Matthew Bisatt

Root numbers of abelian varieties

Consider an elliptic curve over a number field. The set of rational points of is well known to have the structure of a finitely generated abelian group and its rank is famously predicted to equal the order of vanishing of a certain L-function.

This conjecture however presupposes that the L-function has analytic continuation which is not always known. To circumvent this, we introduce the root number which is independent of the L-function and conjecturally controls whether the rank is odd or even. In particular, if this object tells us the rank is odd then this implies that the elliptic curve has infinitely many rational points.

I will discuss how one computes the root number in practice on elliptic curves and generalise this to their higher dimensional analogues: abelian varieties.