## Number Theory Seminar: Nils Bruin

- Date: 09/22/2011
- Time: 16:10

Simon Fraser University

Imaginary quadratic class numbers and Sha on congruent number curves

Abstract:

We consider two classical number theoretic problems that may seem quiteunrelated: * 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.

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