Peter Koymans

Averages of class numbers

In the 1950s Erdos developed a method to give upper and lower bounds of the correct order of magnitude for $d(P(n))$ where $d$ is the divisor function and $P$ is a polynomial. This was greatly extended by Nair and Tenenbaum to a wide class of multiplicative functions and sequences.

In a different direction, Heath-Brown and Fouvry--Kluners used character sum techniques to respectively obtain the average size of the $2$-Selmer group in the quadratic twist family $dy^2 = x^3 - x$ and the average size of the 4-torsion of Q(sqrt(d)).

We combine these two techniques to get the order of magnitude for the average size of the $3 * 2^k$-torsion for every $k >= 1$ and bounded ranks (on average) for the family $P(t) y^2 = x^3 - x$. In this talk, we will explain the aforementioned techniques and how we are able to combine them. This is joint work with Carlo Pagano and Efthymios Sofos.