One has j=E34/Delta. In the region |j|leq1, one is parameterizing elliptic curves with good reduction, and so Delta is a unit. Thus |j|=|E4|3. This will help you
when p=5.
When p=7, one can write j=1278+E26/Delta, hence |E6|2=|j−1728| on
the region |j|leq1.
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
E10=E4E6,
so E610=j2(j−1728)3Delta5, and so when |j|leq1, one has
|E10|6=|j|2|j−1728|3. Perhaps this will help?
No comments:
Post a Comment