Iwasawa Theory for p-adic Representations

Greenbergfor certain constants av, av We write the functional equation in the above way to focus on the behavior of Lv(s) at s= 1, which seems convenient for formulating our conjectures and results.But replacing V by the Tate twist V(l-n)={Vz(l -n)} and noting that Lvo-nh)=Lv(s+n-1) will give appropriate statements about Lv(s) at s=n, for any integer n.Let rv denote the order of pole for I'v(s) at s=l.Since I'v(s) has no zeros, rv>O.Often rv is also the order of vanishing of Lv(s) at s= 1, but not always.(We mention two examples: (1) V ={Qz(l)}, Lv(s)=((s-1),Hasse-Weil L-function LE(s), rv=O, but Lv(l) can vanish.)Now, if cp is any even Dirichlet character, the twisted L-series Lv(s, cp) should also satisfy a functional equation similar to (7), with the same I'-factor, relating cp) to Lv.(s, cp-1 ).If we let cp vary over the characters of I'= Gal (Q00 /Q), regarded as Dirichlet characters, it seems reasonable to conjecture that Lv(s, cp) will have a zero of order exactly rv at s= 1, except possibly for finitely many cp.Sometimes this is easy to verify.A more subtle case is LE(s).Rohrlich [22] has proved the above conjecture in this case (i.e. Lv(l, cp)='s=O for all but finitely many cp e f) if Eis a Weil curve.Now our general philosophy is that the behavior of the L-functions Lv(s, cp) at s= 1 (cp e f) should somehow be reflected in the structure of the Selmer groups Svp;T/Qoo).Thus, the above remarks suggest the following conjecture.We assume p is ordinary for V and TP is any Ga invariant lattice.Conjecture 1. Svp/Tp(Q00 ) has A-corank equal to rv.We will be able to prove the following weaker result by making use of Tate's calculation of Euler-Poincare characteristics and also the conjectural description of r v in terms of quantities attached to the representation space VP.We will have to also assume that Vis pure in the sense that it arises from a motive of pure weight.We believe the above conjecture even without this assumption, but its seems to be a more subtle question then.Theorem 1.If Vis pure, then corankA (Svp;T/Qoo))>rv, It is especially interesting to consider the case where rv=rv.=0.Then LvU) and Lv.(1) are critical values in the sense of Deligne [4].Deligne