Asst. Prof. Enis Kaya from the Department of Mathematics at Bilkent University has been awarded funding under the TÜBİTAK 2232-B International Young Researchers Program for a period of three years, with a total budget exceeding 17 million TL, for his project titled “The quadratic Chabauty method at primes of bad reduction & rational points on modular curves.”
The main goal of this project is to develop an algorithm to apply the quadratic Chabauty method at primes of bad reduction in order to determine rational points on algebraic curves. Rational points are a fundamental concept in arithmetic geometry, connecting algebraic geometry with number theory and sometimes with other areas such as p-adic analysis. In general, finding rational points on a given curve can be a highly non-trivial task.
The quadratic Chabauty method has been used successfully by various researchers for the same purpose, but only at primes of good reduction. Hence, the extension to primes of bad reduction is new and is expected to be more natural and efficient in certain cases. The methods to be used combine rigid analytic spaces, Berkovich analytic spaces, p-adic integration theories, models of curves, tropical geometry, toric geometry, and programming in the computer algebra systems SageMath and Magma.