Friday, 20 March 2009

how do you evaluate the p-adic modular form E_p-1 in the region |j|

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=|j1728| 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(j1728)3Delta5, and so when |j|leq1, one has
|E10|6=|j|2|j1728|3. Perhaps this will help?

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