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Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

A general mean value theorem for Dirichlet series, with a sharp error estimate near the boundary of the critical strip, is proved. Applications of this theorem to various automorphic $L$-functions are discussed. Moreover, sharp upper bounds of mean square values of $L$-functions are obtained when they are attached to lifted forms, such as Doi–Naganuma and Ikeda lifts in the theory of Siegel modular forms.

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Cusp forms on the exceptional group of type $E_{7}$

Liftings and mean value theorems for automorphic L-functions

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2 results

Cusp forms on the exceptional group of type $E_{7}$

Liftings and mean value theorems for automorphic L-functions