Eisenstein Series in Kohnen Plus Space for Hilbert Modular Forms

Abstract: In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, says $r+(1/2),$ whose $n$-th Fourier coefficient $H(n)$ only occurs when $(-1)^r n$ is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the Kohnen plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. If one such Hilbert modular form $f$ of parallel weight $\kappa+(1/2)$ lying in a generalized Kohnen plus space has $\xi$-th Fourier coefficients $c(\xi)$, then $c(\xi)$ does not vanish only if $(-1)^\kappa\xi$ is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These Eisenstein series give a generalization of the modular forms introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over $\mathbb{C}.$