Weights of modular forms on ${\mathrm{SO}}^{+}(2,l)$ and congruences between Eisenstein series and cusp forms of half-integral weight on ${\mathrm{SL}}_2$

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 $E$.