Adam Morgan

4-ranks of class groups of biquadratic fields

Let K be a quadratic number field, and consider the family of biquadratic fields K_n= K(\sqrt{n}) for n a squarefree integer.
I will discuss joint work with Peter Koymans and Harry Smit in which we study, as n varies, the 4-rank of the class group of K_n,
showing in particular that for 100 % of squarefree n, the 4-rank is given by an explicit formula involving the number of prime divisors of n that are inert in K. If time permits I will discuss an elliptic curve analogue of this work, which is joint with Ross Paterson.