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L-functions and isogenies of abelian varieties

Posted in
Harry Justus Smit
Universiteit Utrecht
Wed, 2020-01-15 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Faltings's isogeny theorem states that two abelian varieties
over a number field are isogenous precisely when the characteristic
polynomials associated to the reductions of the abelian varieties at all
prime ideals are equal. This implies that two abelian varieties defined
over the rational numbers with the same L-function are necessarily
isogenous, but this is false over a general number field.

In order to still use the L-function to determine the underlying field
and abelian variety, we extract more information from the L-function by
"twisting": a twist of an L-function is the L-function of the tensor of
the underlying representation with a character. We discuss a theorem
stating that abelian varieties over a general number field are
characterized by their L-functions twisted by Dirichlet characters of
the underlying number field.

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