Question

background/motivation

let E_k 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 and b_k 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 E_p–1 defines a modular form over ℤ_p. that is, E_p–1 ∈ H⁰(X₀(1), ℤ_p ⊗ 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 E_p–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 ℂ_p for E_p–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 |E_p–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 |E_p–1| is used to define the "overconvergent" region of the modular curve. this is (roughly) defined as the region of the $j$-line satisfying |E_p–1|>r for |r|<1. this region will (always) include part of the |j|<1 region. in order to understand what |E_p–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:

how can i explicitly evaluate |E_p–1| at elliptic curves with | j | < 1?

Answer 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 (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?