Date: Sat, 23 Jan 1999 10:47:07 -0800
From: Shuzo Takahashi <shuzo@abel.MATH.HARVARD.EDU>
To: "Kenneth A. Ribet" <ribet@math.berkeley.edu>
Cc: was@math.berkeley.edu
Subject: Mestre-Oesterl\'e's theorem

Dear Ken,

I am still on the Berkeley NTS mailing list (which I like) and I saw that
you talked or are going to talk about Mestre-Oesterl\'e's theorem on the
surjectivity of the map on the groups of connected components.
I was wondering whether you might know if the statement below (a slight
generalization of their theorem) is true.

Let J be J_0(pq) (p and q are different primes). Consider an optimal
quotient J --> E and the map Phi(J,p) --> Phi(E,p) on the groups of
connected components induced from J --> E by the mod p reduction.

Statement: For any p, Phi(J,p) --> Phi(E,p) is surjective
           if q = 2, 3, 5, 7, or 13.

Mestre-Oesterl\'e's theorem can be considered as the case q = 1.
The statement is numerically true for p < 40 (thanks to data provided by
David Kohel) and if E is unique in its Q-isogeny class up to isomorphism,
this is obviously true because in this case c_p = #Phi(E,p) = 1 by your
level-lowering theorem. Also, for example, if q = 11, the statement is
generally false, but for some p's, it is still true.

I am interested in this because for q = 2, 3, 5, 7, or 13 this statement is
equivalent to the statement
    c_p c_q = c'_p c'_q
where c'_p and c'_q are the orders of the groups of connected components of
the elliptic curve (isogenous to E) which is an optimal quotient of Shimura
curve associated with the maximal order in the indefinite quaternion
algebra of discriminant pq.

Best Regards,
Shuzo