14 research outputs found
Root numbers of elliptic curves in residue characteristic 2
To determine the global root number of an elliptic curve defined over a
number field, one needs to understand all the local root numbers. These have
been classified except at places above 2, and in this paper we attempt to
complete the classification. At places above 2, we express the local root
numbers in terms of norm residue symbols (resp. root numbers of explicit
1-dimensional characters) in case when wild inertia acts through a cyclic
(resp. quaternionic) quotient.Comment: 10 page
Algebraicity of L-values for elliptic curves in a false Tate curve tower
Let E be an elliptic curve over , and τ an Artin representation over that factors through the non-abelian extension , where p is an odd prime and n, m are positive integers. We show that L(E,τ,1), the special value at s=1 of the L-function of the twist of E by τ, divided by the classical transcendental period Ω+ d+ |Ω− d− |ε(τ) is algebraic and Galois-equivariant, as predicted by Deligne's conjecture
On a BSD-type formula for L-values of Artin twists of elliptic curves
This is an investigation into the possible existence and consequences of
a Birch–Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely arithmetic predictions
for elliptic curves, and show that these force some peculiar properties of the Tate–Shafarevich
group, which do not appear to be tractable by traditional Selmer group techniques. In particular,
we exhibit settings where the different p-primary components of the Tate–Shafarevich group
do not behave independently of one another. We also give examples of “arithmetically identical” settings for elliptic curves twisted by Artin representations, where the associated L-values
can nonetheless differ, in contrast to the classical Birch–Swinnerton-Dyer conjecture
Computations in non-commutative Iwasawa theory
We study special values of L-functions of elliptic curves over Q twisted by
Artin representations that factor through a false Tate curve extension
. In this setting, we explain how to
compute L-functions and the corresponding Iwasawa-theoretic invariants of
non-abelian twists of elliptic curves. Our results provide both theoretical and
computational evidence for the main conjecture of non-commutative Iwasawa
theory.Comment: 60 pages; with appendix by John Coates and Ramdorai Sujath
Regulator constants and the parity conjecture
The p-parity conjecture for twists of elliptic curves relates multiplicities
of Artin representations in p-infinity Selmer groups to root numbers. In this
paper we prove this conjecture for a class of such twists. For example, if E/Q
is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p
extension, then the p-parity conjecture holds for twists of E by all orthogonal
Artin representations of Gal(K^\infty/Q). We also give analogous results when
K/Q is non-abelian, the base field is not Q and E is replaced by an abelian
variety. The heart of the paper is a study of relations between permutation
representations of finite groups, their "regulator constants", and
compatibility between local root numbers and local Tamagawa numbers of abelian
varieties in such relations.Comment: 50 pages; minor corrections; final version, to appear in Invent. Mat
Estimating the growth in Mordell-Weil ranks and Shafarevich-Tate groups over Lie extensions
Let E/Q be an elliptic curve, p > 3 a good ordinary prime for E, and K∞ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for E over a tower of extensions K ₙ/ₖ inside K∞; we obtain lower bounds on the former, and upper bounds on the latter’s size