background/motivation
let Ek denote the modular form of level one and weight k with q-expansion given by $E_k(q)=1- \frac{2k}{b_k}\sum_n \sigma_{k-1}(n)q^n$ where σi is the divisor sum and bk is the k-th bernoulli number. for k = p – 1 the the denominators of this series don't contain any powers of p and hence by the q-expansion principle Ep–1 defines a modular form over ℤp. that is, Ep–1 ∈ H0(X0(1), ℤp ⊗ ω⊗k) where X0(1) is the compactified modular curve for the full modular group, and ω is the pushforward of the canonical bundle on the universal elliptic curve over ℤ.
now i can use p-adic uniformisation to evaluate Ep–1 via its q-expansion at elliptic curves with j-invariant satisfying | j | > 1; in concrete terms this means formally inverting the power series for 1/j(q), and then evaluating *Ep–1(q) on the result.
i don't suppose one should expect to get a well-defined value in ℂp for Ep–1 evaluated at elliptic curves for which | j | < 1 (as you would for an modular form over ℂ) since this would mean choosing a non-vanishing differential on the universal elliptic curve over ℤp. the norm |Ep–1 |, however, is well defined, since any two choices of basis for the canonial bundle on a given elliptic curve can only differ by a unit.
of course the context i am interested in is when |Ep–1| is used to define the "overconvergent" region of the modular curve. this is (roughly) defined as the region of the $j$-line satisfying |Ep–1|>r for |r|<1. this region will (always) include part of the |j|<1 region. in order to understand what |Ep–1|>r "means" in concrete terms, i was hoping to numerically compare |E4| and |j| for some specific ℂp-values of j near 0 (for primes p = 5, 7, 11 etc. for which j = 0 is supersingular).
anyway, my question is:
how can i explicitly evaluate |Ep–1| at elliptic curves with | j | < 1?