Averages of class numbers

Peter Koymans (ETH Zurich)

Wednesday 29th May 16:00-17:00 Maths 110

Abstract

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.

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