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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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On properties of univariate max functions at local maximizers
Abstract More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that is useful in several applications that we describe. We also investigate whether this smoothness property extends to max functions more generally. We show that the pointwise maximum of a finite set ofq-times continuously differentiable univariate functions must have zero derivative at a maximizer for$$q=1$$ , but arbitrarily close to the maximizer, the derivative may not be defined, even when$$q=3$$ and the maximizer is isolated.
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- Award ID(s):
- 2012250
- PAR ID:
- 10364473
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Optimization Letters
- Volume:
- 16
- Issue:
- 9
- ISSN:
- 1862-4472
- Page Range / eLocation ID:
- p. 2527-2541
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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