26 research outputs found
New variables of separation for particular case of the Kowalevski top
We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in
special case of zero square integral. An explicit procedure to find variables
of separation and separation relations is considered in detail.Comment: 11 pages, LaTeX with Ams font
On Charge-3 Cyclic Monopoles
We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3
cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a
(Toda) spectral curve of genus 2. A well adapted homology basis is presented
enabling the theta functions and monopole data of the genus 4 curve to be given
in terms of genus 2 data. The Richelot correspondence, a generalization of the
arithmetic mean, is used to solve for this genus 2 curve. Results of other
approaches are compared.Comment: 34 pages, 16 figures. Revision: Abstract added and a few small
change
Computing supersingular isogenies on Kummer surfaces
We apply Scholten\u27s construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over and corresponding abelian surfaces over . Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2,2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography
Leonard Euler: addition theorems and superintegrable systems
We consider the Euler approach to construction and to investigation of the
superintegrable systems related to the addition theorems. As an example we
reconstruct Drach systems and get some new two-dimensional superintegrable
Stackel systems.Comment: The text of the talk at International Conference Geometry, Dynamics,
Integrable Systems, September 2-7, 2008, Belgrade, Serbia, LaTeX, 18 page