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Abelian varieties
, 2008
"... These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in ver ..."
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Cited by 160 (8 self)
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These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still
ABELIAN VARIETIES
"... A canonical reference for the subject is Mumford’s book [6], but Mumford generally works over an algebraically closed field (though his arguments can be modified to give results over an arbitrary base field). Milne’s article [4] is also a good source and allows a general base field. These notes borr ..."
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borrow heavily from van der Geer and Moonen [5], and differ in the main from [6] and [4] in that we give a more natural approach to the theory of the dual abelian variety. 1. Basic properties of abelian varieties Let k be a field. A kvariety is a geometrically integral separated kscheme of finite type
Faltings, Degeneration of abelian varieties
, 1990
"... An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We fo ..."
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Cited by 139 (8 self)
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An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We
of Abelian varieties ✩
, 2005
"... www.elsevier.com/locate/jnt On a question of Frey and Jarden about the rank ..."
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www.elsevier.com/locate/jnt On a question of Frey and Jarden about the rank
Supersingular abelian varieties in cryptology
 Advances in Cryptology  CRYPTO 2002
"... Abstract. For certain security applications, including identity based encryption and short signature schemes, it is useful to have abelian varieties with security parameters that are neither too small nor too large. Supersingular abelian varieties are natural candidates for these applications. This ..."
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Cited by 51 (7 self)
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Abstract. For certain security applications, including identity based encryption and short signature schemes, it is useful to have abelian varieties with security parameters that are neither too small nor too large. Supersingular abelian varieties are natural candidates for these applications
Hypersymmetric Abelian Varieties
, 2006
"... We introduce the notion of a hypersymmetric abelian variety over a field of positive characteristic p. We show that every symmetric Newton polygon admits a hypersymmetric abelian variety having that Newton polygon; see 2.5 and 4.8. Isogeny classes of absolutely simple hypersymmetric abelian variet ..."
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Cited by 24 (14 self)
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We introduce the notion of a hypersymmetric abelian variety over a field of positive characteristic p. We show that every symmetric Newton polygon admits a hypersymmetric abelian variety having that Newton polygon; see 2.5 and 4.8. Isogeny classes of absolutely simple hypersymmetric abelian
Moduli of abelian varieties
, 2010
"... We start with a discussion of CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CMlifted to characteristic zero? Does there exist an abelian variety, say over Q a, or over Fp, of d ..."
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Cited by 14 (9 self)
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We start with a discussion of CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CMlifted to characteristic zero? Does there exist an abelian variety, say over Q a, or over Fp
Moduli of CM abelian varieties
"... Abstract. We discuss CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CMlifted to characteristic zero? Does there exist an abelian variety of dimension g> 3 not isogenous with th ..."
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Abstract. We discuss CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CMlifted to characteristic zero? Does there exist an abelian variety of dimension g> 3 not isogenous
Mirror symmetry for abelian varieties
 J. Algebraic Geom
"... 0.1. We define the relation of mirror symmetry on the class of pairs (complex abelian variety A + an element of the complexified ample cone of A) and study its properties. More precisely, let A be a complex abelian variety, C a A ⊂ NSA(R) – the ample cone of A and put ..."
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Cited by 28 (4 self)
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0.1. We define the relation of mirror symmetry on the class of pairs (complex abelian variety A + an element of the complexified ample cone of A) and study its properties. More precisely, let A be a complex abelian variety, C a A ⊂ NSA(R) – the ample cone of A and put
Syzygies of abelian varieties
 J. Amer. Math. Soc
"... This is the first in a series of papers meant to introduce a notion of regularity on abelian varieties and more general irregular varieties. This notion, called Mukai regularity, is based on Mukai’s concept of Fourier transform, and in a very particular form (called Theta regularity) it parallels an ..."
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Cited by 63 (13 self)
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This is the first in a series of papers meant to introduce a notion of regularity on abelian varieties and more general irregular varieties. This notion, called Mukai regularity, is based on Mukai’s concept of Fourier transform, and in a very particular form (called Theta regularity) it parallels
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