Quadratic Chabauty for Atkin-Lehner quotients of modular curves and Shimura curves

Date and place

21–27 August 2022 at Universität Bayreuth, Bayreuth, Germany

Participants

• Nikola Adžaga (University of Zagreb)
• Lea Beneish (UC Berkeley, remote)
• Shiva Chidambaram (MIT, remote)
• Stevan Gajović (Max-Planck-Institut Bonn, remote)
• Pip Goodman (Max-Planck-Institut Bonn)
• Timo Keller (Leibniz Universität Hannover)
• Angelos Koutsianas (Aristotle University of Thessaloniki)
• Oana Padurariu (Boston University, remote)
• Ciaran Schembri (Dartmouth College)
• Himanshu Shukla (Universität Bayreuth)
• John Voight (Dartmouth College, remote)
• Borna Vukorepa (University of Zagreb, remote)
• Boya Wen (University of Wisconsin-Madison, remote)

Aim of the conference

We work on extending our computation Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6 of the rational points on the modular curves $X_0^+(p)$ of genus 4, 5, and 6, and the one on hyperelliptic Atkin-Lehner quotients from Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings, to cover more general

• Atkin-Lehner quotients, and
• Shimura curves.

To this end, we extend the quadratic Chabauty algorithm to

• compute local heights away from $p$, and
• determine the quadratic Chabauty function when there are not many rational points,

for examples of Atkin-Lehner quotients of $X_0(N)$ with $N$ square-free.