Averages of class numbers [NT seminar]

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.