Number Theory 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.