Under some extra assumptions on the polynomial g we give an explicit description of all the isomorphism classes which can be computed in terms of fractional ideals of an order in a finite product of number fields.

In the ordinary case, we also give a module-theoretic description of the polarizations of A. When we move to the wilder realm of positive characteristic we cannot have such a functorial description due to the existence of objects like supersingular elliptic curves whose endomorphisms form a quaternionic algebra which does not admit a 2-dimensional representation, as pointed out by Serre.

In the ordinary case, Howe has included in this equivalence the notions of dual variety and polarizations, see [ 16 ]. In [ 24 ] we have used such descriptions to produce algorithms to compute the isomorphism classes of abelian varieties with square-free characteristic polynomial of Frobenius and, when applicable, the polarizations and the corresponding automorphism groups. In the ordinary case we translate the notion of a polarization to this context.

When the order R is Bass there is a classification of such modules, see [ 2 ] and [ 22 ], and we can explicitly compute representatives of the isomorphism classes of the abelian varieties. There are other categorical descriptions of the category of abelian varieties isogenous to a power of elliptic curves in terms of modules with extra-structure, see the Appendix in [ 1921 ] and [ 18 ].

The paper is structured as follows.

In Sect. These equivalences are based on the theorems of Deligne and Centelegheâ€”Stix cited above and the target category of the functors realizing them is well suited for computational purposes. Finally, in Sect. The aim of the paper is to provide an effective algorithm to perform computations of isomorphism classes of abelian varieties. We get also statements about polarized abelian varieties, see Corollary 5.

All rings considered are commutative and unital. All morphisms between abelian varieties A and B over a field k are also defined over kunless otherwise specified.

Also, an abelian variety A is simple if it is so over the field of definition. Let g be an integral square-free monic polynomial, say of degree n. Note that K is a finite product of distinct number fields. An over-order of R is an order S in K containing R. A fractional ideal of R is a finitely generated sub- R -module of K containing a non-zero-divisor. Observe that the underlying additive subgroup of any fractional ideal I is a free abelian group of rank nthat is, I is a lattice in K.

Given any full lattice I in K the set I : I is an order. This order is called the multiplicator ring of I. An order R is called Bass if every over-order of R is Gorenstein. Since in this paper we will extensively use the properties of Bass orders we will list here other equivalent definitions. The study of such orders started with the paper [ 2 ] on Gorenstein rings.

There are many sources where one can find a proof of Proposition 2. Since every fractional ideal of a quadratic order can be generated by 2 elements as an abelian group, they are examples of Bass orders. In general we have that.Ian Blake 1, V. Kumar Murty 2and Hamid Usefi 3. Aubry, Reed-Muller codes associated to projective algebraic varieties, in Coding Theory and Algebraic Geometry, 4.

Google Scholar. Bartoli, M. De Boeck, S. Fanali and L. Storme, On the functional codes defined by quadrics and Hermitian varieties, Des. Codes Crypt. Bose, On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements, Calcutta Math. Bose and I. Calderbank and W. Kantor, The geometry of two weight codes, Bull. London Math. Cherdieu and R. Edoukou, A. Hallex, F. Rodier and L. Storme, The small weight codewords of the functional codes associated to non-singular Hermitian varieties, Des.

Edoukou, S. Ling and C. Xing, Structure of functional codes defined on non-degenerate Hermitian varieties, J.

Theory Ser. A, Ghorpade and G.In this note we investigate minimal linear codes arising from Hermitian varieties and quadrics. We study their parameters and formulate some open problems about their weight distribution. From now on, we will call C a [ nk ] linear code endowed with the Hamming metric, i.

A codeword c is said to be minimal if its support determines c up to a scalar multiplication, i. Minimal codewords were employed by Massey see [ 1011 ] for the construction of a perfect and ideal SSS, in which the access structure is determined by the set of the minimal codewords of a linear code.

Unfortunately, the determination of the set of minimal codewords of a given code is a difficult task. This fact led to the study of codes for which every non-zero codeword is minimal, that are called minimal codes. A useful tool to construct minimal codes was given by Ashikhmin and Barg see [ 2 ]. Families of these codes were first constructed in [ 7 ], whereas [ 68 ] give the first infinite family of minimal codes for the binary and ternary case, respectively.

Afterwards, in [ 3 ] the first examples of minimal linear codes for every field of odd characteristic were constructed. Following the geometrical approach of [ 3 ], in [ 4 ] it was proved that it is possible to construct families of minimal codes through the study of cutting blocking sets.

Recently, in [ 113 ] the authors found out independently that cutting blocking sets not only determine minimal codes, but they are actually in bijection with them. Therefore, minimal codes and algebraic varieties over finite fields are very related objects.

In this note we investigate families of minimal codes arising from celebrated objects in finite geometry, i. Hermitian varieties and quadrics, giving some information about their weight distribution.

In the following, we recall some useful basic definitions on blocking sets; for an exhaustive reference we point to [ 12Chapter 3].

The dimension of a k -blocking set corresponds to the dimension of the subspace generated by its elements. As we already mentioned, in order to construct minimal linear codes as in [ 4 ] we need to introduce a particular class of blocking set.

The resulting code does not depend on this choice. The following result gives a sufficient condition to construct minimal codes see [ 4 ] for details. Our aim is to show that properties a and b of Theorem 1 hold in the case of Hermitian varieties and quadrics. Our notations and terminologies are standard, see [ 9 ].

The following result is a corollary of [ 9Lemma 2. Propositions 1 and 2 allow to prove that Theorem 1 can be applied to non-singular Hermitian varieties. Condition a of Theorem 1 holds for non-singular Hermitian varieties. Then Condition b of Theorem 1 holds.

Since the number of points of a non-singular projective Hermitian variety of dimension n is. We collect here the weight of some codewords, as a first approach to the weight distribution problem. The next proposition is from Remark 1. Let q be odd.

## Abelian Varieties

If f is non-degenerate, i. Up to projective equivalence there are one or two distinct non-singular quadrics according to n being even or odd; see [ 9Chapter 1]. We say that two non-singular quadrics have the same character if they are both parabolic character 1both hyperbolic character 2 or both elliptic character 0.

This results allow us to prove properties a and b of Theorem 1 hold for non-singular quadrics in canonical form. We prove the claim separately for elliptic, parabolic and hyperbolic quadrics. In this paper we proved that Hermitian varieties and quadrics are cutting blocking sets, and hence they give place to minimal codes.Thanks for helping us catch any problems with articles on DeepDyve.

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Also, a family of caps of H n,q2 is constructed from Wq2-maximal curves. A natural generalization of an ovoid is a cap also called a partial ovoid. Advances in Geometry â€” de Gruyter. Enjoy affordable access to over 18 million articles from more than 15, peer-reviewed journals. Get unlimited, online access to over 18 million full-text articles from more than 15, scientific journals. See the journals in your area. Save searches from Google Scholar, PubMed.

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They were placed on your computer when you launched this website. You can change your cookie settings through your browser.Finite Fields are fundamental structures in mathematics. They lead to interesting deep problems in number theory, play a major role in combinatorics and finite geometry, and have a vast amount of applications in computer science. Papers in this volume cover these aspects of finite fields as well as applications in coding theory and cryptography.

Topics in Finite Fields. Gohar KyureghyanGary L. MullenAlexander Pott. American Mathematical Soc. Collineation groups strongly irreducible on an oval in a projective plane of odd order. On the solvability of certain equations over finite fields. On automorphism groups of binary linear codes. Extended differential properties of cryptographic functions.

## ANNALES DE L'INSTITUT FOURIER

A divisibility criterion for exceptional APN functions. Non weakly regular bent polynomials from vectorial quadratic functions. Strongly regular graphs arising from Hermitian varieties. A survey of permutation binomials and trinomials over finite fields. Computing class groups of function fields using stark units. Finding primitive elements in finite fields of small characteristic. The coset leader and list weight enumerator.

Wieferich past and future. Field reduction and linear sets in finite geometry. Bent functions from spreads. On the characterization of a semimultiplicative analogue of planar functions over finite fields. A duality statement. An upper bound for the number of Galois points for a plane curve.

**Fields Medal â€” Peter Scholze â€” ICM2018**

A generalization of the nonlinear combination generator. Dedekind sums with a parameter in function fields. Numbers of points of hypersurfaces without lines over finite fields. Optimal binary subspace codes of length 6 constant dimension 3 and minimum subspace distance 4.

A solution of an equivalence problem for semisimple cyclic codes.The main reference book is [1]. See also [2] and [3]. Please let me know if you find any typos or mistakes!

Euler discovered an addition formula for elliptic integrals where and is a certain algebraic function. In modern language, the affine equation defines an elliptic curve and the group structure on it gives the addition formula.

### Hermitian symmetric space

More generally, let be an algebraic curve with genusthen integration gives a map and an isomorphism is called the Jacobian of and has a natural group structure. Notice that the group law on is compatible with its algebraic variety structure. This motivates us to make the following definition. So we can associate an abelian variety to each algebraic curve with. Another example of abelian varieties comes from number theory.

Let be a totally real extension of degree and be am imaginary quadratic extension, i. Then has elements and the complex conjugation acts on it. Thus a CM-type gives an isomorphism by evaluation. For any lattice in a 1-dimensional complex vector space, is an abelian variety.

We define Then is positive definite and restricts to an integral pairing: and. The answer to the first question is false: even forfor almost all latticesthe complex torus is not an abelian variety. However, the converse is true: every abelian variety must be a complex torus. It follows that abelian varieties are complex tori. The following holds for any complex torus, hence any abelian variety. Now we introduce the general notion of abelian varieties over an arbitrary field.

By a variety overwe mean a geometrically integral, separated and finite type -scheme. Since is surjective, we know that is torsion-free.

Over complex numbers, is a free abelian group of finite rank, so we know that is a finite generated abelian group. More generally, over an arbitrary field. Let be a prime different fromthen. It is a -module as is defined over. The Tate module can be viewed as an analog of the homology group.Let X be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field F q 2 of order q 2. If the number of F q 2 -rational points of X satisfies the Hasseâ€”Weil upper bound, then X is said to be F q 2 -maximal.

Download to read the full article text. Google Scholar. Ballico E. Pure Appl. Algebra Cossidente, A. Algebra 28 10 Eisenbud, D. Fuhrmann, R. Number Theory 67 Palermo 51 Garcia, A. Mat N. Hefez, A. Palermo, Suppl. Hirschfeld, J.

Homma, M. Algebra 20 Kaji, H. Lang, S. Rathmann, J. Reine Angew. Segre, B. Pura Appl. London Math. Tate, J. Download references. Sauro 85,Potenza, Italy. You can also search for this author in PubMed Google Scholar.

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