An integrality criterion for elliptic modular forms (Q914736)

In the paper under review the author states - with motivations and hints of proof - necessary and sufficient conditions for a (classical) elliptic modular form f of weight k and level \(N\geq 3\) to be already defined (in the sense of \textit{N. M. Katz} [Lect. Notes Math. 350, 69-190 (1973; Zbl 0271.10033)]) over a valuation ring \(O_ v\) of a number field K, containing a primitive N-root \(\zeta_ N\) of unity, thus refining the so-called q-expansion principle (loc. cit.). To be precise, let E be a (suitably chosen) elliptic curve over K with period \(\Omega_ E\in {\mathbb{C}}\), \(E\otimes {\mathbb{C}}\cong {\mathbb{C}}/{\mathbb{Z}}+{\mathbb{Z}}\tau\), Im \(\tau\) \(>0\), and \[ \delta_ k=(- 1/4\pi)((2id/dz)+k/Im z) \] the Maaß operator, \(\delta_ k^{(r)}\) its r-th iterate. Then f is defined over \(O_ v\) if and only if all the numbers \(c_ r(f):=((-4\pi)^ r/\Omega_ E^{k+2r})(\delta_ k^{(r)}f)(\tau)\) belong to \(O_ v\) and satisfy the inequality \(v(\sum^{r}_{i=1}b_{i,r}c_ i(f))\geq v(r!)\) for \(r\in {\mathbb{N}}\). Here the coefficients \(b_{i,r}\) are defined by \(\sum^{r}_{i=1}b_{i,r}x^ i=r!\left( \begin{matrix} x\\ r\end{matrix} \right).\) Necessarily the proof is based on the theory of Deligne-Katz of the space of moduli of elliptic curves with level structure and an interpretation of the operator \(\delta_ k\) in this setting.