2,595 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
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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
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
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
The effect of convolving families of L-functions on the underlying group symmetries
L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M -->
oo, the statistical behavior (1-level density) of the low-lying zeros of
L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of
matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity,
where each G_i is one of the classical compact groups (unitary, symplectic or
orthogonal). Assuming that the convolved families of L-functions F_N x G_M are
automorphic, we study their 1-level density. (We also study convolved families
of the form f x G_M for a fixed f.) Under natural assumptions on the families
(which hold in many cases) we can associate to each family L of L-functions a
symmetry constant c_L equal to 0 (resp., 1 or -1) if the corresponding
low-lying zero statistics agree with those of the unitary (resp., symplectic or
orthogonal) group. Our main result is that c_{F x G} = c_G * c_G: the symmetry
type of the convolved family is the product of the symmetry types of the two
families. A similar statement holds for the convolved families f x G_M. We
provide examples built from Dirichlet L-functions and holomorphic modular forms
and their symmetric powers. An interesting special case is to convolve two
families of elliptic curves with rank. In this case the symmetry group of the
convolution is independent of the ranks, in accordance with the general
principle of multiplicativity of the symmetry constants (but the ranks persist,
before taking the limit N,M --> oo, as lower-order terms).Comment: 41 pages, version 2.1, cleaned up some of the text and weakened
slightly some of the conditions in the main theorem, fixed a typ
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