7,719 research outputs found
A conditional determination of the average rank of elliptic curves
Under a hypothesis which is slightly stronger than the Riemann Hypothesis for
elliptic curve -functions, we show that both the average analytic rank and
the average algebraic rank of elliptic curves in families of quadratic twists
are exactly . As a corollary we obtain that under this last
hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all
curves in our family, and that asymptotically one half of these curves have
algebraic rank , and the remaining half . We also prove an analogous
result in the family of all elliptic curves. A way to interpret our results is
to say that nonreal zeros of elliptic curve -functions in a family have a
direct influence on the average rank in this family. Results of Katz-Sarnak and
of Young constitute a major ingredient in the proofs.Comment: 27 page
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
A database of genus 2 curves over the rational numbers
We describe the construction of a database of genus 2 curves of small
discriminant that includes geometric and arithmetic invariants of each curve,
its Jacobian, and the associated L-function. This data has been incorporated
into the L-Functions and Modular Forms Database (LMFDB).Comment: 15 pages, 7 tables; bibliography formatting and typos fixe
Pseudospherical surfaces with singularities
We study a generalization of constant Gauss curvature -1 surfaces in
Euclidean 3-space, based on Lorentzian harmonic maps, that we call
pseudospherical frontals. We analyze the singularities of these surfaces,
dividing them into those of characteristic and non-characteristic type. We give
methods for constructing all non-degenerate singularities of both types, as
well as many degenerate singularities. We also give a method for solving the
singular geometric Cauchy problem: construct a pseudospherical frontal
containing a given regular space curve as a non-degenerate singular curve. The
solution is unique for most curves, but for some curves there are infinitely
many solutions, and this is encoded in the curvature and torsion of the curve.Comment: 26 pages, 11 figures. Version 3: examples added (new Section 6).
Introduction section revise
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