Introduction to algebraic approaches for solving isogeny path-finding problems (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

The isogeny path-finding is a computational problem that finds an isogeny connecting two given isogenous elliptic curves. The hardness of the isogeny path-finding problem supports the fundamental security of isogeny-based cryptosystems. In this paper, we introduce an algebraic approach for solving the isogeny path-finding problem. The basic idea is to reduce the isogeny problem to a system of algebraic equations using modular polynomials, and to solve the system by Gröbner basis computation. We report running time of the algebraic approach for solving the isogeny path-finding problem of 3-power isogeny degrees on supersingular elliptic curves. This is a brief summary of [16] with implementation codes

Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with non-regular drift

International audienceWe consider an Euler-Maruyama type approximation method for a stochastic differential equation (SDE) with a non-regular drift and regular diffusion coefficient. The method regu-larizes the drift coefficient within a certain class of functions and then the Euler-Maruyama scheme for the regularized scheme is used as an approximation. This methodology gives two errors. The first one is the error of regularization of the drift coefficient within a given class of parametrized functions. The second one is the error of the regularized Euler-Maruyama scheme. After an optimization procedure with respect to the parameters we obtain various rates, which improve other known results

On Weak Approximation of Stochastic Differential Equations with Discontinuous Drift Coefficient

This is an abridged version submitted in a conference proceedings.International audienceIn this paper, weak approximations of multi-dimensional stochastic differential equations with discontinuous drift coefficients are considered. Here as the approximated process, the Euler-Maruyama approximation of SDEs with approximated drift coefficients is used, and we provide a rate of weak convergence of them. Finally we present a rate of weak convergence of the Euler-Maruyama approximation of the original SDEs with constant diffusion coefficients

A New Constraint on the Lyα\alpha Fraction of UV Very Bright Galaxies at Redshift 7

We study the extent to which very bright (-23.0 < MUV < -21.75) Lyman-break selected galaxies at redshifts z~7 display detectable Lya emission. To explore this issue, we have obtained follow-up optical spectroscopy of 9 z~7 galaxies from a parent sample of 24 z~7 galaxy candidates selected from the 1.65 sq.deg COSMOS-UltraVISTA and SXDS-UDS survey fields using the latest near-infrared public survey data, and new ultra-deep Subaru z'-band imaging (which we also present and describe in this paper). Our spectroscopy has yielded only one possible detection of Lya at z=7.168 with a rest-frame equivalent width EW_0 = 3.7 (+1.7/-1.1) Angstrom. The relative weakness of this line, combined with our failure to detect Lya emission from the other spectroscopic targets allows us to place a new upper limit on the prevalence of strong Lya emission at these redshifts. For conservative calculation and to facilitate comparison with previous studies at lower redshifts, we derive a 1-sigma upper limit on the fraction of UV bright galaxies at z~7 that display EW_0 > 50 Angstrom, which we estimate to be < 0.23. This result may indicate a weak trend where the fraction of strong Lya emitters ceases to rise, and possibly falls between z~6 and z~7. Our results also leave open the possibility that strong Lya may still be more prevalent in the brightest galaxies in the reionization era than their fainter counterparts. A larger spectroscopic sample of galaxies is required to derive a more reliable constraint on the neutral hydrogen fraction at z~7 based on the Lya fraction in the bright galaxies.Comment: 20 pages, 7 figures, accepted for publication in Ap