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

Abstract

Let E E be an elliptic curve over a number field F F , and let F ∞ F_\infty be a Galois extension of F F whose Galois group G G is a p p -adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E E over F ∞ F_\infty , and the global root numbers attached to the twists of the complex L L -function of E E by Artin representations of G G .

On the Birch-Swinnerton-Dyer quotients modulo squares

“…In contrast, the p-parity conjecture over ‫ޑ‬ was known in almost all cases, thanks to Birch, Stephens, Greenberg and Guo [3], [15], [16] (E CM), Kramer, Monsky [22], [26] (p D 2), Nekovář [28] (p potentially ordinary or potentially multiplicative) and Kim [18] (p supersingular). The results for Selmer groups in dihedral and false Tate curve extensions are similar to those recently obtained by Mazur-Rubin [23] and Coates-Fukaya-Kato-Sujatha [7], [8] Finally, we will need a slight modification of c.E=K/. Fix an invariant differential !…”

“…1 Finally, let us mention how the applications of our theory connect to earlier work. To our best knowledge, over number fields Theorem 1.3 is the first general result of this kind, except for the work [7], [11] on curves with a p-isogeny. In contrast, the p-parity conjecture over ‫ޑ‬ was known in almost all cases, thanks to Birch, Stephens, Greenberg and Guo [3], [15], [16] (E CM), Kramer, Monsky [22], [26] (p D 2), Nekovář [28] (p potentially ordinary or potentially multiplicative) and Kim [18] (p supersingular).…”

“…Arithmetic of elliptic curves (ordinary at p) in such extensions has been studied in the context of noncommutative Iwasawa theory; see e.g. [17], [7], [8].…”

Self-points on elliptic curves

“…This is due to changing signs in the functional equations and the corresponding parity results on the corank of Selmer groups. See [Coates et al 2009;Mazur and Rubin 2008]. These results predict, under the assumption of the finiteness of the Tate-Shafarevich group, that there should be more points of infinite order in the division tower that are not accounted for by higher self-points.…”

Variation of Tamagawa numbers of semistable abelian varieties in field extensions

“…Most results on the p-parity conjecture concern elliptic curves; in particular the first formula is now known to hold for all elliptic curves over Q and in many cases for elliptic curves over totally real fields [4,12]. The situation with general abelian varieties is more difficult, and the main results are those of [2] that establish the first formula assuming that the abelian variety admits a suitable isogeny, and of [3] that proves the second formula for a class of representations τ . Our results on the behaviour of Tamagawa numbers let us strengthen the results of the latter paper as follows.…”