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Let be an absolutely simple abelian variety over a number field; we study whether the reductions tend to be simple, too. We show that if is a definite quaternion algebra, then the reduction is geometrically isogenous to the self-product of an absolutely simple abelian variety for in a set of positive density, while if is of Mumford type, then is simple for almost all . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...
We give an explicit construction of an integral basis for a radical function field , where , under the assumptions and . The field discriminant of is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the -signatures of a radical function field are also discussed in this paper.
We extend Champernowne’s construction of normal numbers to base b to the case and obtain an explicit construction of a generic point of the shift transformation of the set .
We prove that the error term differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.
We prove that for every x > q ≥ 1, and similar estimates for the Liouville function. We also give better constants when x/q is large.,
For generator of the multiplicative group of the field , the discrete logarithm of an element of the field to the base , is that integer , . The -ary digits which represent can be described with extremely simple polynomial forms.
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