On the parity of ranks of Selmer groups
Jan Nekovář, Andrew James Plater · Asian Journal of Mathematics · 2000
Introduction.Let / = X)n>i a n(f)Q n € Sko (To{N)) be a normalized newform of even weight fco > 2. Let F be the number field generated by the coefficients of / and p a prime of F lying above a rational prime p.There is a two-dimensional representation V(f) of GQ = Gal (Q/Q) over Fp associated to /, characterized by the conditions It (FWgeomW/)) =1 cin(/)^~s satisfies the functional equation Aoo(/, s) := ( -J r(s)L 00 (/, s) = Woo(/)Aoo(/, fco -5), where w 00 (f) = ±1 = (-l) eo0 for eoo = 0 or 1. Bloch and Kato [BI-Ka] defined a generalized "Selmer group" Hj(Q,V ko ) C i? 1 (Q, Vk 0 ) and conjectured that ord^/aLooCf,*) = dim Fp H}(Q, Vi 0 ).We are interested in a (mod 2) version of this conjecture:The Parity Conjecture for ranks of Selmer groups ord a=ibo/ 2Loo(/,s) = dim Fp if}(Q,^0) (mod 2).Assume that p > 3 and that / is ordinary at p, i.e. that ap(f) 6 Fp is a p-adic unit.According to Hida's theory, there is a p-adic family of ordinary modular forms of varying weights containing / (we ignore the phenomenon of "p-stabilization" in this Introduction).In concrete terms, this means that there is an integer c > 0 such that for every integer k > 2 satisfying k = ko (mod (p -l)p c ), there is an ordinary newform fk of weight k on ro(iV) such that /fc 0 = / and .k = k' (mod (p -l)p n+c ) implies /* = /*' (modp"). LetAT, k = 2, a p (f) = 1, f 1 ifp\ \ 0 other otherwise.