scholarly journals Computing abelian varieties over finite fields isogenous to a power

2019 ◽  
Vol 5 (4) ◽  
Author(s):  
Stefano Marseglia
Keyword(s):  
Characteristic Polynomial ◽  
Finite Fields ◽  
Abelian Varieties ◽  
Number Fields ◽  
Explicit Description ◽  
Finite Product ◽  
Real Roots ◽  
Isomorphism Classes ◽  
Ordinary Case ◽  
Theoretic Description

Abstract In this paper we give a module-theoretic description of the isomorphism classes of abelian varieties A isogenous to $$B^r$$Br, where the characteristic polynomial g of Frobenius of B is an ordinary square-free q-Weil polynomial, for a power q of a prime p, or a square-free p-Weil polynomial with no real roots. 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.

scholarly journals Finiteness theorems for K3 surfaces and abelian varieties of CM type

2018 ◽  
Vol 154 (8) ◽  
pp. 1571-1592 ◽  
Author(s):  
Martin Orr ◽  
Alexei N. Skorobogatov
Keyword(s):  
Abelian Varieties ◽  
Algebraic Closure ◽  
Complex Multiplication ◽  
Uniform Boundedness ◽  
Invariant Subgroup ◽  
Smooth Projective Variety ◽  
Number Fields ◽  
K3 Surfaces ◽  
Isomorphism Classes ◽  
Finiteness Theorems

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

scholarly journals Polarizations on abelian varieties

2002 ◽  
Vol 133 (2) ◽  
pp. 223-233 ◽  
Author(s):  
A. SILVERBERG ◽  
YU. G. ZARHIN
Keyword(s):  
Abelian Variety ◽  
Finite Fields ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Number Fields ◽  
General Context ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic

Every isogeny class over an algebraically closed field contains a principally polarized abelian variety ([10, corollary 1 to theorem 4 in section 23]). Howe ([3]; see also [4]) gave examples of isogeny classes of abelian varieties over finite fields with no principal polarizations (but not with the degrees of all the polarizations divisible by a given non-zero integer, as in Theorem 1·1 below). In [17] we obtained, for all odd primes [lscr ], isogeny classes of abelian varieties in positive characteristic, all of whose polarizations have degree divisible by [lscr ]2. We gave results in the more general context of invertible sheaves; see also Theorems 6·1 and 5·2 below. Our results gave the first examples for which all the polarizations of the abelian varieties in an isogeny class have degree divisible by a given prime. Inspired by our results in [17], Howe [5] recently obtained, for all odd primes [lscr ], examples of isogeny classes of abelian varieties over fields of arbitrary characteristic different from [lscr ] (including number fields), all of whose polarizations have degree divisible by [lscr ]2.

scholarly journals Public-key cryptosystem based on invariants of diagonalizable groups

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
František Marko ◽  
Alexandr N. Zubkov ◽  
Martin Juráš
Keyword(s):  
Linear Algebra ◽  
Finite Fields ◽  
Number Fields ◽  
Euclidean Algorithm ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

scholarly journals Cycles in the de Rham cohomology of abelian varieties over number fields

2018 ◽  
Vol 154 (4) ◽  
pp. 850-882
Author(s):  
Yunqing Tang
Keyword(s):  
Endomorphism Ring ◽  
Abelian Varieties ◽  
Number Fields ◽  
De Rham Cohomology ◽  
Tate Conjecture ◽  
De Rham ◽  
Prime Dimension

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of$\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

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