130 research outputs found
Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions
Let A be an abelian variety over a number field k and F a finite cyclic
extension of k of p-power degree for an odd prime p. Under certain technical
hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number
conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru-
ences involving values of derivatives of the Hasse-Weil L-functions of twists
of A, normalised by completely explicit twisted regulators. This
reinterpretation makes the eTNC amenable to numerical verification and
furthermore leads to explicit predictions which refine well-known conjectures
of Mazur and Tate
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
A heuristic for boundedness of ranks of elliptic curves
We present a heuristic that suggests that ranks of elliptic curves over the
rationals are bounded. In fact, it suggests that there are only finitely many
elliptic curves of rank greater than 21. Our heuristic is based on modeling the
ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies
on a theorem counting alternating integer matrices of specified rank. We also
discuss analogues for elliptic curves over other global fields.Comment: 41 pages, typos fixed in torsion table in section
Self-duality of Selmer groups
The first part of the paper gives a new proof of self-duality for Selmer
groups: if A is an abelian variety over a number field K, and F/K is a Galois
extension with Galois group G, then the Q_pG-representation naturally
associated to the p-infinity Selmer group of A/F is self-dual. The second part
describes a method for obtaining information about parities of Selmer ranks
from the local Tamagawa numbers of A in intermediate extensions of F/K.Comment: 12 pages; to appear in Proc. Cam. Phil. So
On the Birch-Swinnerton-Dyer quotients modulo squares
Let A be an abelian variety over a number field K. An identity between the
L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation
between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo
finiteness of Sha, and give an analogous statement for Selmer groups. Based on
this, we develop a method for determining the parity of various combinations of
ranks of A over extensions of K. As one of the applications, we establish the
parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and
some restrictions on the reduction at primes above 2 and 3: the parity of the
Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as
determined by the root number. We also prove the p-parity conjecture for all
elliptic curves over Q and all primes p: the parities of the p^\infty-Selmer
rank and the analytic rank agree.Comment: 29 pages; minor changes; to appear in Annals of Mathematic
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
On Greenberg's -invariant of the symmetric sixth power of an ordinary cusp form
We derive a formula for Greenberg's -invariant of Tate twists of the
symmetric sixth power of an ordinary non-CM cuspidal newform of weight ,
under some technical assumptions. This requires a "sufficiently rich" Galois
deformation of the symmetric cube which we obtain from the symmetric cube lift
to \GSp(4)_{/\QQ} of Ramakrishnan--Shahidi and the Hida theory of this group
developed by Tilouine--Urban. The -invariant is expressed in terms of
derivatives of Frobenius eigenvalues varying in the Hida family. Our result
suggests that one could compute Greenberg's -invariant of all symmetric
powers by using appropriate functorial transfers and Hida theory on higher rank
groups.Comment: 20 pages, submitte
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