# Poincar\'e square series for the Weil representation

## TL;DR

This paper develops explicit formulas for certain modular forms related to the Weil representation, enabling efficient computation and applications in automorphic product construction.

## Contribution

It introduces new coefficient formulas for modular forms associated with the Weil representation and provides an efficient computational algorithm.

## Key findings

- Derived explicit coefficient formulas for modular forms Q_{k,m,β}
- Developed an efficient p-adic computation method for Eisenstein series
- Enabled construction of automorphic products using these formulas

## Abstract

We calculate the Jacobi Eisenstein series of weight $k \ge 3$ for a certain representation of the Jacobi group, and evaluate these at $z = 0$ to give coefficient formulas for a family of modular forms $Q_{k,m,\beta}$ of weight $k \ge 5/2$ for the (dual) Weil representation on an even lattice. The forms we construct always contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for $Q_{k,m,\beta}$ and the Eisenstein series of Bruinier and Kuss $p$-adically to get an efficient algorithm. The main application is in constructing automorphic products.