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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.

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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). –  David Loeffler Jul 3 '12 at 7:08
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. –  user29743 Jul 3 '12 at 14:38
Thanks, countinghaus. That was very helpful! –  Jenifer Jul 3 '12 at 17:02

1 Answer 1

It might help to go back to the definition of Hecke operators in level $1$ in Serre's Course in arithmetic. For a prime $p$ and a lattice $\Lambda$, the $p$the Hecke corresondence (I forget if Serre uses exactly this terminology) takes $\Lambda$ to $\sum \Lambda'$, where $\Lambda'$ runs over all index $p$ sublattices of $\Lambda$.

This is a multi-valued function from lattices to lattices (it is $1$-to-$p+1$-valued).

Now lattices (mod scaling) are just elliptic curves: $\Lambda \mapsto \mathbb C/\lambda$. And so we can also think of this as a multi-valued map from the moduli space of ellitic curves (i.e. the $j$-line, or $Y_0(1)$ if you like) to itself.

How to describe a multi-valued map more geometrically? Think about its graph inside $Y_0(1) \times Y_0(1)$. The graph of a function has the property that its projection onto the first factor is an isomorphism. The graph of a $p+1$-valued function has the property that its projection onto the first factor is of degree $p+1$.

This graph has an explicit description: it is just $Y_0(p)$ (the modular curve of level $\Gamma_0(p)$). Remember that $Y_0(p)$ parameterizes pairs $(E,E')$ of $p$-isogenous curves. We embed it into $Y_0(1) \times Y_0(1)$ in the obvious way, by mapping the pair $(E,E')$ (thought of as an element of $Y_0(p)$) to $(E,E')$ (thought of as an element of the product).

In terms of the upper half-plane variable $\tau$, one can think of this map as being $\tau \bmod \Gamma_0(p)$ maps to $\bigl(\tau \bmod SL_2(\mathbb Z), p\tau \bmod SL_2(\mathbb Z) \bigr).$

So we have recast Serre's description of the $p$th Hecke operator in terms of a correspondence on lattices in the geometric language of correspondences on curves: i.e. the $p$th Hecke operator is given by a mutli-valued morphism from $Y_0(1)$ to itself, rigorously encoded by its graph thought of as a curve in the product surface $Y_0(1) \times Y_0(1)$, which is in fact isomorphic to $Y_0(p)$.

We can easily compactify the situation, to get $X_0(p)$ embedding as the graph of a correspondence on $X_0(1) \times X_0(1)$.

[Caveat: Actually the map $Y_0(p) \to Y_0(1) \times Y_0(1)$ need not be an embedding; it is a birational map onto its image, but the image can be singular (and the same applied with $X$'s instead of $Y$'s). This is because the point on $Y_0(p)$ is not just the pair $(E,E')$, but the additional data of the $p$-isogeny $E\to E'$, which is not uniquely determined up to isomorphism in some exceptional cases. But this is a technical point which is not worth fussing about at the beginning.]

The advantage of having a geometric correspondence in sight is that whenever we apply any kind of linearization functor to our curve, the correspondence will turn into a genuine single valued operator.

The point is that if we have a multi-valued function from one abelian group to another, we can just add up the values to get a single-valued function.

So the correspondence $T_p$ induces genuine maps from the Jacobian of $X_0(1)$ to itself, or from the cohomology of $X_0(1)$ to itself, or from the space of holomorphic differentials on $X_0(1)$ to itself.

Now actually in the case of $X_0(1)$, which has genus zero, the Jacobian and the space of holomorphic differentials are trivial. But we can do everything with $X_0(N)$ or $X_1(N)$ in place of $X_0(1)$ for any $N$, and all the same remarks apply.

Remembering that the holomorphic differentials on $X_0(N)$ are the weight two cuspforms of level $N$, one can compute that the $p$th Hecke correspondence gives rise to the usual $p$th Hecke operator on cuspforms in this way.

What's the point of considering the correspondence? There are many; here's one:

if we reduce everything mod $p$, we get a mod $p$ correspondence on the mod $p$ reduction of $X_0(N)$, whose graph is the mod $p$ reduction of $X_0(Np)$. But this latter reduction is well-known to be singular, and in fact reducible; it is the union of two copies of $X_0(N)$. Thus the $p$th Hecke correspondence mod $p$ decomposes as the sum of two simpler correspondences, which one checks to be the Frobenius morphism from $X_0(N)$ Mod $p$ to iself, and its dual.

This is the Eichler--Shimura congruence relation (in some form it actually goes back to Kronecker), and it underlies the relationship between $T_p$-eigenvalues and the trace of Frobenius in the $2$-dimensional Galois reps. attached to Hecke eigenforms.

Some MO posts which are vaguely relevant:

The map on differentials induced by a correspondence

The Eichler --Shimura relation

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