Quadratic Chabauty for integral points

• Applicant: Professor Dr. Jan Steffen Müller
• Subject Area: Mathematics
• Term: from 2016 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 325713478

The explicit computation of rational or integral points on algebraic curves of genus at least two defined over the rationals has many important applications, but is a notoriously difficult problem in general. Chabauty's method often succeeds in determining the rational points using p-adic analysis, but it is restricted to curves whose Jacobians have Mordell-Weil rank strictly less than the genus. To overcome this issue, M. Kim proposed a framework for extending Chabauty's method based on p-adic Hodge theory. His theory predicts that rational (or at least integral) points should be zeros of combinations of iterated p-adic integrals. However, it seems very difficult to use Kim's approach directly for explicit computations.In the spirit of Kim's philosophy, the applicant used p-adic heights in earlier joint work with J. Balakrishnan and A. Besser to explicitly write down such integrals vanishing in integral points when the rank equals the genus and the curve is hyperelliptic of odd degree. In the proposed project, we will extend this technique to hyperelliptic curves of even degree, superelliptic curves and smooth plane quartics whose Jacobians have rank equal to the genus. We will also combine the technique with a new, but related approach also based on p-adic heights, which will simplify the method and make it applicable for some larger rank examples as well.After developing the necessary theory, we will implement complete algorithms for the computation of integral points on such curves.

• DFG Programme: Research Grants
• International Connection: Netherlands