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Abstract Let$$\mathbb {F}_q^d$$ be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ and a fixed nonzero$$t\in \mathbb {F}_q$$ , let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ , where$$h_y:E\rightarrow \{0,1\}$$ is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ . Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ that if$$|E|\ge Cq^{\frac{11}{4}}$$ andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ isdwhenever$$E\subseteq \mathbb {F}_q^d$$ with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ .
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Abelian varieties of prescribed order over finite fields
Abstract Given a prime powerqand$$n \gg 1$$ , we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ is$$\#A({\mathbb {F}}_q)$$ for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ . As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ for some abelian varietyAover$${\mathbb {F}}_q$$ . Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ , then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ is realizable.
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- PAR ID:
- 10575670
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 392
- Issue:
- 1
- ISSN:
- 0025-5831
- Format(s):
- Medium: X Size: p. 1167-1202
- Size(s):
- p. 1167-1202
- Sponsoring Org:
- National Science Foundation
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