6,604 research outputs found
Regulators of rank one quadratic twists
We investigate the regulators of elliptic curves with rank 1 in some families
of quadratic twists of a fixed elliptic curve. In particular, we formulate some
conjectures on the average size of these regulators. We also describe an
efficient algorithm to compute explicitly some of the invariants of an odd
quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich
group, etc.) and we discuss the numerical data that we obtain and compare it
with our predictions.Comment: 28 pages with 32 figure
Towards an 'average' version of the Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer conjecture states that the rank of the
Mordell-Weil group of an elliptic curve E equals the order of vanishing at the
central point of the associated L-function L(s,E). Previous investigations have
focused on bounding how far we must go above the central point to be assured of
finding a zero, bounding the rank of a fixed curve or on bounding the average
rank in a family. Mestre showed the first zero occurs by O(1/loglog(N_E)),
where N_E is the conductor of E, though we expect the correct scale to study
the zeros near the central point is the significantly smaller 1/log(N_E). We
significantly improve on Mestre's result by averaging over a one-parameter
family of elliptic curves, obtaining non-trivial upper and lower bounds for the
average number of normalized zeros in intervals on the order of 1/log(N_E)
(which is the expected scale). Our results may be interpreted as providing
further evidence in support of the Birch and Swinnerton-Dyer conjecture, as
well as the Katz-Sarnak density conjecture from random matrix theory (as the
number of zeros predicted by random matrix theory lies between our upper and
lower bounds). These methods may be applied to additional families of
L-functions.Comment: 20 pages, 2 figures, revised first draft (fixed some typos
Discretisation for odd quadratic twists
The discretisation problem for even quadratic twists is almost understood,
with the main question now being how the arithmetic Delaunay heuristic
interacts with the analytic random matrix theory prediction. The situation for
odd quadratic twists is much more mysterious, as the height of a point enters
the picture, which does not necessarily take integral values (as does the order
of the Shafarevich-Tate group). We discuss a couple of models and present data
on this question.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix
Theory and Elliptic Curve
Effective equidistribution and the Sato-Tate law for families of elliptic curves
Extending recent work of others, we provide effective bounds on the family of
all elliptic curves and one-parameter families of elliptic curves modulo p (for
p prime tending to infinity) obeying the Sato-Tate Law. We present two methods
of proof. Both use the framework of Murty-Sinha; the first involves only
knowledge of the moments of the Fourier coefficients of the L-functions and
combinatorics, and saves a logarithm, while the second requires a Sato-Tate
law. Our purpose is to illustrate how the caliber of the result depends on the
error terms of the inputs and what combinatorics must be done.Comment: Version 1.1, 24 pages: corrected the interpretation of Birch's moment
calculations, added to the literature review of previous results
Rank distribution in a family of cubic twists
In 1987, Zagier and Kramarz published a paper in which they presented
evidence that a positive proportion of the even-signed cubic twists of the
elliptic curve should have positive rank. We extend their data,
showing that it is more likely that the proportion goes to zero
Chaos and stability in a two-parameter family of convex billiard tables
We study, by numerical simulations and semi-rigorous arguments, a
two-parameter family of convex, two-dimensional billiard tables, generalizing
the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A
17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard
tables are continuously deformed from the integrable circular billiard to
different versions of completely-chaotic stadia. In particular, we conjecture
that a new class of ergodic billiard tables is obtained in certain regions of
the two-dimensional parameter space, when the billiards are close to skewed
stadia. We provide heuristic arguments supporting this conjecture, and give
numerical confirmation using the powerful method of Lyapunov-weighted dynamics.Comment: 19 pages, 13 figures. Submitted for publication. Supplementary video
available at http://sistemas.fciencias.unam.mx/~dsanders
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