17 research outputs found
On the cyclicity of the rational points group of abelian varieties over finite fields
We propose a simple criterion to know if an abelian variety defined over
a finite field is cyclic, i.e., it has a cyclic group of
rational points; this criterion is based on the endomorphism ring
End. We also provide a criterion to know if an isogeny
class is cyclic, i.e., all its varieties are cyclic; this criterion is based on
the characteristic polynomial of the isogeny class. We find some asymptotic
lower bounds on the fraction of cyclic -isogeny classes among
certain families of them, when tends to infinity. Some of these bounds
require an additional hypothesis. In the case of surfaces, we prove that this
hypothesis is achieved and, over all -isogeny classes with
endomorphism algebra being a field and where is an even power of a prime,
we prove that the one with maximal number of rational points is cyclic and
ordinary.Comment: 13 pages, this is a preliminary version, comments are welcom
Primes in short arithmetic progressions
Let and be three parameters. We show that, for most moduli
and for most positive real numbers , every reduced arithmetic
progression has approximately the expected number of primes from
the interval , provided that and satisfies
appropriate bounds in terms of and . Moreover, we prove that, for most
moduli and for most positive real numbers , there is at least
one prime lying in every reduced arithmetic progression , provided that .Comment: 21 pages. Final version, published in IJNT. Some minor change