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\title{BSD Notebook}
\author{William Stein}
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\section{September 24, 2009}

{\bf From Sheldon:}

For each prime $N$ between 19 and 97 (inclusive) we need to choose a
$p$-Eisenstein quotient $A$ of $J_1(N)$ (with $p$ odd), and find $4$
weight two newforms attached to $A$ such that the vectors consisting
of their first four Fourier coefficients are linearly independent over
$\Z[D]$ in characteristic 2 (or possibly in char 3 if the search in char
2 fails), where $D$ is the group of diamond operators at level $N$.  Also,
if possible, I'd like to know if the $p$-Eisenstein prime is unramified
in the Hecke algebra.  Finally, I don't really need to know the above
for $N= 19, 23, 29, 31, 41$, or $59$.  I have a direct geometric proof of
the non-existence of degree $4$-torsion points of order $N$ in these
cases.  However, if it's not very difficult it would be nice to know
the answer in these case too.

Anyway, barring more of my stupid mistakes here's a list of levels $N$,
primes $p$, and $n$=order of the character through which a generator of
the diamond operators acts on the $p$-Eisenstein quotient.  The levels
marked with an asterisk aren't really necessary, but would be nice if
it's not extra work for you.  If the calculation doesn't produce
independence in some case, then we can probably just try a different
$p$.

\begin{verbatim}
19*, 487, 9
23*, 37181, 11
29*, 43, 7
31*, 2302381, 15
37, 19, 9
41*, 431, 5
43, 463, 7
47, 139, 23
53, 96331, 13
59*, 9988553613691393812358794271, 29
61, 2801, 5
67, 661, 11
71, 211, 5
73, 241, 6
79, 199, 3
83, 17210653, 41
89, 37, 4
97, 367, 3
\end{verbatim}

{\bf From William:}

I wrote some relevant code and ran it.  Here is the code.
\begin{verbatim}
data = [(19, 487, 9), (23, 37181, 11), (29, 43, 7), (31, 2302381, 15),
(37, 19, 9), (41, 431, 5), (43, 463, 7), (47, 139, 23), (53, 96331, 13), 
(59, 9988553613691393812358794271, 29), (61, 2801, 5), (67, 661, 11), 
(71, 211, 5), (73, 241, 6), (79, 199, 3), (83, 17210653, 41), 
(89, 37, 4), (97, 367, 3)]

def char(N, p, d):
    G = DirichletGroup(N,CyclotomicField(d))
    return [eps[0] for eps in G.galois_orbits() if
            eps[0].order()==d][0].minimize_base_ring()

def ms(N,p,d):
    return ModularSymbols(char(N,p,d), 2, +1).cuspidal_subspace()

def modp_reductions(f,p):
    # compute mod-p reductions of a newform
    K = f.parent().base_ring()
    fac = K.factor(p)
    v = []
    F0 = None
    for P, e in K.factor(p):
        F = K.residue_field(P) 
        if F0 is None: 
            F0 = F
            phi = lambda x: x
        else:
            # fix map F --> F0
            phi = F.hom([F.polynomial().roots(F0)[0][0]])
        R = F0[['q']]           
        for i in range(F.degree()):
            v.append(R([phi(F(a)^(p^i)) for a in f.list()]))
    return v            

\end{verbatim}

Then I ran the following:
\begin{verbatim}
for N,p,d in data:
    print N,p,d
    M = ms(N,p,d)
    for A in M.decomposition():
        f = A.q_eigenform(5,'a')
        G = [g.list()[1:5] for g in modp_reductions(f, 2)]
        K = G[0][0].parent()
        V = K**4
        z = [V(x) for x in G]
        S = V.span(z)
        print S.rank()
\end{verbatim}
which output
\begin{verbatim}
19 487 9
4
23 37181 11
4
29 43 7
4
31 2302381 15
4
37 19 9
4
4
41 431 5
4
43 463 7
4
4
47 139 23
...
\end{verbatim}

This computation means the following.  
\begin{proposition}
  Let $A$ be any simple abelian variety factor of $J_1(N)$ with
  character $(N,p,d)$ in the list above (with $N<47$, since that is
  all I've done so far) and $A$ having character of order $d$. Let
  $f_1,\ldots, f_r$ be the newforms associated to $A$, so $r=\dim(A)$.
  Let $\overline{f}_1, \ldots,\overline{f}_r$ be the images of these
  newforms in $\Fbar_2[[q]]$ under reduction modulo primes over $2$.
  For each, let $v_i$ be the vector in $\Fbar_2^4$ of the coefficients
  of $q, q^2, q^3, q^4$ of $\overline{f}_i$.  Then the span of the
  $v_i$'s has dimension $4$.
\end{proposition}

Is that equivalent to what we need or not?  It may take hours for the
data to run up to $100$.  Let me know if this is headed in the right
direction, if I'm completely misunderstanding everything, if something
is unclear, if you want to see things more explicitly, etc.

\end{document}