Number Theory Seminar: Nils Bruin

Abstract:

We consider two classical number theoretic problems that may seem quite
unrelated:
 
* What is the power of 2 dividing the class number of Q(sqrt(-n))
* Which n are congruent numbers (n called congruent if it occurs as the
  area of a right-angled triangle with rational length sides)
 
The second question is equivalent to determining whether the elliptic curve E_n: y^2=x^3-n^2*x has positive rank. This observation suggest we might want to consider:
 
* What is the power of 2 in the order of Sha(E_n).
 
If we restrict to prime values n=p, it is already known that partial answers to these questions can be related to the splitting of p in the quartic number field Q(sqrt(1+i)).
 
In this talk we will discuss the next step in the classification.