Su: quadratic imaginary modular symbols
From: William A. Stein <was@math.harvard.edu>
 To: John Cremona <John.Cremona@nottingham.ac.uk>
 Cc: mazur@math.harvard.edu
 Date: Wed, 14 Nov 2001 20:00:43 -0500
 
Dear John,

You should receive a thesis by a student of Darmon on a Mazur-Tate
type conjecture involving quadratic imaginary modular symbols soon.
(The student thanks you in his introduction.)  I'm the outside reader
for the thesis, and I hope to discuss it with you sometime.

I have another question about a totally different notion of quadratic
imaginary modular symbols, which arose when discussing Heegner points
with Mazur today.  Fix a positive integer N and let K be a quadratic
imaginary extension of Q.  Let S be the set P^1(Q) union the elements
in K with positive imaginary part.  Let R denote the real numbers.
Consider the additive subgroup H(K) of H_1(X_0(N),R) generated by all
("extended") modular symbols of the form {alpha, beta}, where alpha
and beta are in S.  Do you know whether or not anybody has written
down a systematic theory for these H(K)'s?  In particular:

  (1) Is H(K) finitely generated?  (Perhaps incorrectly, I think 
      that it is.)   If so, what is the rank of H(K) over Z?

  (2) Do the Hecke operators on H_1(X_0(N),R) preserve H(K)?  (Probably.)

  (3) Do you know a (finite) presentation for H(K)?


Best regards,

  William