The Quadratic Chabauty Method for Affine Hyperbolic Curves

• Applicant: Dr. Marius Leonhardt
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552795810

The fundamental question in number theory is finding the integer solutions to polynomial equations. In 1983, Faltings showed that equations that geometrically describe a curve only have a finite number of solutions. However, the determination of these finitely many solutions is still an open problem. A promising approach to solving this problem is the Chabauty-Kim method, which forms the basis of my research. My research project has two goals: 1. I want to compute the solutions of equations of so-called affine curves using the quadratic Chabauty method. With M. Lüdtke and J.S. Müller I have shown that this method can theoretically calculate a candidate list for the solutions, and we can determine the maximum length of this candidate list for a given equation. With the help of Jennifer Balakrishnan, an expert in the field of algorithmic number theory, I will extend this result to an algorithm that calculates the candidate list and thus the solutions of the equation. We will use techniques of the quadratic Chabauty method of projective curves, which was developed by Prof Balakrishnan. I will then calculate the solutions for certain curves, especially for modular curves, since their solutions are of interest in other parts of arithmetic geometry. 2. I want to determine an upper bound on the number of solutions of so-called CM curves. With L.A. Betts and D. Corwin I have found such a bound for arbitrary curves assuming the Bloch-Kato conjectures. This project will not assume the Bloch-Kato conjectures, but instead I will exploit additional symmetries (on the cohomological invariants) of CM curves. This would be the first family of examples for which the Chabauty-Kim method provides explicit bounds on the solutions. Understanding these examples will provide deep insights into the mechanism of the method and allow me to improve it for a broader class of curves as well.

• DFG Programme: WBP Fellowship
• International Connection: USA
• Host: Professorin Dr. Jennifer S. Balakrishnan