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.