Mathematical Conversations

In arithmetic geometry, there are lots of examples of natural density zero sets of primes raised from the geometry of elliptic curves or more generally abelian varieties. One may ask whether such thin set is finite or not. For example, given any two elliptic curves defined over rational numbers, consider the set of primes p such that after modulo p and passing to certain field extension, there exists a non-trivial group homomorphism between these two curves. This set is often of density zero, but it is a recent result of Charles that this set is infinite using Arakelov intersection theory and the equidistribution of Hecke orbits. In this talk, I will discuss a strategy to study some other thin sets of primes by Charles’ method. A simplified version of Arakelov intersection theory will be explained as the intersection theory for $\mathrm{Spec}(\mathbb Z)$. This is joint work with Ananth Shankar.