# Weight One Jacobi Forms and Umbral Moonshine

## TL;DR

This paper studies weight one holomorphic Jacobi forms, revealing their scarcity and connecting their properties to umbral moonshine, thereby simplifying the understanding of associated conjectures and series.

## Contribution

It proves the non-existence of certain weight one Jacobi forms and characterizes umbral moonshine series using Rademacher sums.

## Key findings

- Non-zero holomorphic Jacobi forms of weight one do not exist for many index-level combinations.
- Characterization of umbral moonshine McKay--Thompson series via Rademacher sums.
- Simplification of umbral moonshine conjectures statements.

## Abstract

We analyze holomorphic Jacobi forms of weight one with level. One such form plays an important role in umbral moonshine, leading to simplifications of the statements of the umbral moonshine conjectures. We prove that non-zero holomorphic Jacobi forms of weight one do not exist for many combinations of index and level, and use this to establish a characterization of the McKay--Thompson series of umbral moonshine in terms of Rademacher sums.