Root numbers, Selmer groups, and non-commutative Iwasawa theory

Global root numbers have played an important role in the study of rational points on abelian varieties since the discovery of the conjecture of Birch and Swinnerton-Dyer. The aim of this paper is to throw some new light on this intriguing and still largely conjectural relationship. The simplest avatar of this phenomenon is the parity conjecture which asserts that for an abelian variety A over a number field F , and for each prime number p, the Zp-corank of the Selmer group of A over F should have the same parity as the root number of the complex L-function of A. Many affirmative results in this direction have been established when A is an elliptic curve, notably by B. Birch and N. Stephens [4], T. and V. Dokchitser [17], [18], R. Greenberg and L. Guo [24], B-D. Kim [28], P. Monsky [33], and J. Nekovář [34], [35]. In §1, we use ideas due to Cassels, Fisher [15, Appendix], Shuter [43], and T.Dokchitser and V. Dokchitser [17] to prove with some technical restrictions, the parity conjecture for the prime p and and an abelian variety A of dimension g over a number field F having an isogeny of degree p. In the rest of the paper, we give some fragmentary evidence that there is a close connexion between root numbers and the Selmer group of an elliptic curve E over certain non-commutative p-adic Lie extensions of the base field F . The non-commutativity of the Galois group of these p-adic Lie extensions is important for us, because we are interested in cases in which there are infinite families of irreducible self-dual Artin representations of this group. In fact, for two non-commutative p-adic Lie extensions, we prove analogues of the parity conjecture for twists of both the Selmer group and the complex L-function by all irreducible, orthogonal Artin representations of the Galois group. Our results have some overlap with the recent work of Mazur and Rubin [30], [31], and T. and V. Dokchitser [19] although our viewpoint is rather different in that our proofs use methods and invariants arising from Iwasawa theory. Because of our use of Iwasawa theory, our results at present require much stronger hypotheses than these authors. However one advantage of our approach is that it provides both upper and lower bounds for the Zp-corank of the Selmer group, and surprisingly in some cases the two bounds coincide (see also [16, Appendix]). One advantage of our approach is that it provides both upper and lower bounds for the Zp-corank of the Selmer group, and surprisingly in some cases the two bounds coincide