2022/05/25 by Hauffe-Waschbüsch, Adrian, Krieg, Aloys, Williams, Brandon
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2205.12492
We consider the Hermitian Eisenstein series E^(\mathbbK)k of degree 2 and weight k associated with an imaginary-quadratic number field \mathbbK and determine the influence of \mathbbK on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity E^(\mathbbK)24 = E^(\mathbbK)8, which is well-known for Siegel modular forms of degree 2, if and only if \mathbbK = ℚ (√(-3)). As an application, we show that the Eisenstein series E^(\mathbbK)k, k=4,6,8,10,12 are algebraically independent whenever \mathbbK≠ ℚ(√(-3)). The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maass 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 -∞. Finally, we show that these forms generate the space of cusp forms in the Maass Spezialschar as a module over the Hecke algebra as \mathbbK varies over imaginary-quadratic number fields.