50 research outputs found
Elliptic curves of large rank and small conductor
For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the
smallest conductor known, improving on the previous records by factors ranging
from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search
methods, and tabulate, for each r=5,6,...,11, the five curves of lowest
conductor, and (except for r=11) also the five of lowest absolute discriminant,
that we found.Comment: 16 pages, including tables and one .eps figure; to appear in the
Proceedings of ANTS-6 (June 2004, Burlington, VT). Revised somewhat after
comments by J.Silverman on the previous draft, and again to get the correct
page break
Roots of the derivative of the Riemann zeta function and of characteristic polynomials
We investigate the horizontal distribution of zeros of the derivative of the
Riemann zeta function and compare this to the radial distribution of zeros of
the derivative of the characteristic polynomial of a random unitary matrix.
Both cases show a surprising bimodal distribution which has yet to be
explained. We show by example that the bimodality is a general phenomenon. For
the unitary matrix case we prove a conjecture of Mezzadri concerning the
leading order behavior, and we show that the same follows from the random
matrix conjectures for the zeros of the zeta function.Comment: 24 pages, 6 figure
The n-level spectral correlations for chaotic systems
We study the -level spectral correlation functions of classically chaotic
quantum systems without time-reversal symmetry. According to Bohigas, Giannoni
and Schmit's universality conjecture, it is expected that the correlation
functions are in agreement with the prediction of the Circular Unitary Ensemble
(CUE) of random matrices. A semiclassical resummation formalism allows us to
express the correlation functions as sums over pseudo-orbits. Using an extended
version of the diagonal approximation on the pseudo-orbit sums, we derive the
-level correlation functions identical to the determinantal
correlation functions of the CUE.Comment: 20 pages, no figure, minor corrections mad
Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral fluctuations
To treat the spectral statistics of quantum maps and flows that are fully
chaotic classically, we use the rigorous Riemann-Siegel lookalike available for
the spectral determinant of unitary time evolution operators . Concentrating
on dynamics without time reversal invariance we get the exact two-point
correlator of the spectral density for finite dimension of the matrix
representative of , as phenomenologically given by random matrix theory. In
the limit the correlator of the Gaussian unitary ensemble is
recovered. Previously conjectured cancellations of contributions of
pseudo-orbits with periods beyond half the Heisenberg time are shown to be
implied by the Riemann-Siegel lookalike
Random matrix theory, the exceptional Lie groups, and L-functions
There has recently been interest in relating properties of matrices drawn at
random from the classical compact groups to statistical characteristics of
number-theoretical L-functions. One example is the relationship conjectured to
hold between the value distributions of the characteristic polynomials of such
matrices and value distributions within families of L-functions. These
connections are here extended to non-classical groups. We focus on an explicit
example: the exceptional Lie group G_2. The value distributions for
characteristic polynomials associated with the 7- and 14-dimensional
representations of G_2, defined with respect to the uniform invariant (Haar)
measure, are calculated using two of the Macdonald constant term identities. A
one parameter family of L-functions over a finite field is described whose
value distribution in the limit as the size of the finite field grows is
related to that of the characteristic polynomials associated with the
7-dimensional representation of G_2. The random matrix calculations extend to
all exceptional Lie groupsComment: 14 page
Derivation of determinantal structures for random matrix ensembles in a new way
There are several methods to treat ensembles of random matrices in symmetric
spaces, circular matrices, chiral matrices and others. Orthogonal polynomials
and the supersymmetry method are particular powerful techniques. Here, we
present a new approach to calculate averages over ratios of characteristic
polynomials. At first sight paradoxically, one can coin our approach
"supersymmetry without supersymmetry" because we use structures from
supersymmetry without actually mapping onto superspaces. We address two kinds
of integrals which cover a wide range of applications for random matrix
ensembles. For probability densities factorizing in the eigenvalues we find
determinantal structures in a unifying way. As a new application we derive an
expression for the k-point correlation function of an arbitrary rotation
invariant probability density over the Hermitian matrices in the presence of an
external field.Comment: 36 pages; 2 table
A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor
We propose a random matrix model for families of elliptic curve L-functions
of finite conductor. A repulsion of the critical zeros of these L-functions
away from the center of the critical strip was observed numerically by S. J.
Miller in 2006; such behaviour deviates qualitatively from the conjectural
limiting distribution of the zeros (for large conductors this distribution is
expected to approach the one-level density of eigenvalues of orthogonal
matrices after appropriate rescaling).Our purpose here is to provide a random
matrix model for Miller's surprising discovery. We consider the family of even
quadratic twists of a given elliptic curve. The main ingredient in our model is
a calculation of the eigenvalue distribution of random orthogonal matrices
whose characteristic polynomials are larger than some given value at the
symmetry point in the spectra. We call this sub-ensemble of SO(2N) the excised
orthogonal ensemble. The sieving-off of matrices with small values of the
characteristic polynomial is akin to the discretization of the central values
of L-functions implied by the formula of Waldspurger and Kohnen-Zagier.The
cut-off scale appropriate to modeling elliptic curve L-functions is
exponentially small relative to the matrix size N. The one-level density of the
excised ensemble can be expressed in terms of that of the well-known Jacobi
ensemble, enabling the former to be explicitly calculated. It exhibits an
exponentially small (on the scale of the mean spacing) hard gap determined by
the cut-off value, followed by soft repulsion on a much larger scale. Neither
of these features is present in the one-level density of SO(2N). When N tends
to infinity we recover the limiting orthogonal behaviour. Our results agree
qualitatively with Miller's discrepancy. Choosing the cut-off appropriately
gives a model in good quantitative agreement with the number-theoretical data.Comment: 38 pages, version 2 (added some plots
A few remarks on colour-flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals
The Humboldt Foundation is acknowledged for the financial support of that visit. The research in Nottingham was supported by EPSRC grant EP/C515056/1 'Random Matrices and Polynomials: a tool to understand complexity'