2006/10/31 by Tim Dokchitser, Vladimir Dokchitser · 60 citations
Mathematics · #Abelian group #Abelian variety #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic number field #Analytic Number Theory Research #Conjecture #Elliptic curve #Modulo #Parity (physics) #Quotient #math.NT #msc:11G05 #msc:11G10 #msc:11G40
paper · pdf · doi:10.4007/annals.2010.172.567
published in Annals of Mathematics 172(1), 567-596 (Princeton University) · 29 pages; minor changes; to appear in Annals of Mathematics
arxiv created 2008/10/20 · openalex publication_date 2010/06/27 · arxiv updated 2013/09/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let A be an abelian variety over a number field K. An identity between the L-functions L.A=K i ; s/ for extensions K i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of X, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of X.E=K.EOE2//OE6 1 and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E=K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over and all primes p: the parities of the p 1 -Selmer rank and the analytic rank agree.