PreprintCC BY 4.0arXiv
Modular and Shimura curves X0D(N) embedded in ℙ1 × ℙ1
Abstract
We classify the modular curves X0(N) that admit a smooth embedding in ℙ1 × ℙ1 and obtain a partial classification for the Shimura curves X0D(N). We also prove that, for D>1, a Shimura curve X0D(N) admits a smooth plane model if and only if its genus is at most one.
Related
- The Shimura Lift and Hilbert Modular Forms
- Computing the cohomology of Shimura curves in quasi-linear time
- Modular Heights of Unitary Shimura Varieties II: Arithmetic Generating Series of Divisors
- Modular Heights of Unitary Shimura Varieties III: Proof of the Main Theorem
- Shimura curves of discriminant 14 and 15 and associated Heun Functions
BibTeX
@misc{padurariu2026modular,
title = {{Modular and Shimura curves $X_0^D(N)$ embedded in $\mathbb P^1 \times \mathbb P^1$}},
author = {Oana Padurariu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13967},
eprint = {2609.13967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13967},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Modular and Shimura curves X_0^D(N) embedded in ℙ^1 × ℙ^1
AU - Oana Padurariu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13967
UR - https://arxiv.org/abs/2609.13967
AB - We classify the modular curves X_0(N) that admit a smooth embedding in ℙ^1 × ℙ^1 and obtain a partial classification for the Shimura curves X_0^D(N). We also prove that, for D>1, a Shimura curve X_0^D(N) admits a smooth plane model if and only if its genus is at most one.
ER -