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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
Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
We give examples over arbitrary fields of rings of invariants that are not
finitely generated. The group involved can be as small as three copies of the
additive group, as in Mukai's examples over the complex numbers. The failure of
finite generation comes from certain elliptic fibrations or abelian surface
fibrations having positive Mordell-Weil rank.
Our work suggests a generalization of the Morrison-Kawamata cone conjecture
from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in
dimension 2 in the case of minimal rational elliptic surfaces.Comment: 26 pages. To appear in Compositio Mathematic
Addition law structure of elliptic curves
The study of alternative models for elliptic curves has found recent interest
from cryptographic applications, once it was recognized that such models
provide more efficiently computable algorithms for the group law than the
standard Weierstrass model. Examples of such models arise via symmetries
induced by a rational torsion structure. We analyze the module structure of the
space of sections of the addition morphisms, determine explicit dimension
formulas for the spaces of sections and their eigenspaces under the action of
torsion groups, and apply this to specific models of elliptic curves with
parametrized torsion subgroups
Rational S^1-equivariant elliptic cohomology
For each elliptic curve A over the rational numbers we construct a 2-periodic
S^1-equivariant cohomology theory E whose cohomology ring is the sheaf
cohomology of A; the homology of the sphere of the representation z^n is the
cohomology of the divisor A(n) of points with order dividing n. The
construction proceeds by using the algebraic models of the author's AMS Memoir
``Rational S^1 equivariant homotopy theory.'' and is natural and explicit in
terms of sheaves of functions on A.
This is Version 5.2 of a paper of long genesis (this should be the final
version). The following additional topics were first added in the Fourth
Edition:
(a) periodicity and differentials treated
(b) dependence on coordinate
(c) relationship with Grojnowksi's construction and, most importantly,
(d) equivalence between a derived category of O_A-modules and a derived
category of EA-modules. The Fifth Edition included
(e) the Hasse square and
(f) explanation of how to calculate maps of EA-module spectra
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