Anna Rio

Determining the 2-Sylow subgroup of an elliptic curve over a finite field

In this paper we describe an algorithm that outputs the order and the structure, including generators, of the 2-Sylow subgroup of an elliptic curve over a finite field. To do this, we do not assume any knowledge of the group order. The results that lead to the design of this algorithm are of inductive type. Then a right choice of points allows us to reach the end within a linear number of successive halvings. The algorithm works with abscissas, so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field. Efficient methods for this computation determine the efficiency of our algorithm.

Computing the ℓ-power torsion of an elliptic curve over a finite field

The algorithm we develop outputs the order and the structure, including generators, of the l-Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive l-divisions leads to generators of the subgroup under consideration. After the computation of a couple of polynomials, each division step relies on finding rational roots of polynomials of degree l. We specify in complete detail the case l = 3, when the complexity of each trisection is given by the computation of cubic roots in finite fields.

On the Galois correspondence theorem in separable Hopf Galois theory

There are fourteen ne gradings on the exceptional Lie algebra e6 over an algebraically closed eld of zero characteristic. We provide their descriptions and a proof that any ne grading is equivalent to one of them. 2010 Mathematics Subject Classication: 17B25, 17B70.There are fourteenfine gradings on the exceptional Lie algebra e6 over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that any fine grading is equivalent to one of them.In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra.In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.We show that the product BMO space can be characterized by iterated commutators of a large class of Calderon-Zygmund operators. This result followsfrom a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytonens representation theorem of Calderon-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.In this paper we survey some results on the Dirichlet problem for nonlocal operators of the form. We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the field.

The Hopf Galois property in subfield lattices

Let K/k be a finite separable extension, n its degree and its Galois closure. For n ≤ 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/k according to the Galois group (or the degree) of . In this paper we study the case n = 6, and intermediate extensions F/k such that , for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of ℚ of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.

Explicit 2-power torsion of genus 2 curves over finite fields

We give an efficient explicit algorithm to find the structure and generators of the maximal 2-subgroup of the Jacobian of a genus 2 curve over a finite field of odd characteristic. We use the 2-torsion points as seeds to successively perform a chain of halvings to find divisors of increasing 2-power order. The halving loop requires a solution to certain degree 16 polynomials over the base field, and the termination of the algorithm is based on the description of the graph structure of the maximal 2-subgroup. The structure of our algorithm is the natural extension of the even characteristic case.

Generalization of Vélu’s formulae for isogenies between elliptic curves

Given an elliptic curve E and a finite subgroup G, Velu’s formulae concern to a separable isogeny IG : E → E′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG(P) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstras coefficients of E′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.

Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

We give a deterministic polynomial time algorithm to find the structure of the 2-Sylow subgroup of the Jacobian of a genus 2 curve over a finite field of characteristic 2. Our procedure starts with the points of order 2 and then performs a chain of successive halvings while such an operation makes sense. The stopping condition is triggered when certain polynomials fail to have roots in the base field, as previously shown by I. Kitamura, M. Katagi and T. Takagi. The structure of our algorithm is similar to the already known case of genus 1 and odd characteristic.