Abelian and Hamiltonian varieties of groupoids

We study certain groupoids generating Abelian, strongly Abelian, and Hamiltonian varieties. An algebra is Abelian if t( a,[`(c)] ) = t( a,[`(d)] ) ? t( b,[`(c)] ) = t( b,[`(d)] ) t\left( {a,\bar{c}} \right) = t\left( {a,\bar{d}} \right) \to t\left( {b,\bar{c}} \right) = t\left( {b,\bar{d}} \right) for any polynomial operation on the algebra and for all elements a, b, [`(c)] \bar{c} , [`(d)] \bar{d} . An algebra is strongly Abelian if t( a,[`(c)] ) = t( b,[`(d)] ) ? t( e,[`(c)] ) = t( e,[`(d)] ) t\left( {a,\bar{c}} \right) = t\left( {b,\bar{d}} \right) \to t\left( {e,\bar{c}} \right) = t\left( {e,\bar{d}} \right) for any polynomial operation on the algebra and for arbitrary elements a, b, e, [`(c)] \bar{c} , [`(d)] \bar{d} . An algebra is Hamiltonian if any subalgebra of the algebra is a congruence class. A variety is Abelian (strongly Abelian, Hamiltonian) if all algebras in a respective class are Abelian (strongly Abelian, Hamiltonian). We describe semigroups, groupoids with unity, and quasigroups generating Abelian, strongly Abelian, and Hamiltonian varieties.     

Power residues on Abelian varieties

Given a prime l and an elliptic curve E defined over a number field k, we show that a non-zero point P] E(k) lies in lE(k) if and only if P lies in lE(k)(mod ) for almost all finite primes  of k. We give conditions on l under which analogous results hold for Abelian varieties and with one point replaced by a finite number of points. We also construct examples to show that these conditions are essential.     

Abelian varieties in characteristic p

In this paper Tate's finiteness conjecture for isogenies of polarized Abelian varieties in characteristic p>2 is proved. From this conjecture it is deduced that Tate modules are semisimple and that Tate's conjecture on the homomorphisms of Abelian varieties is valid.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 393–400, March, 1976.     

Abelian varieties and the general Hodge conjecture

We investigate the relationship between the usual and general Hodgeconjectures for abelian varieties. For certain abelian varieties A, weshow that the usual Hodge conjecture for all powers of A implies thegeneral Hodge conjecture for A.     

Base number theorem for Abelian varieties

This work was done while the author was visiting the Brown University Providence, R.I., USA.     

Torsion of Abelian varieties of finite characteristic

The finiteness of the torsion of Abelian varieties with a complete real field of endomorphisms in the maximal Abelian extension of the field of definition is proven. This assertion is formally deduced from the finiteness hypothesis for isogenic Abelian varieties, proven for characteristic p > 2. The structure is studied of the Lie algebra of Galois groups acting in a Tate module; in particular, for fields of characteristic greater than two there is proven one-dimensionality of the center of the Lie algebra.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 3–11, July, 1977.     

Abelian varieties over fields of finite characteristic

The aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.     

Efficient pairing computation on supersingular Abelian varieties

We present a general technique for the efficient computation of pairings on Jacobians of supersingular curves. This formulation, which we call the eta pairing, generalizes results of Duursma and Lee for computing the Tate pairing on supersingular elliptic curves in characteristic 3. We then show how our general technique leads to a new algorithm which is about twice as fast as the Duursma–Lee method. These ideas are applied to elliptic and hyperelliptic curves in characteristic 2 with very efficient results. In particular, the hyperelliptic case is faster than all previously known pairing algorithms.      

Abelian varieties with finite abelian group action

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan’s decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group. An early version of this article was inadvertently published before all the revisions had been completed and then retracted [https://doi.org/10.1007/s00013-018-1244-3]. This article is the final peer reviewed version.