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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