The groups of points on abelian varieties over finite fields

Rybakov, Sergey

doi:10.2478/s11533-010-0003-x

Cited by 11 publications

(19 citation statements)

References 9 publications

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“…m r be nonnegative integers, and let H = r i=1 Z/ mi Z. The Hodge polygon Hp (H, r) is the convex polygon with vertices (i, r−i j=1 m j ) for 0 i < r. Given an abelian group G, we let G denote the -primary component of G. The following is the main result of [10]. Theorem 2.3 (Rybakov).…”

Section: Weil Polynomials and Groups Of Abelian Surfacesmentioning

confidence: 99%

“…The preceding theorem is the original formulation of [10], but it is easy to restate the theorem without any reference to the Newton polygon or the Hodge polygon. Namely, we can state the theorem in terms of the divisibility of the derivatives of f A (T ) at T = 1.…”

Section: Weil Polynomials and Groups Of Abelian Surfacesmentioning

confidence: 99%

“…We then want to study which of the groups G(n 1 , n 2 , n 3 , n 4 ) actually occur when we vary over all finite fields F q and over all abelian surfaces A/F q . For fixed q, a characterization of the groups occurring as the group of points on a general abelian variety was recently found by Rybakov [10,11]. Rybakov's elegant criterion relates the Newton polygon of the characteristic polynomial of the variety to the Hodge polygon of the group.…”

Section: Introductionmentioning

confidence: 99%

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Abelian surfaces over finite fields with prescribed groups

David

Garton

Scherr

et al. 2014

Bulletin of the London Mathematical Society

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Let A be an abelian surface over Fq, the field of q elements. The rational points on A/Fq form an abelian group A(F q ) Z/n1Z × Z/n1n2Z × Z/n1n2n3Z × Z/n1n2n3n4Z. We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups Z/n 1Z × Z/n1n2Z × Z/n1n2n3Z × Z/n1n2n3n4Z do not occur if n 1 is very large with respect to n2, n3, n4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).

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Section: Weil Polynomials and Groups Of Abelian Surfacesmentioning

confidence: 99%

Section: Weil Polynomials and Groups Of Abelian Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Abelian surfaces over finite fields with prescribed groups

David

Garton

Scherr

et al. 2014

Bulletin of the London Mathematical Society

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“…Tsfasman described groups of points for dim X = 1 in [Tsf85]. The classification for dim X = 1, 2 and partially in case dim X = 3 was obtained by Rybakov in [Ryb10], [Ryb12], and [Ryb15]. In this work we are going to apply the following result:…”

Section: Introductionmentioning

confidence: 98%

“…Theorem 1.3 ( [Ryb10]). Suppose f X (t) is separable and given an l-group H such that Np(f X (1 − t)) lies on or above Hp(H, 2g) and their endpoints coincide.…”

Section: Introductionmentioning

confidence: 99%

Groups of points on abelian threefolds over finite fields

Kotelnikova

2019

Finite Fields and Their Applications

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In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given Fq-isogeny class. This work completes partial classification given in [Ryb15].We denote by Fr X the Frobenius map acting on T l (X). The characteristic polynomial f X (t) of Fr X is a q-Weil polynomial, i. e. f X (t) is monic, defined over Z and absolute values of its roots are all equal to √ q. The following statement holds:Proposition 1.1 ([Ryb10]). The group of points X(F q ) l is isomorphic to coker(1 − Fr X ).We call the exponents of the group coker(1 − Fr X ) Smith's invariants of the operator (1 − Fr X ) as well as of a Z l [1 − Fr X ]-module T l (X) (see section 2.3 for a thorough definition).To state the results by Rybakov concerning groups of points on ordinary varieties it is convenient to introduce the notion of Newton polygon and Hodge polygon. Both of them are thought of as subsets of plane with coordinates (x, y).

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On some categories of structured sets

Chiaselotti,

Gentile,

Infusino

2024

European Journal of Mathematics

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Given an arbitrary set $$\Omega $$ Ω , we consider the collections $$\textrm{SS}\hspace{0.55542pt}(\Omega )$$ SS ( Ω ) , $$\textrm{SR}\hspace{0.55542pt}(\Omega )$$ SR ( Ω ) and $$\textrm{SO}\hspace{0.55542pt}(\Omega )$$ SO ( Ω ) of all the set systems, the binary set relations and the set operators on $$\Omega $$ Ω . We introduce the notion of linking map on $$\Omega $$ Ω as any map whose domain and codomain may be chosen between the above collections. After providing a descriptive overview useful for framing the notion of linking map in a broad non-specialized context, we explain how linking maps occur in a very natural way in two specific results. The first of these results concerns the classic identification between the subfamily $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) of all the equivalence relations on $$\Omega $$ Ω and the subfamily $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) of all the set partitions on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) of closure operators on $$\Omega $$ Ω and four linking maps whose restrictions to the subfamilies $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) , $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) and $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) are bijections. The second result concerns the identification between the subfamily $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) of all closure set operators on $$\Omega $$ Ω and the subfamily $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω ) of all closure set systems on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{DSR}\hspace{0.55542pt}(\Omega )$$ DSR ( Ω ) of binary set relations and four linking maps whose restrictions to the subfamilies $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) , $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω ) and $$\textrm{DSR}\hspace{0.55542pt}(\Omega )$$ DSR ( Ω ) are again bijections. In an attempt to extend in a natural way the above linking maps to categorical isomorphisms, after fixing a nonnegative integer k, we introduce three categories $$\mathbf{SS^k}$$ SS k , $$\mathbf{SR^k}$$ SR k and $$\mathbf{SO^k}$$ SO k , whose detailed study mainly occupies the first part of the present work. Objects and arrows of these three categories are obtained by means of k-iterations of the powerset functor "Equation missing", and they generalize the notions of set systems, set relations and set operators, respectively. In the second part of the paper, we extend the linking maps previously described at a categorical level in terms of isomorphisms between specific categories of set systems, binary set relations and set operators generalizing the occurring collections introduced before, and also prove numerous other results concerning the main properties of all these categories, such as completeness, cocompleteness and Cartesian closedness.

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The groups of points on abelian varieties over finite fields

Cited by 11 publications

References 9 publications

Abelian surfaces over finite fields with prescribed groups

Abelian surfaces over finite fields with prescribed groups

Groups of points on abelian threefolds over finite fields

On some categories of structured sets

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