Date: Tue, 21 Sep 1999 17:56:01 -0700 (PDT)
From: Jordan Ellenberg <ellenber@msri.org>
To: William Stein <was@msri.org>
Subject: Component groups of motives

Dear William,

The other day I asked you "whether the component group of the optimal
quotient of the Jacobian could be read off the Galois representation."
I thought about this a little more, and I decided that the answer is "yes
and no."  The point is that the Galois representation attached to a
modular form is customarily considered to be a representation

G_Q -> GL_2(Q_ell)

defined up to conjugacy, and this is indeed an "isogeny-invariant object"
which contains no information about component groups.  It is a "motive
with coefficients in Q."

If we want information about component groups, we need a "motive with
coefficients in Z."  We then get a well-defined Galois representation

G_Q -> GL_2(Z_ell)

defined up to conjugacy _by GL_2(Z_ell)_.  Choosing an "integral model"
for our motive with coefficients in Q is like choosing a representative of
an isogeny class of abelian varities.

Indeed, if f is weight 2, then we have a well-defined choice of integral
model for the motive associated to f; namely, the optimal quotient.

QUESTION:  Does Scholl's construction produce a motive with coefficients
in Z or in Q?  If in Q, is there a natural "optimal" choice of integral
model for this motive, which agrees with the optimal quotient in the
weight 2 case?

Comment:  I actually think "integral model" is bad terminology because it
suggests we're mucking around with the base scheme, which we are not.
Maybe I should say "a lattice"....?

See you tomorrow,

Jordan