A quaternionic Saito–Kurokawa lift and cusp forms on G2
A quaternionic Saito–Kurokawa lift and cusp
forms on G2
复制标题
四元数 Saito-Kurokawa 升力和尖点
DOI：
10.2140/ant.2021.15.1213
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发表时间：
2019
影响因子：
1.3
通讯作者：
Aaron Pollack
中科院分区：
文献类型：
--
作者：
Aaron Pollack
我们考虑了$\theta(f)$从$\ mathm {Sp}_4$上的斜向西格尔模形式$f$到$\ mathm {SO}(4,4)$上的“模形式”$\theta(f)$的特殊提升。这个升降机可以被认为是Saito-Kurokawa升降机的类似物，现在升降机的图像是$\ mathm {SO}(4,4)$的表示，它们在无穷远处是四元数。我们将$\ (f)$的傅里叶系数与$f$的傅里叶系数联系起来，特别地证明$\ (f)$是非零的并且如果$f$是非零的，则$\ (f)$具有代数傅里叶系数。将$\theta(f)$限定为$G_2 \subseteq \mathrm{SO}(4,4)$，得到了任意大权$G_2$上具有所有代数傅里叶系数的倒模形式。在一级的情况下，我们得到了用f的傅里叶系数表示的精确公式。特别地，我们在一级的$G_2$上构造了具有全整数傅立叶系数的非零倒模形式。
We consider a special theta lift $\theta(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $\theta(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the image of the lift is representations of $\mathrm{SO}(4,4)$ that are quaternionic at infinity. We relate the Fourier coefficients of $\theta(f)$ to those of $f$, and in particular prove that $\theta(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $\theta(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, 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 $\theta(f)$ 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.