# A note on Schwartz functions and modular forms

## TL;DR

This paper extends the construction of Schwartz functions using modular forms to improve sphere packing bounds in certain dimensions, matching known optimal solutions in dimensions 8 and 24.

## Contribution

It introduces a generalized method for constructing Schwartz functions via quasi-modular and modular forms, enhancing bounds in sphere packing problems.

## Key findings

- Constructs infinite families of Schwartz functions for sphere packing bounds.
- Achieves optimal bounds in dimensions 8 and 24, matching previous solutions.
- Provides a unified framework using modular forms for sphere packing bounds.

## Abstract

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.