The Galois closure of the Garcia-Stichtenoth tower

We describe the Galois closure of the Garcia-Stichtenoth tower and prove that it is optimal.Comment: 10 page

Generalization of Deuring Reduction Theorem

In this paper we generalize the Deuring theorem on a reduction of elliptic curve with complex multiplication. More precisely, for an Abelian variety $A$, arising after reduction of an Abelian variety with complex multiplication by a CM field $K$ over a number field at a pace of good reduction. We establish a connection between a decomposition of the first truncated Barsotti-Tate group scheme $A[p]$ and a decomposition of p\cO_{K} into prime ideals. In particular, we produce these explicit relationships for Abelian varieties of dimensions $1, 2$ and 3

Optimal curves of low genus over finite fields

The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves.Comment: a proof of the theorem 5.6 was correcte

On the Zeta Functions of an optimal tower of function fields over \FF_4

In this paper we derive a recursion for the zeta function of each function field in the second Garcia-Stichtenoth tower when $q=2$. We obtain our recursion by applying a theorem of Kani and Rosen that gives information about the decomposition of the Jacobians. This enables us to compute the zeta functions explicitly of the first six function fields.Comment: 14 page

The Number of Rational Points On Genus 4 Hyperelliptic Supersingular Curves in Characteristic 2

One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd

Computing algorithm for reduction type of CM abelian varieties

Let $\mathcal{A}$ be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number $p$. Denote by ${\rm A}$ an abelian variety over a finite field of characteristic $p$, obtained by the reduction of $\mathcal{A}$ at the prime ideal. In this paper we derive an algorithm which allows to decompose the group scheme ${\rm A}[p]$ into indecomposable quasi-polarized ${\rm BT}_1$-group schemes. This can be done for the unramified $p$ on the basis of its decomposition into prime ideals in the endomorphism algebra of ${\rm A}$. We also compute all types of such correspondence for abelian varieties of dimension up to $5$. As a consequence we establish the relation between the decompositions of prime $p$ and the corresponding pairs of $p$-rank and $a$-number of an abelian variety ${\rm A}$.Comment: arXiv admin note: text overlap with arXiv:1209.520

Large Scale Variable Fidelity Surrogate Modeling

Engineers widely use Gaussian process regression framework to construct surrogate models aimed to replace computationally expensive physical models while exploring design space. Thanks to Gaussian process properties we can use both samples generated by a high fidelity function (an expensive and accurate representation of a physical phenomenon) and a low fidelity function (a cheap and coarse approximation of the same physical phenomenon) while constructing a surrogate model. However, if samples sizes are more than few thousands of points, computational costs of the Gaussian process regression become prohibitive both in case of learning and in case of prediction calculation. We propose two approaches to circumvent this computational burden: one approach is based on the Nystr\"om approximation of sample covariance matrices and another is based on an intelligent usage of a blackbox that can evaluate a~low fidelity function on the fly at any point of a design space. We examine performance of the proposed approaches using a number of artificial and real problems, including engineering optimization of a rotating disk shape.Comment: 21 pages, 4 figures, Ann Math Artif Intell (2017

Characteristic Polynomial of Supersingular Abelian Varieties over Finite Fields

In this article, we give a complete description of the characteristic polynomials of supersingular abelian varieties over finite fields. We list them for the dimensions upto 7

Multifidelity Bayesian Optimization for Binomial Output

The key idea of Bayesian optimization is replacing an expensive target function with a cheap surrogate model. By selection of an acquisition function for Bayesian optimization, we trade off between exploration and exploitation. The acquisition function typically depends on the mean and the variance of the surrogate model at a given point. The most common Gaussian process-based surrogate model assumes that the target with fixed parameters is a realization of a Gaussian process. However, often the target function doesn't satisfy this approximation. Here we consider target functions that come from the binomial distribution with the parameter that depends on inputs. Typically we can vary how many Bernoulli samples we obtain during each evaluation. We propose a general Gaussian process model that takes into account Bernoulli outputs. To make things work we consider a simple acquisition function based on Expected Improvement and a heuristic strategy to choose the number of samples at each point thus taking into account precision of the obtained output

Solvability of equations in elementary functions

We prove that the equation $\tan(x) - x = a$ is unsolvable in elementary functionsComment: 8 pages, SZU, MIPT, HS