I think that what I ought to do now is to write a THIRD essay with the real
nitty-gritty(= great practical detail,there must be something in idiomatic
French?)on many of these computations,since Mueller at S-B also wants to
implement it all,& perhaps in the world there may be as many as 10 people
who can do these things as well as talk about them.
Meanwhile,here is the raison-d'estre of my original methods. I had
(have)a very fast QEULDI/QEULMU for multiplying by Euler's series. If one
takes(e.g.)q=41 and loads eta.eta(q) as x**(7/4) (1,-1,-1,0,0,1,0,1,
(which is a binary theta-series based on a quadratic form),the the Hecke
operator T5 acting on this starts off x**(3/4) (-1,0,-1,0,..,so dividing
this by eta.eta(q) we get x**-1 (1,1,3,4,....) as a Hauptmodul of GZ*(41).
Note that the T5 is the terms 15/4,35/4,55/4,.etc plus the extra bit
x**(5*7/4) (1,0,0,0,0,-1,0,0,0,0,0,-1,...) with sign to be carefully
determined. Of course this is ALSO a theta series,and could be done directly,
but since I have the Heckop routine it saves my time to compute 5*N terms of
eta*eta(q) rather than N terms of a tiresome new quadratic form(essentially
one in an order of Q(sqrt(-41))).
That above is XONHK1.
(X2NHK1 takes two distinct heckops subtracts them to get a bigger zero and
divides by eta.eta(q))
Sometimes however, the lowest valence of a function on GZ*(q) is bigger
than (q+1)/24. Now eta.eta(q).function has a POLE and is no good. Then
quite often I take eta^2.eta^2(q) and heckop that,&c,which is XONHK2.
Now the supersingular method plus laundering can be communicated with
very little data. Every function on GZ*(q) regular inside the upperhalfplane
with a pole at infinity and an integral power series(such as my 'A')is
congruent modulo q to a polynomial in j. This polynomial can be sent
very cheaply(with a noncanonical constant term chosen for convenience)
. Now your program should do this:
(degree of polynomial is v)
(1) Expand the polynomial in j in a power series in x up to x**N mod q.
(2) Multiply this by eta.eta(q) up to x**N mod q, least absolute residues.
(3) Assume (in most cases) that these coefficients are correct(over Z)
(4) For any Chinese prime P, divide last series by eta.eta(q) mod P to get
'A' mod P.
How big is N here? Enough to find ME between j and 'A'
which is about q*(2v+1)+3(for checking).
How does one prove that it works? By having done it correctly
using XONHK,&c, and knowing that the assumption (3) is right for our N.
I will now fix a program to get these polynomials printed(of
course it is being done,but not printed,now),& send a few. Probably
OK with all except 11,17,19,43,67,163 (and 37,but this is good by eta).
You may get another note soon. i am keeping copies of this
current series of communications,so easy to refer. oa
Please report errors.