We carry out "Hecke summation" for the classical Eisenstein series E k in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In the process we determine the automorphic representations generated by the E k , in particular for k = 2, where the representation is neither a pure tensor nor has finite length. We also consider Eisenstein series of weight 2 with level, and Eisenstein series with character.
We find the number s k ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ( 4 , A Q ) \mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.
We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4.In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of GSp(4, 𝔸) whose local component at 𝑝 = 2 admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at 𝑝 = 2, we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4. M S C 2 0 2 0 11F46, 11F70 (primary) Contents
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