Harry Smit

Characterizing number fields using L-series

The celebrated Neukirch-Uchida theorem states that two number fields with isomorphic absolute Galois group must be isomorphic themselves. This result has since been extended to quotients of this Galois group such as the solvable closure and (very recently, by Saidi and Tamagawa) the 3-step solvable closure. The abelianization does not, however, have this characterizing property. In fact, many imaginary quadratic number fields have isomorphic abelianized Galois group.

One way to supplement the abelianized Galois group is by adding some information on the (Dirichlet) L-series of the number fields. We show that in this way it is possible to not only characterize the number field, but also the isomorphisms and homomorphisms between number fields. If time allows, we discuss how similar techniques can be used to characterize isogeny classes of abelian varieties using twists of the L-series attached to the abelian variety.