On the parity of ranks of Selmer groups

Abstract

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/)) =<*/(/)det(FV(^) geom |y(/)) = ^-1 for all primes £ { piV. The Tate twist Vk 0 = V(f)(ko/2) is self dual: there is a skew-symmetric bilinear formThe complex L-function L 00 (f, s) = ^2n>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.

On the $p$-converse of the Kolyvagin–Gross–Zagier theorem

“…the determinant representation of T f is given by the p-adic cyclotomic character. As explained in [NP00], this implies that there exists a skew-symmetric morphism of…”

“…Denote by A (W ) ⊂ Q p k − 2 the subring of formal power series in k − 2 which converge for every k ∈ W . As explained in [GS93] (see also [NP00]), there exist an open neighbourhood U = U f ⊂ Z p of 2, and a natural morphism (the Mellin transform centred at φ f )…”

Root numbers and parity of ranks of elliptic curves

“…The purpose of this section is to extend the latter result to p = 2 (Theorem 2.4). Its proof goes along similar lines to that of Monsky [19] for elliptic curves over Q and uses potential modularity as in [20].…”

“…Instead of confronting residue characteristic 2, we use a 'global-to-local' approach. The point is that the p-parity conjecture can be seen to hold over totally real fields for all elliptic curves with non-integral j-invariant, by considerations that exploit modularity (Nekovář [20] for odd p and Theorem 2.4 for p = 2). Therefore we can prove the local formula by reversing the argument above: if K and F are totally real and we know that ( * ) holds at all places v but one, it must hold over the remaining place as well.…”

Sur la dualité et la descente d’Iwasawa

“…Dans cette optique, il est naturel d'étudier les applications de descente j n : A n → (A ∞ ) Γn , de nombreux exemples qui peuvent servir de guide à une théorie générale. Inspiré par les méthodes de [16] et [31], on propose dans cet article un cadre pour une telle théorie, puis on établit quelques résultats pratiques qui permettent l'unification et la généralisation de nombreux résultats connus. Pour avoir une première idée de ce qu'on peut espérer, on s'inspire essentiellement de deux études parallèles (celle de [27] pour le groupe de classes et celle de [33] pour les unités cyclotomiques) dans le cas où F ∞ = F n est la Z p -extension cyclotomique d'un corps de nombres F (Γ Z p ).…”