25 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
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
L-series and Feynman Integrals
Integrals from Feynman diagrams with massive particles soon outgrow polylogarithms. We consider the simplest situation in which this occurs, namely for diagrams with two vertices in two space-time dimensions, with scalar particles of unit mass. These comprise vacuum diagrams, on-shell sunrise diagrams and diagrams obtained from the latter by cutting internal lines. In all these cases, the Feynman integral is a moment of n = a + b Bessel functions, of the form M(a,b,c) := â« â0 1a0(t)Kb0(t)tcdt. The corresponding L-series are built from Kloosterman sums over finite fields. Prior to the Creswick conference, the first author obtained empirical relations between special values of L-series and Feynman integrals with up to n = 8 Bessel functions. At the conference, the second author indicated how to extend these. Working together we obtained empirical relations involving Feynman integrals with up to 24 Bessel functions, from sunrise diagrams with up to 22 loops. We have related results for moments that lie beyond quantum field theory
Computing L-series of hyperelliptic curves
We discuss the computation of coefficients of the L-series associated to a
hyperelliptic curve over Q of genus at most 3, using point counting, generic
group algorithms, and p-adic methods.Comment: 15 pages, corrected minor typo
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 Kâ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the MordellâWeil rank and ShafarevichâTate group for E over a tower of extensions K â/â inside Kâ; we obtain lower bounds on the former, and upper bounds on the latterâs size
Ranks of twists of elliptic curves and Hilbert's Tenth Problem
In this paper we investigate the 2-Selmer rank in families of quadratic
twists of elliptic curves over arbitrary number fields. We give sufficient
conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer
rank, and we give lower bounds for the number of twists (with bounded
conductor) that have a given 2-Selmer rank. As a consequence, under appropriate
hypotheses we can find many twists with trivial Mordell-Weil group, and
(assuming the Shafarevich-Tate conjecture) many others with infinite cyclic
Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our
results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth
Problem has a negative answer over the ring of integers of every number field.Comment: Minor changes. To appear in Inventiones mathematica