Summary: Some calculations regarding torsion and component groups
Note: What I am calling observations below are really hunches based on part of the data; they
have to be tested on more data. Let f be a newform of weight 2 on 0(N), and let A = Af be
the quotient of J0(N) associated to f. For the moment, suppose N is square-free, and let K be
the full cuspidal subgroup of J0(N), which is necessarily rational.
Observation 0.1. A(Q)tor K.
I suppose to start with, one can check if |A(Q)tor| divides |K|, say for elliptic curves, us-
ing your table http://modular.ucsd.edu/Tables/cuspgroup/index.html. I would not be too
surprised if things do not work out at the primes 2, 3 and the primes dividing N.
When N is not square-free, I guess the above may be true with K replaced by the rational part
of the full cuspidal subgroup (i.e., A(Q)tor K J0(N)(Q)). But I have not looked at the data.
Suppose now on that N is any integer (but at times one may need to assume N is square-free).
The rational divisor (0) - () generates a finite subgroup of J0(N)(Q), which we denote C.
The image of C under the quotient map J0(N) A is a cyclic subgroup of A(Q)tor; we denote
this subgroup by CA and call it the cuspidal subgroup of A (note that this is not the subgroup
generated by the images of all the cuspidal divisors, but by the image of just (0) - ()). Let wp
denote the eigenvalue of the Atkin-Lehner involution Wp acting on f. The product of the Wp's
for p|N is the Fricke involution WN , whose eigenvalue is denoted wN .
The following observations are based on Cremona's data and your table
http://modular.fas.harvard.edu/Tables/non zeroinf tor.txt.