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### Minimal abelian varieties of algebras, I [article]

Keith A. Kearnes, Emil W. Kiss, Agnes Szendrei
2020 arXiv   pre-print
We show that any 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.  ...

### An Introduction to Tame Congruence Theory [chapter]

Emil W. Kiss
1997 Algebraic Model Theory
To define the concept 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

Matthew A. Valeriote
1990 Proceedings of the American Mathematical Society
It is shown that every finite simple 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  ...

### Finite simple abelian algebras are strictly simple

Matthew A. Valeriote
1990 Proceedings of the American Mathematical Society
It is shown that every finite simple 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  ...

### FINITENESS PROPERTIES OF LOCALLY FINITE ABELIAN VARIETIES

KEITH A. KEARNES, ROSS WILLARD
1999 International journal of algebra and computation
We show that any locally finite 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.  ...

### Varieties whose finitely generated members are free [article]

Keith A. Kearnes, Emil W. Kiss, Agnes Szendrei
2016 arXiv   pre-print
We prove that a 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.  ...

### RESIDUAL SMALLNESS AND WEAK CENTRALITY

KEITH A. KEARNES, EMIL W. KISS
2003 International journal of algebra and computation
Conversely, we show in two special cases that a weakly 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.  ...

### A note on supersingular abelian varieties [article]

Chia-Fu Yu
2017 arXiv   pre-print
The proof uses 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.  ...

### On function field Mordell-Lang: the semiabelian case and the socle theorem

Franck Benoist, Elisabeth Bouscaren, Anand Pillay
2017 Proceedings of the London Mathematical Society
The main result is a reduction, using model-theoretic tools, 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.  ...

### A gap theorem for abelian varieties over differential fields

Alexandru Buium, Anand Pillay
1997 Mathematical Research Letters
Since any regular map 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.  ...

### 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.  ...

### A note on supersingular abelian varieties

Chia-Fu Yu
2020 Bulletin of the Institute of Mathematics Academia Sinica NEW SERIES
In this note we show that every supersingular 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.  ...

### On finiteness of endomorphism rings of abelian varieties [article]

Chia-Fu Yu
2014 arXiv   pre-print
The endomorphism ring End(A) 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  and uses the notion of minimal isogenies whose significance is pointed out in Li-Oort  .  ...

### A note on o-minimal flows and the Ax--Lindemann--Weierstrass theorem for abelian varieties over C [article]

Ya'acov Peterzil, Sergei Starchenko
2016 arXiv   pre-print
In this short note we present an elementary proof 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.  ...

### Simple Abelian algebras

Ágnes Szendrei
1992 Journal of Algebra
power 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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