On smooth plane models for modular curves of Shimura type

TL;DR

This paper proves finiteness results for smooth plane models of modular curves, showing that only finitely many such curves exist with low degree, and classifies Shimura type curves with degree 8 models.

Contribution

It establishes new bounds on the degrees of smooth plane models for modular and Shimura type curves, and classifies certain low-degree cases.

Findings

Finitely many modular curves admit smooth plane models.

No modular curve of Shimura type admits degree 5, 6, or 7 models.

If degree ≥ 19, no such smooth plane models exist.

Abstract

In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we show that it must be a twist of one of four curves.