Coleman, Robert F.; Gouvêa, Fernando Q.; Jochnowitz, Naomi \(E_ 2\), \(\Theta\), and overconvergence. (English) Zbl 0846.11027 Int. Math. Res. Not. 1995, No. 1, 23-41 (1995). Consider a \(p\)-adic modular form (in Katz’s sense) defined as a global section of certain line bundles over the rigid-analytic space, i.e. over the modular curve minus curves with supersingular reduction (supersingular disks). For many purposes it is essential to consider the so-called overconvergent forms, i.e. those which have imposed a growth condition on the coefficients of the Laurent expansion around the supersingular disk. The authors prove the following results.The \(p\)-adic modular form \(E_2\) is not overconvergent for \(p \geq 5\) (theorem 1). (The cases \(p = 2,3\) were answered negatively by N. Koblitz in 1977 [Bull. Lond. Math. Soc. 9, 188-192 (1977; Zbl 0354.10021)]). It is shown to be a corollary of a recent result of Coleman;They give two proofs of the fact that the \(\theta\) operator destroys overconvergence on forms of non-zero weight (corollaries 5 and 10);The ring generated by \(E_2\) over the ring of overconvergent modular forms is stable under the \(\theta\) operator, the \(U\) operator, and the Hecke operators \(T_\ell\) for all \(\ell \neq p\) (proposition 8, theorem 4). Reviewer: A.Dabrowski (Szczecin) Cited in 11 Documents MSC: 11F33 Congruences for modular and \(p\)-adic modular forms Keywords:\(p\)-adic modular form; supersingular disks; \(\theta\) operator; overconvergent forms; Laurent expansion Citations:Zbl 0354.10021 PDFBibTeX XMLCite \textit{R. F. Coleman} et al., Int. Math. Res. Not. 1995, No. 1, 23--41 (1995; Zbl 0846.11027) Full Text: DOI