Zeta statistics and Hadamard functions

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Zeta statistics and Hadamard functions. / Bilu, Margaret; Das, Ronno; Howe, Sean.

In: Advances in Mathematics, Vol. 407, 108556, 08.10.2022, p. 1-68.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Bilu, M, Das, R & Howe, S 2022, 'Zeta statistics and Hadamard functions', Advances in Mathematics, vol. 407, 108556, pp. 1-68. https://doi.org/10.1016/j.aim.2022.108556

APA

Bilu, M., Das, R., & Howe, S. (2022). Zeta statistics and Hadamard functions. Advances in Mathematics, 407, 1-68. [108556]. https://doi.org/10.1016/j.aim.2022.108556

Vancouver

Bilu M, Das R, Howe S. Zeta statistics and Hadamard functions. Advances in Mathematics. 2022 Oct 8;407:1-68. 108556. https://doi.org/10.1016/j.aim.2022.108556

Author

Bilu, Margaret ; Das, Ronno ; Howe, Sean. / Zeta statistics and Hadamard functions. In: Advances in Mathematics. 2022 ; Vol. 407. pp. 1-68.

Bibtex

@article{837f77b6fa4249a18f124b58bc55e967,
title = "Zeta statistics and Hadamard functions",
abstract = "We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any “natural” sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.",
keywords = "Arithmetic statistics, Batyrev-Manin conjecture, Cohomological stability, Configuration spaces, Grothendieck ring of varieties, Zeta functions",
author = "Margaret Bilu and Ronno Das and Sean Howe",
note = "Publisher Copyright: {\textcopyright} 2022 The Authors",
year = "2022",
month = oct,
day = "8",
doi = "10.1016/j.aim.2022.108556",
language = "English",
volume = "407",
pages = "1--68",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press",

}

RIS

TY - JOUR

T1 - Zeta statistics and Hadamard functions

AU - Bilu, Margaret

AU - Das, Ronno

AU - Howe, Sean

N1 - Publisher Copyright: © 2022 The Authors

PY - 2022/10/8

Y1 - 2022/10/8

N2 - We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any “natural” sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

AB - We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any “natural” sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

KW - Arithmetic statistics

KW - Batyrev-Manin conjecture

KW - Cohomological stability

KW - Configuration spaces

KW - Grothendieck ring of varieties

KW - Zeta functions

UR - http://www.scopus.com/inward/record.url?scp=85134401908&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2022.108556

DO - 10.1016/j.aim.2022.108556

M3 - Journal article

AN - SCOPUS:85134401908

VL - 407

SP - 1

EP - 68

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 108556

ER -

ID: 342928538