Rankin-Cohen brackets of Eisenstein series. (English) Zbl 1532.11063
The question of what is the linear span of (and the relations among) products \(E_k E_l\) for elliptic Eisenstein series \(k, l\) is historically tied to the development of the Rankin-Selberg method and thus remains relevant till today. In modern mathematics this type of question has found applications in the computation of modular forms, but has also sparked studies of its generalizations. In light of Gan-Gross-Prasad, where the product appears in the branching from \(\mathrm{O}(2,1) \times \mathrm{O}(2,1) \cong \mathrm{O}(2,2)\) to \(\mathrm{O}(2,1)\), both generalizations to higher levels, including oldforms, and higher weights, including the action of differential operators, are natural. The author conducts an analysis of the latter case, and examines the span of Rankin-Cohen brackets of order up to \(n = 3\), for which they show that the span of \([E_k, E_l]_n\) equals the space of all modular forms in sufficiently large weights.
Reviewer: Martin Raum (Gothenburg)
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
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