1,727 research outputs found
Descent Via Isogeny on Elliptic Curves with Large Rational Torsion Subgroups.
We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. We apply these techniques to several examples: in one case we use descent via 9-isogeny to determine the rank of an elliptic curve; in another case we find nontrivial members of the 9-part of the Tate-Shafarevich group, and in a further case, nontrivial members of the 13-part of the Tate-Shafarevich group
Amicable pairs and aliquot cycles for elliptic curves
An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good
reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study
elliptic amicable pairs and analogously defined longer elliptic aliquot cycles.
We show that there exist elliptic curves with arbitrarily long aliqout cycles,
but that CM elliptic curves (with j not 0) have no aliqout cycles of length
greater than two. We give conjectural formulas for the frequency of amicable
pairs. For CM curves, the derivation of precise conjectural formulas involves a
detailed analysis of the values of the Grossencharacter evaluated at a prime
ideal P in End(E) having the property that #E(F_P) is prime. This is especially
intricate for the family of curves with j = 0.Comment: 53 page
Fast algorithms for computing isogenies between elliptic curves
We survey algorithms for computing isogenies between elliptic curves defined
over a field of characteristic either 0 or a large prime. We introduce a new
algorithm that computes an isogeny of degree ( different from the
characteristic) in time quasi-linear with respect to . This is based in
particular on fast algorithms for power series expansion of the Weierstrass
-function and related functions
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
Computing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class
polynomial . The first is a p-adic lifting algorithm for inert primes p
in the order of discriminant D < 0. The second is an improved Chinese remainder
algorithm which uses the class group action on CM-curves over finite fields.
Our run time analysis gives tighter bounds for the complexity of all known
algorithms for computing , and we show that all methods have comparable
run times
Constructing Permutation Rational Functions From Isogenies
A permutation rational function is a rational function
that induces a bijection on , that is, for all
there exists exactly one such that . Permutation
rational functions are intimately related to exceptional rational functions,
and more generally exceptional covers of the projective line, of which they
form the first important example.
In this paper, we show how to efficiently generate many permutation rational
functions over large finite fields using isogenies of elliptic curves, and
discuss some cryptographic applications. Our algorithm is based on Fried's
modular interpretation of certain dihedral exceptional covers of the projective
line (Cont. Math., 1994)
Isogenies of Elliptic Curves: A Computational Approach
Isogenies, the mappings of elliptic curves, have become a useful tool in
cryptology. These mathematical objects have been proposed for use in computing
pairings, constructing hash functions and random number generators, and
analyzing the reducibility of the elliptic curve discrete logarithm problem.
With such diverse uses, understanding these objects is important for anyone
interested in the field of elliptic curve cryptography. This paper, targeted at
an audience with a knowledge of the basic theory of elliptic curves, provides
an introduction to the necessary theoretical background for understanding what
isogenies are and their basic properties. This theoretical background is used
to explain some of the basic computational tasks associated with isogenies.
Herein, algorithms for computing isogenies are collected and presented with
proofs of correctness and complexity analyses. As opposed to the complex
analytic approach provided in most texts on the subject, the proofs in this
paper are primarily algebraic in nature. This provides alternate explanations
that some with a more concrete or computational bias may find more clear.Comment: Submitted as a Masters Thesis in the Mathematics department of the
University of Washingto
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