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

Transformations of Spherical Blocks

We further explore the correspondence between N=2 supersymmetric SU(2) gauge theory with four flavors on epsilon-deformed backgrounds and conformal field theory, with an emphasis on the epsilon-expansion of the partition function natural from a topological string theory point of view. Solving an appropriate null vector decoupling equation in the semi-classical limit allows us to express the instanton partition function as a series in quasi-modular forms of the group Gamma(2), with the expected symmetry Weyl group of SO(8) semi-direct S_3. In the presence of an elementary surface operator, this symmetry is enhanced to an action of the affine Weyl group of SO(8) semi-direct S_4 on the instanton partition function, as we demonstrate via the link between the null vector decoupling equation and the quantum Painlev\'e VI equation.Comment: 31 pages, 1 figure; v2: typos corrected, references adde

Symmetry breaking and manipulation of nonlinear optical modes in an asymmetric double-channel waveguide

We study light-beam propagation in a nonlinear coupler with an asymmetric double-channel waveguide and derive various analytical forms of optical modes. The results show that the symmetry-preserving modes in a symmetric double-channel waveguide are deformed due to the asymmetry of the two-channel waveguide, yet such a coupler supports the symmetry-breaking modes. The dispersion relations reveal that the system with self-focusing nonlinear response supports the degenerate modes, while for self-defocusingmedium the degenerate modes do not exist. Furthermore, nonlinear manipulation is investigated by launching optical modes supported in double-channel waveguide into a nonlinear uniform medium.Comment: 10 page

Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation

In this work the correspondence between the semiclassical limit of the DOZZ quantum Liouville theory on the torus and the Nekrasov-Shatashvili limit of the N=2* (Omega-deformed) U(2) super-Yang-Mills theory is used to propose new formulae for the accessory parameter of the Lame equation. This quantity is in particular crucial for solving the problem of uniformization of the one-punctured torus. The computation of the accessory parameters for torus and sphere is an open longstanding problem which can however be solved if one succeeds to derive an expression for the so-called classical Liouville action. The method of calculation of the latter has been proposed some time ago by Zamolodchikov brothers. Studying the semiclassical limit of the four-point function of the quantum Liouville theory on the sphere they have derived the classical action for the Riemann sphere with four punctures. In the present work Zamolodchikovs idea is exploited in the case of the Liouville field theory on the torus. It is found that the Lame accessory parameter is determined by the classical Liouville action on the one-punctured torus or more concretely by the torus classical block evaluated on the saddle point intermediate classical weight. Secondly, as an implication of the aforementioned correspondence it is obtained that the torus accessory parameter is related to the sum of all rescaled column lengths of the so-called "critical" Young diagrams extremizing the instanton "free energy" for the N=2* gauge theory. Finally, it is pointed out that thanks to the known relation the sum over the "critical" column lengths can be expressed in terms of a contour integral in which the integrand is built out of certain special functions.Comment: 41 pages, published versio

BPS Geodesics in N=2 Supersymmetric Yang-Mills Theory

We introduce some techniques for making a more global analysis of the existence of geodesics on a Seiberg-Witten Riemann surface with metric $ds^2 = |\lambda_{SW}|^2$. Because the existence of such geodesics implies the existence of BPS states in N=2 supersymmetric Yang-Mills theory, one can use these methods to study the BPS spectrum in various phases of the Yang-Mills theory. By way of illustration, we show how, using our new methods, one can easily recover the known results for the N=2 supersymmetric SU(2) pure gauge theory, and we show in detail how it also works for the N=2, SU(2) theory coupled to a massive adjoint matter multiplet.Comment: 23 pages, harvmac, epsf, 8 figure

How to compute an isogeny on the extended Jacobi quartic curves?

Computing isogenies between elliptic curves is a significantpart of post-quantum cryptography with many practicalapplications (for example, in SIDH, SIKE, B-SIDH, or CSIDHalgorithms). Comparing to other post-quantum algorithms, themain advantages of these protocols are smaller keys, the similaridea as in the ECDH, and a large basis of expertise aboutelliptic curves. The main disadvantage of the isogeny-basedcryptosystems is their computational efficiency - they are slowerthan other post-quantum algorithms (e.g., lattice-based). That iswhy so much effort has been put into improving the hithertoknown methods of computing isogenies between elliptic curves.In this paper, we present new formulas for computing isogeniesbetween elliptic curves in the extended Jacobi quartic formwith two methods: by transforming such curves into the shortWeierstrass model, computing an isogeny in this form and thentransforming back into an initial model or by computing anisogeny directly between two extended Jacobi quartics