Rational Points on Modular Curves

ag.algebraic-geometry nt.number-theory
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International Centre for Theoretical Sciences 
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The study of rational points on varieties is a field of special interest to arithmetic geometers. Over the past few decades, many techniques have been used to decide whether a variety over a number field has a rational point or not, and even to describe those points completely. In this program, we are mainly interested in the study of rational points on modular curves.

Elliptic curves, modular forms and modular curves are central objects in arithmetic geometry. Modular curves can be thought of as moduli spaces for elliptic curves with extra level structures. The objective of this program is to understand the theoretical and computational aspects of determining K-rational points on modular curves X_H(K) for various fields K and subgroups H of GL_2(ℤ/Nℤ) for any natural number N.

In the mini-courses, we give an advanced introduction to the theory of rational points on modular curves, under both theoretical and computational aspects. These courses would include an introduction to the geometry of modular curves, their ℚ-rational points, classical and non-abelian Chabauty methods, and related computational aspects. We also attempt to strike a balance between the more advanced topics and the down-to-earth examples.

In the discussion meeting, we wish to bring many experts across the world in the area of arithmetic geometry together to share their ideas and the current state of research that will facilitate future research in this direction. We strongly encourage participation of young researchers.


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