20 research outputs found
Some heuristics about elliptic curves
We give some heuristics for counting elliptic curves with certain properties.
In particular, we re-derive the Brumer-McGuinness heuristic for the number of
curves with positive/negative discriminant up to , which is an application
of lattice-point counting. We then introduce heuristics (with refinements from
random matrix theory) that allow us to predict how often we expect an elliptic
curve with even parity to have . We find that we expect there to
be about curves with with even parity
and positive (analytic) rank; since Brumer and McGuinness predict
total curves, this implies that asymptotically almost all even parity curves
have rank 0. We then derive similar estimates for ordering by conductor, and
conclude by giving various data regarding our heuristics and related questions
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
Counting elliptic curves of bounded Faltings height
We give an asymptotic formula for the number of elliptic curves over
with bounded Faltings height. Silverman has shown that the
Faltings height for elliptic curves over number fields can be expressed in
terms of modular functions and the minimal discriminant of the elliptic curve.
We use this to recast the problem as one of counting lattice points in a
particular region in .Comment: 12 pages, 2 figures, 1 table. To be published in Acta Arithmetic
Moments of the critical values of families of elliptic curves, with applications
We make conjectures on the moments of the central values of the family of all
elliptic curves and on the moments of the first derivative of the central
values of a large family of positive rank curves. In both cases the order of
magnitude is the same as that of the moments of the central values of an
orthogonal family of L-functions. Notably, we predict that the critical values
of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves
in the positive rank family.
Furthermore, as arithmetical applications we make a conjecture on the
distribution of a_p's amongst all rank 2 elliptic curves, and also show how the
Riemann hypothesis can be deduced from sufficient knowledge of the first moment
of the positive rank family (based on an idea of Iwaniec).Comment: 24 page
Nonvanishing of twists of -functions attached to Hilbert modular forms
We describe algorithms for computing central values of twists of
-functions associated to Hilbert modular forms, carry out such computations
for a number of examples, and compare the results of these computations to some
heuristics and predictions from random matrix theory.Comment: 19 page
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