We consider a special theta lift $𝜃\left(f\right)$ from cuspidal Siegel modular forms $f$ on ${Sp}_4$ to “modular forms” $𝜃\left(f\right)$ on $SO\left(4,4\right)$ in the sense of our prior work (Pollack 2020a). This lift can be considered an analogue of the Saito–Kurokawa lift, where now the image of the lift is representations of $SO\left(4,4\right)$ that are quaternionic at infinity. We relate the Fourier coefficients of $𝜃\left(f\right)$ to those of $f$, and in particular prove that $𝜃\left(f\right)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $𝜃\left(f\right)$ to $G_2\subseteq SO\left(4,4\right)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily large weight with all algebraic Fourier coefficients. In the case of level one, we obtain precise formulas for the Fourier coefficients of $𝜃\left(f\right)$ in terms of those of $f$. In particular, we construct nonzero cuspidal modular forms on $G_2$ of level one with all integer Fourier coefficients.

我们考虑一个特殊的 theta 电梯 $𝜃\left(F\right)$ 来自尖牙 Siegel 模块化形式 $F$ 上 ${斯普}_4$ 到“模块化形式” $𝜃\left(F\right)$ 上 $所以\left(4,4\right)$就我们之前的工作而言（Pollack 2020a）。这个电梯可以被认为是 Saito-Kurokawa 电梯的类似物，现在电梯的图像是$所以\left(4,4\right)$在无穷远处是四元数。我们将傅立叶系数联系起来$𝜃\left(F\right)$ 对那些 $F$，特别是证明 $𝜃\left(F\right)$ 非零且具有代数傅立叶系数，如果 $F$做。限制$𝜃\left(F\right)$ 至 $G_2\subseteq 所以\left(4,4\right)$，我们在 $G_2$具有所有代数傅立叶系数的任意大权重。在第一级的情况下，我们获得了傅立叶系数的精确公式$𝜃\left(F\right)$ 就那些 $F$. 特别地，我们构造了非零尖牙模形式$G_2$ 具有所有整数傅立叶系数的一级。