2019/01/21 by Smit, Harry
#11G40 #11R37 #11R42 #14K02 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1901.06894
Faltings's isogeny theorem states that two abelian varieties are isogenous over a number field precisely when the characteristic polynomials of the reductions at almost all prime ideals of the number field agree. This implies that two abelian varieties over ℚ with the same L-series are necessarily isogenous, but this is false over a general number field. Let A and A' be two abelian varieties, defined over number fields K and K' respectively. Our main result is that A and A' are isogenous after a suitable isomorphism between K and K' if and only if the Dirichlet character groups of K and K' are isomorphic and the L-series of A and A' twisted by the Dirichlet characters match.