HIDA FAMILIES (letters from Coleman)
William,
        Thanks.  I've thought of a computational problem similiar to
and maybe easier than the X_0(49) problem you have been thinking about.  I
asked Kevin whether Hida families (i.e. the slope zero components of
the eigencurves) are smooth (in char 0).  He tried to make an example
using weight one forms.  The idea is that if f(q)=\sum a_nq^n is a
weight one cuspidal eigenform of tame level N then there exists a
prime l\ne p such that a_l = 2 (i.e., such that the image of Frobenius
at l is the identity).  Suppose f is ordinary at p and H is the Hida
family through f. Assume H is just spec(\Lambda). When one goes up to
level Nl one gets a double cover J of H ramified at f.  Kevin thought
J wouldn't be smooth there but I don't know.  People have thought
about Hida families for a while.  This question should be more
accessible.  Let me know if you are interested in pursuing this.
        Robert                                   

  William,
        I think you should look at Hida's book and therrte is a paper
of Gouvea that seems relevant. It's lying around somewhere.  I'LL try
to find it.  You might also look at Buzzard-Taylor.
        Robert                         

William,
        For some good examples of weight one cusp forms on Gamma_1(p),
all one needs is a prime p congruent to -1 mod 4 such that the class
number of Q(\sqrt{-p}) is > 1.  See remark B4.5 of P-adic Banach
Spaces and ...
        Robert     

William,
        It is "On the Ordinary Hecke Algebra."  Journal of Number
theory, vol 41. no. 2 1992.  It seems to avoid the cases we're
interested in though.  One of the examples I mentioned is p=23,
(h=3).  One can take l = 59 (=(36+23)).  Then l splits completely in
the Hilbert class field of Q(sqrt{-23}).  So one can try to understand
this example.
        Robert