Arithmetic Statistics

Levent Alpoge

I will give an introduction to the techniques that allow one to count number fields of small degree and bounded discriminant, to bound the average ranks of elliptic curves and of higher genus hyperelliptic curves, and to bound the number of integral/rational points on said curves. The methods for counting orbits in certain cases are well-enough developed that I hope to impart on the audience how to turn the crank, so to speak. It will transpire that in these cases we will always reduce to counting integral points of bounded height in a lattice. I will indicate some first results towards counting orbits with nontrivial polynomial relations among their invariants. I will try to present the parametrizations and other techniques with as much intuition as I can give (for example: how one could have come up with these parametrizations), and the material will be accessible to all participants of the program.