March 2023 On Hermitian Eisenstein series of degree $2$
Adrian Hauffe-Waschbüsch, Aloys Krieg, Brandon Williams
Funct. Approx. Comment. Math. 68(1): 127-141 (March 2023). DOI: 10.7169/facm/2047


We consider the Hermitian Eisenstein series $E^{(\mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $\mathbb{K}$ and determine the influence of $\mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(\mathbb{K})}^2}_4 = E^{{(\mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $\mathbb{K} = \mathbb{Q}(\sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(\mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $\mathbb{K}\neq \mathbb{Q}(\sqrt{-3})$. The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maa{\ss} space that does not vanish identically for sufficiently large weight; however, when the weight is fixed, we will see that it tends to $0$ as the discriminant tends to $-\infty$. Finally, we show that these forms generate the space of cusp forms in the Maaß Spezialschar as a module over the Hecke algebra as $\K$ varies over imaginary-quadratic number fields.


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Adrian Hauffe-Waschbüsch. Aloys Krieg. Brandon Williams. "On Hermitian Eisenstein series of degree $2$." Funct. Approx. Comment. Math. 68 (1) 127 - 141, March 2023.


Published: March 2023
First available in Project Euclid: 15 December 2022

MathSciNet: MR4564867
Digital Object Identifier: 10.7169/facm/2047

Primary: 11F46 , 11F55

Keywords: Hermitian Eisenstein series , Maaß space , Siegel Eisenstein series

Rights: Copyright © 2023 Adam Mickiewicz University


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Vol.68 • No. 1 • March 2023
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