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Explicit bounds for split reductions of simple abelian varieties

Jeffrey D. Achter (2012)

Journal de Théorie des Nombres de Bordeaux

Let X / K be an absolutely simple abelian variety over a number field; we study whether the reductions X 𝔭 tend to be simple, too. We show that if End ( X ) is a definite quaternion algebra, then the reduction X 𝔭 is geometrically isogenous to the self-product of an absolutely simple abelian variety for 𝔭 in a set of positive density, while if X is of Mumford type, then X 𝔭 is simple for almost all 𝔭 . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...

Explicit construction of integral bases of radical function fields

Qingquan Wu (2010)

Journal de Théorie des Nombres de Bordeaux

We give an explicit construction of an integral basis for a radical function field K = k ( t , ρ ) , where ρ n = D k [ t ] , under the assumptions [ K : k ( t ) ] = n and c h a r ( k ) n . The field discriminant of K is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the P -signatures of a radical function field are also discussed in this paper.

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

We extend Champernowne’s construction of normal numbers to base b to the d case and obtain an explicit construction of a generic point of the d shift transformation of the set 0 , 1 , . . . , b - 1 d .

Explicit form for the discrete logarithm over the field GF ( p , k )

Gerasimos C. Meletiou (1993)

Archivum Mathematicum

For a generator of the multiplicative group of the field G F ( p , k ) , the discrete logarithm of an element b of the field to the base a , b 0 is that integer z : 1 z p k - 1 , b = a z . The p -ary digits which represent z can be described with extremely simple polynomial forms.

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