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2019

A note on Schwartz functions and modular forms

Rolen, Larry, Wagner, Ian

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We generalize the recent work of Viazovska by constructing infinite families of Schwartz functions, suitable for Cohn-Elkies style linear programming bounds, using quasi-modular and modular forms.

In particular for dimensions $d \equiv 0 \pmod{8}$ we give the constructions that lead to the best sphere packing upper bounds via modular forms.
In dimension $8$ and $24$ these exactly match the functions constructed by Viazovska and Cohn, Kumar, Miller, Radchenko, and Viazovska which resolved the sphere packing problem in those dimensions.

A note on Schwartz functions and modular forms · Around

Built on

Universal optimality of the E 8 E_{8} and Leech lattices and interpolation formulas
Original
H. Cohn, A. Kumar, S. Miller, D. Radchenko, and M. Viazovska · 1902
Earlier work this paper cites.
Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene
A. Thue · 1910
Earlier work this paper cites.
Bounds for packings on a sphere and in space
G.A. Katabiansky and V.I. Levenshtein · 1978
Earlier work this paper cites.
New upper bounds on sphere packings I
H. Cohn and N. Elkies · 2003
Earlier work this paper cites.

Similar

A proof of the Kepler conjecture
T. Hales · 2005
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Universally optimal distribution of points on spheres
H. Cohn and A. Kumar · 2007
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A note on sphere packings in high dimension
A. Venkatesh · 2013
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An optimal uncertainty principle in twelve dimensions via modular forms
Original
H. Cohn and F. Gonçalves
Cited in the paper.

Then

Harmonic Maass forms and mock modular forms: theory and applications
K. Bringmann, A. Folsom, K. Ono, and L. Rolen · 2017
Later among the works it cites.
The sphere packing problem in dimension 24 24
H. Cohn, A. Kumar, S. Miller, D. Radchenko, and M. Viazovska · 2017
Later among the works it cites.
The sphere packing problem in dimension 8 8
M. Viazovska · 2017
Later among the works it cites.

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