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Minimal abelian varieties of algebras, I
[article]

2020
*
arXiv
*
pre-print

We show that any

arXiv:1912.05653v4
fatcat:uqdjqoupsjhlfnqjhrev4o3eyq
*abelian**variety*that is not affine has a nontrivial strongly*abelian*subvariety. In later papers in this sequence we apply this result to the study*of**minimal**abelian**varieties*. ... Background This paper concerns the classification*of**minimal**varieties**of**algebras*. ... Applying this to the situation where V is a*minimal**variety*, we obtain that any*minimal**abelian**variety*is affine or strongly*abelian*. ...##
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An Introduction to Tame Congruence Theory
[chapter]

1997
*
Algebraic Model Theory
*

To define the concept

doi:10.1007/978-94-015-8923-9_5
fatcat:x5avartgknhn7lnpdxzg3bj6zq
*of*a polynomial, recall that a term*of*an*algebra*is any composition*of*its basic operations. ... During this localization process we shall concentrate on the polynomial structure*of*an*algebra*A. ... An application:*minimal**varieties*As an illustration, we outline the proof*of*the structure theorem*of*locally finite,*abelian*,*minimal**varieties*. ...##
###
Finite Simple Abelian Algebras are Strictly Simple

1990
*
Proceedings of the American Mathematical Society
*

It is shown that every finite simple

doi:10.2307/2047692
fatcat:zmh2ml6apffd5nhbl6tixsseiu
*Abelian*universal*algebra*is strictly simple. This generalizes a well-known fact about*Abelian*groups and modules. ... An*algebra*is said to be*Abelian*if for every term t(x,y) and for all elements a, b, c, d, we have the following implication: t(a,c) = t(a,d) -> t(b,c) = t(b,d) . ... License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-*of*-use FINITE SIMPLE*ABELIAN**ALGEBRAS*ARE STRICTLY SIMPLE ...##
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Finite simple abelian algebras are strictly simple

1990
*
Proceedings of the American Mathematical Society
*

It is shown that every finite simple

doi:10.1090/s0002-9939-1990-0990434-2
fatcat:yijdu43l7bdshh5i64b7czw464
*Abelian*universal*algebra*is strictly simple. This generalizes a well-known fact about*Abelian*groups and modules. ... An*algebra*is said to be*Abelian*if for every term t(x,y) and for all elements a, b, c, d, we have the following implication: t(a,c) = t(a,d) -> t(b,c) = t(b,d) . ... License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-*of*-use FINITE SIMPLE*ABELIAN**ALGEBRAS*ARE STRICTLY SIMPLE ...##
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FINITENESS PROPERTIES OF LOCALLY FINITE ABELIAN VARIETIES

1999
*
International journal of algebra and computation
*

We show that any locally finite

doi:10.1142/s0218196799000114
fatcat:2vul2q5u5jg6jbetrrc6mdinny
*abelian**variety*is generated by a finite*algebra*. We solve a problem posed by D. Hobby and R. McKenzie by exhibiting a nonfinitely based finite*abelian**algebra*. ... A*variety**of**algebras*is said to be*abelian*if all*of*its members are*abelian*. For any ring R the*variety**of*left R-modules is*abelian*. For any monoid M the*variety**of*left M-sets is*abelian*. ... By the first part*of*the paper, the*variety**of*the second part is generated by a finite*abelian**algebra*. ...##
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Varieties whose finitely generated members are free
[article]

2016
*
arXiv
*
pre-print

We prove that a

arXiv:1508.03807v2
fatcat:saxzk4ip3ba3vom3dchabcdbuq
*variety**of**algebras*whose finitely generated members are free must be definitionally equivalent to the*variety**of*sets, the*variety**of*pointed sets, a*variety**of*vector spaces over a division ... ring, or a*variety**of*affine vector spaces over a division ring. ... Namely, a*minimal*subvariety*of*a*variety*generated by a finite strongly*abelian**algebra*is definitionally equivalent to a matrix power*of*the*variety**of*sets or the*variety**of*pointed sets. ...##
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RESIDUAL SMALLNESS AND WEAK CENTRALITY

2003
*
International journal of algebra and computation
*

Conversely, we show in two special cases that a weakly

doi:10.1142/s0218196703001237
fatcat:yjcbo7vtrrbbhpwnfdgnnrrbd4
*abelian**variety*is residually bounded by a finite number: when it is generated by an E-*minimal*, or by a finite strongly nilpotent*algebra*. ... We develop a method*of*creating skew congruences on subpowers*of*finite*algebras*using groups*of*twin polynomials, and apply it to the investigation*of*residually small*varieties*generated by nilpotent ... To prove (3) =⇒(2) let S be a finite subdirectly irreducible*algebra*in V(A). It is well-known and easy to see that all finite*algebras*in V(A) are E-*minimal*, and therefore so is S. ...##
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A note on supersingular abelian varieties
[article]

2017
*
arXiv
*
pre-print

The proof uses

arXiv:1412.7107v2
fatcat:cc74kqj5n5hjhozndmyf7jopqa
*minimal*isogenies and the Galois descent. We then construct a superspecial*abelian**variety*which not directly defined over a finite field. ... In this note we show that any supersingular*abelian**variety*is isogenous to a superspecial*abelian**variety*without increasing field extensions. ... The author is partially supported by the grants MoST 100-2628-M-001-006-MY4 and 103-2918-*I*-001-009. ...##
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On function field Mordell-Lang: the semiabelian case and the socle theorem

2017
*
Proceedings of the London Mathematical Society
*

The main result is a reduction, using model-theoretic tools,

doi:10.1112/plms.12076
fatcat:ftdil3izxna4vbyyv6sf3cemve
*of*the semiabelian case to the*abelian*case. ... semiabelian*varieties*. ... If G is itself an*abelian**variety*, then G is a finite sum*of*simple*abelian**varieties*, G*i*. Each G*i*is either isogenous to some H defined over K 0 , or has K 0 -trace zero. ...##
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A gap theorem for abelian varieties over differential fields

1997
*
Mathematical Research Letters
*

Since any regular map

doi:10.4310/mrl.1997.v4.n2.a4
fatcat:duexs4ow3zdq3btdkpux5ytxbi
*of*an*abelian**variety*into a commutative*algebraic*group is a homomorphism composed with a translation it follows that G*i*contains a non trivial*abelian*subvariety B*i*. ... Replacing A by an*abelian**variety*isogenous to it we may assume A = A 1 × ... × A m , where A*i*are simple*abelian**varieties*. ...##
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Page 108 of Mathematical Reviews Vol. 58, Issue 1
[page]

1979
*
Mathematical Reviews
*

Here k is a function field in one variable over C,

*I*, and J, are*abelian**varieties*over k, k is an*algebraic*closure*of*k, and 7;(/;) is the Tate module*of*J,,*i*= 1, 2. ... (This last condition also turned up in part*I*[op.cit.]; J; is a*minimal*Néron model for J, over k,*i*=1, 2.) (ii) The*abelian**variety*7, has completely degenerate reduction at some place*of*k. ...##
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A note on supersingular abelian varieties

2020
*
Bulletin of the Institute of Mathematics Academia Sinica NEW SERIES
*

In this note we show that every supersingular

doi:10.21915/bimas.2020102
fatcat:zbk4cui4dzbslpikaqlg5uquza
*abelian**variety*is isogenous to a superspecial*abelian**variety*without increasing field extensions. The proof uses*minimal*isogenies and Galois descent. ... Endomorphism*algebras**of*supersingular elliptic curves over an arbitrary field are investigated. We correct the main result*of*the author's paper [Math. ... The author was partially supported by the grants MoST 100-2628-M-001-006-MY4 and 103-2918-*I*-001-009. He thanks the referee for a careful reading and helpful comments. ...##
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On finiteness of endomorphism rings of abelian varieties
[article]

2014
*
arXiv
*
pre-print

The endomorphism ring End(A)

arXiv:0905.0019v5
fatcat:r27rpprhsjenhoub75ph6a7y2y
*of*an*abelian**variety*A is an order in a semi-simple*algebra*over Q. The co-index*of*End(A) is the index to a maximal order containing it. ... We show that for*abelian**varieties**of*fixed dimension over any*algebraically*closed field*of*characteristic p>0, the p-exponents*of*the co-indices*of*their endomorphism rings are bounded. ... Obviously the present work relies on the work*of*Manin [9] and uses the notion*of**minimal*isogenies whose significance is pointed out in Li-Oort [8] . ...##
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A note on o-minimal flows and the Ax--Lindemann--Weierstrass theorem for abelian varieties over C
[article]

2016
*
arXiv
*
pre-print

In this short note we present an elementary proof

arXiv:1611.04673v1
fatcat:tzf3sunmejcqzfxxikut4dcdk4
*of*the Ax-Lindemann-Weierstrass theorem for*abelian*and semi-*abelian**varieties*. ... The proof uses ideas*of*Pila, Ullmo, Yafaev, Zannier and is based on basic properties*of*sets definable in o-*minimal*structures. It does not use the Pila-Wilkie counting theorem. ... Geometric restatements*of*ALW for semi-*abelian**varieties*. ...##
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Simple Abelian algebras

1992
*
Journal of Algebra
*

power

doi:10.1016/0021-8693(92)90121-2
fatcat:izzjzodgqfdqve55h7uvvkin2m
*of*a unary permutational*algebra*. ... An*algebra*is called*Abelian*if all its term operations satisfy the so-called "term condition." By a recent result*of*M. Valeriote (Proc. Amer. Math. ... It turned out that such an*algebra*generates a*minimal**variety*if and only if it is quasiprimal. ...
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