background/motivation
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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 σ*i* is the [divisor sum](http://en.wikipedia.org/wiki/Divisor_function) and *bk* is the *k*-th [bernoulli number](http://en.wikipedia.org/wiki/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 ℤ*p*. that is, *Ep–1* ∈ *H*0(*X*0(1), ℤ*p* ⊗ ω⊗k) where *X*0(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 \*E*p*–1(*q*) on the result.


i don't suppose one should expect to get a well-defined value in ℂ*p* 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 ℤ*p*. 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 ℂ*p*-values of *j* near 0 (for primes *p* = 5, 7, 11 etc. for which *j* = 0 is supersingular).


anyway, my question is: 



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how do you evaluate the p-adic modular form E_p-1 in the region |j|<1
One has $j = E\_4^3/\Delta$. In the region $|j|\leq 1$, one is parameterizing elliptic curves with good reduction, and so $\Delta$ is a unit. Thus $|j| = |E\_4|^3$. This will help you
when $p = 5$. 


When $p = 7,$ one can write $j = 1278 + E\_6^2/\Delta,$ hence $|E\_6|^2 = | j - 1728|$ on
the region $|j| \leq 1$. 


For $p = 11$, these sort of explicit computations are harder (but maybe not much; see 
the added material below), because there are two supersingular $j$-invariants. But the $p = 5$ and 7 cases will already be illustrative.


In the case when $p = 2$, I wrote something about this once, which appeared in an
appendix an article by Fernando Gouvea in a Park City proceedings volume. A slightly
butchered version (missing figures, among other things) can be found on my
[web-page](http://www.math.northwestern.edu/~emerton/preprints.html) (near
the bottom). You might also look at the papers of Buzzard--Calegari
for related computations, as well as my thesis (available on my web-page) and later
work by Kilford and Buzzard--Kilford. (There is, or at least once was, a cottage industry
based on combining these sorts of explicit computations with some more theoretical
estimates, to compute information about slopes of overconvergent $p$-adic modular forms
for various small primes $p$.)


Added in response to the comment below: For $p = 11$, one has 
$E\_{10} = E\_4 E\_6,$
so $E\_{10}^6 = j^2(j-1728)^3 \Delta^5,$ and so when $|j| \leq 1,$ one has
$|E\_{10}|^6 = |j|^2|j-1728|^3.$ Perhaps this will help?


