Characterizing number fields using L-series

Abstract: 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.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

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