Weights of modular forms on $\mathrm{SO}^{+}(2,l)$ and congruences between Eisenstein series and cusp forms of half-integral weight on $\mathrm{SL}_{2}$

TL;DR

This paper establishes congruences between Eisenstein series and cusp forms of half-integral weight on $ ext{SL}_2$, demonstrating the existence of integral coefficient cusp forms congruent to Eisenstein series modulo their constant term.

Contribution

It constructs explicit cusp forms congruent to Eisenstein series of half-integral weight with integral coefficients, advancing understanding of modular form congruences.

Findings

Existence of cusp forms congruent to Eisenstein series modulo constant term

Construction of integral coefficient cusp forms matching Eisenstein series weight

Implications for the structure of modular forms on $ ext{SO}^+(2,l)$

Abstract

Let $E$ be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in $\mathbb{Z}$. We show that there is a level one vector valued cusp form $f$ with the same weight as $E$ and with coefficients in $\mathbb{Z}$, which is congruent to $E$ modulo the constant term of $E$.