17 research outputs found

    On the cyclicity of the rational points group of abelian varieties over finite fields

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    We propose a simple criterion to know if an abelian variety AA defined over a finite field Fq\mathbb{F}_q is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring EndFq(A)_{\mathbb{F}_q}(A). 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 Fq\mathbb{F}_q-isogeny classes among certain families of them, when qq 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 Fq\mathbb{F}_q-isogeny classes with endomorphism algebra being a field and where qq 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

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    Let x,hx,h and QQ be three parameters. We show that, for most moduli q≤Qq\le Q and for most positive real numbers y≤xy\le x, every reduced arithmetic progression amod  qa\mod q has approximately the expected number of primes pp from the interval (y,y+h](y,y+h], provided that h>x1/6+ϵh>x^{1/6+\epsilon} and QQ satisfies appropriate bounds in terms of hh and xx. Moreover, we prove that, for most moduli q≤Qq\le Q and for most positive real numbers y≤xy\le x, there is at least one prime p∈(y,y+h]p\in(y,y+h] lying in every reduced arithmetic progression amod  qa\mod q, provided that 1≤Q2≤h/x1/15+ϵ1\le Q^2\le h/x^{1/15+\epsilon}.Comment: 21 pages. Final version, published in IJNT. Some minor change
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