16
$\begingroup$

I've read (skimmed heartily) basic books on the topic of modular forms. (The last being Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.)

I strive for an understanding which is as mathematically mature as possible. (Read: strive for a Langlands-program-ish understanding.) Alas, I am still far from succeeding.

I have heard on many an occasion people referencing "Hecke correspondences". I am aware of the definition of a correspondence, but I'm at a loss of how to think about Hecke correspondences! What made them arise? How are they helpful? What suggested to anyone that they should define them? How do they relate to Hecke operators? Is this thing helpful towards Langlands?

Ach... Hopefully this is within the realm of mathstackexchange (or is this more appropriate to mathoverflow?). This has been gnawing at me for months.

References are also welcome.

  • 2
    To get a better grip on correspondences in general, and Hecke correspondences in particular, I'd suggest looking at Diamond and Shurman's book on modular forms. This gives a very nice account of how to make the logical step from Hecke operators (which it sounds like you've got to grips with somewhat) to Hecke correspondences (which are "geometric avatars" of Hecke operators).2012-07-03
  • 5
    cliff notes version of relationship between Hecke operators and correspondences: assume weight 2. then modular forms "=" one-forms on modular curve = one-forms on Jacobian. Correspondences between curves = maps between Jacobians. The one-form (on the Jacobian) attached to T_p f is the pullback of the one-form attached to f along the correspondence called T_p.2012-07-03
  • 0
    Thanks, countinghaus. That was very helpful!2012-07-03

1 Answers 1