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Abstract For any subset$$Z \subseteq {\mathbb {Q}}$$, consider the set$$S_Z$$of subfields$$L\subseteq {\overline {\mathbb {Q}}}$$which contain a co-infinite subset$$C \subseteq L$$that is universally definable inLsuch that$$C \cap {\mathbb {Q}}=Z$$. Placing a natural topology on the set$${\operatorname {Sub}({\overline {\mathbb {Q}}})}$$of subfields of$${\overline {\mathbb {Q}}}$$, we show that ifZis not thin in$${\mathbb {Q}}$$, then$$S_Z$$is meager in$${\operatorname {Sub}({\overline {\mathbb {Q}}})}$$. Here,thinandmeagerboth mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fieldsLhave the property that the ring of algebraic integers$$\mathcal {O}_L$$is universally definable inL. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every$$\exists $$-definable subset of an algebraic extension of$${\mathbb Q}$$is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials. 

We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over ${\mathbb{F}}_2$, ${\mathbb{F}}_3$and ${\mathbb{F}}_5$. We produce partial results for abelian varieties over a general finite field  ${\mathbb{F}}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over ${\mathbb{F}}_q$when $q$is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over  ${\mathbb{F}}_2$.