Bielliptic Shimura curves $X_0^D(N)$ with nontrivial level

TL;DR

This work completes a broad classification program for low-degree points on Shimura curves by extending Rotger’s level-1 bielliptic results to all $X_0^D(N)$ with $(D,N)=1$ and $N>1$, using explicit gonality and genus bounds to reduce to a finite set of candidate pairs. It shows that any bielliptic involution must be Atkin–Lehner and then analyzes genus-one Atkin–Lehner quotients via CM-point theory and Ribet’s isogeny to determine which quotients are elliptic over $\mathbb{Q}$ and have positive rank, yielding a near-complete list of geometrically bielliptic curves (with two exceptional pairs). The authors also classify geometrically trigonal Shimura curves in the coprime case and apply the bielliptic classification to sharpen results on sporadic points, including substantial progress on which curves have sporadic CM points. The results hinge on a blend of local–global analyses (real and $p$-adic points), automorphism-group constraints, and CM-point techniques, with extensive computational verification via Magma and Ribet’s isogeny to access ranks of genus-one quotients.

Abstract

We work towards completely classifying all bielliptic Shimura curves $X_0^D(N)$ with nontrivial level $N$ coprime to $D$, extending a result of Rotger that provided such a classification for level one. Combined with prior work, this allows us to determine the list of all relatively prime pairs $(D,N)$ for which $X_0^D(N)$ has infinitely many degree $2$ points. As an application, we use these results to make progress on determining which curves $X_0^D(N)$ have sporadic points. Using tools similar to those that appear in this study, we also determine all of the geometrically trigonal Shimura curves $X_0^D(N)$ with $\gcd(D,N)=1$ (none of which are trigonal over $\mathbb{Q}$).

Theorems & Definitions (51)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Definition 2.4
• Theorem 2.5: Eichler