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\(E_ 2\), \(\Theta\), and overconvergence. (English) Zbl 0846.11027

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).

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms

Citations:

Zbl 0354.10021
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