,t_s are ℚ-linearly independent algebraic numbers and p_1,…,p_n are k-linearly independent algebraic numbers, then the s+n numbers e^t_1,…,e^t_s, ℘(p_1),…,℘(p_n) are algebraically independent over ℚ̅. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.
ER -