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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 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–1H0(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?

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Quick side note: j=0 isn't supersingular at p=7, right? Isn't it only the primes where p = -1 mod 3? I'm afraid I have nothing substantive to add. –  David Speyer Feb 5 '10 at 2:42
oops, yep right you are, i meant p=2,5,11 –  Max Flander Feb 5 '10 at 3:48
This comment is sort of tangentially related to your question, since here $p=3$. Mazur, Tate and I wrote a paper wstein.org/papers/pheight on computing $p$-adic heights. The main problem we ran into was evaluating the $p$-adic modular form $E_2$ efficiently, and we solved the problem using Monsky-Washnitzer cohomology. A lot of the paper is also about studying a notion of "log convergence" that Mazur and Tate got really interested in for some reason. –  William Stein Feb 8 '11 at 19:37
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1 Answer

up vote 5 down vote accepted

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 (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?

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thanks for that emerton, i'm familiar with the work you mention, actually it was p=11 that was my motivation; lawren smithline's paper "bounding slopes" (front.math.ucdavis.edu/0705.3614) relates j and E_p-1 for all the p such that there is only one supersingular curve. i haven't been make that method work for p=11, and so i was hoping to do some explicit computer calculations to see if that might give some sort of hint as how to proceed... –  Max Flander Feb 5 '10 at 3:55
I added something about $p = 11$ to my answer, which may help. –  Emerton Feb 5 '10 at 4:34
No worries. Your "thanks heaps" made me wonder if you are Australian, and then I looked at your profile and saw that you're at Melbourne. It's nice to see that p-adic modular forms are still alive and well in the halls of Richard Berry. –  Emerton Feb 5 '10 at 5:48
I think Kilford did something with p=11 recently (computing a basis for overconvergent forms: the point is that you can use both positive and negative powers of T, if T is a parameter on the annulus of ordinary forms). –  Kevin Buzzard Feb 5 '10 at 20:44
thanks for the suggestion kevin--it was actually kilford's paper that got me thinking about this stuff. matthew's answer is exactly what i wanted. to generalise it to arbitrary p: express E_{p-1} as a polynomial in E4 and E6 (homogeneous with respect to the usual grading), take the sixth power and then make the substitutions E4^3 |-> j*delta, E6^2 |-> (j-1728)*delta; now you have a polynomial in j and delta. –  Max Flander Feb 8 '10 at 3:40
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