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Why is one interested in the mod p reduction of modular curves and Shimura varieties?

From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the Hasse-Weil conjecture for modular curves. Are there similar applications for Shimura varieties?

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Um, because the mod p reduction of modular curves and Shimura varieties is really, really interesting? (Seriously.) –  Pete L. Clark Jan 19 '10 at 12:36
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The Eichler-Shimura relation doesn't just prove the Hasse-Weil conjecture for modular curves. It e.g. attaches Galois representations to modular forms of weight 2. More delicate arguments (using etale cohomology with non-constant coefficients, machinery that wasn't available to Shimura) attaches Galois representations to higher weight modular forms. These ideas have had many applications (e.g. eventually they proved FLT). In summary: computing the mod p reduction of modular curves isn't just for Hasse-Weil.

Doing the same for Shimura varieties is technically much harder because one runs into problems both geometric and automorphic. But the upshot, in some sense, is the same: if one can resolve these issues (which one can for, say, many unitary Shimura varieties nowadays, but by no means all Shimura varieties) then one can hope to attach Galois representations to automorphic forms on other reductive groups, and also to compute the L-function of the Shimura variety in terms of automorphic forms.

Why would one want to do these things? Let me start with attaching Galois reps to auto forms. These sorts of ideas are what have recently been used to prove the Sato-Tate conjecture. Enough was known about the L-functions attached to automorphic forms on unitary groups to resolve the analytic issues, and so the main issue was to check that the symmetric powers of the Galois representations attached to an elliptic curve were all showing up in the cohomology of Shimura varieties. Analysing the reduction mod p of these varieties was just one of the many things that needed doing in order to show this (although it was by no means the hardest step: the main technical issues were I guess in the "proving R=T theorems", similar to the final step in the FLT proof being an R=T theorem; the L-function ideas came earlier).

But to answer your original question, yes: if you're in the situation where you understand the cohomology of the Shimura variety well enough, then analysing the reduction of the variety will tell you non-trivial facts about the L-function of the Shimura variety. Note however that the link isn't completely formal. Mod p reduction of the varieties only gives you an "Eichler-Shimura relation", and hence a polynomial which will annihiliate the Frobenius element acting on the etale cohomology. To understand the L-function you need to know the full characteristic polynomial of this Frobenius element. For GL_2 one is lucky in that the E-S poly is the char poly, simply because there's not enough room for it to be any other way. This sort of argument breaks down in higher dimensions. As far as I know these questions are still very open for most Shimura varieties.

So in summary, for general Shimura varieties, you can still hope for an Eichler-Shimura relation, but you might not actually be able to compute the L-function in terms of automorphic forms as a consequence.

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Kevin's answer gives a very good explanation of the role of mod $p$ reduction in the theory of Galois representations and automorphic forms. In this answer I will try to say something a little more technical about one way in which understanding the mod $p$ reduction of modular curves can be applied in arithmetic. The precise application that I will discuss is that of constructing congruences of modular forms.

If $f$ is a Hecke eigenform (of weight 2, to fix ideas), then associated to $f$ is a Galois representation $\rho\_f:G\_{\mathbb Q} \to GL\_2(\overline{\mathbb Q}\_{\ell})$ (for any prime $\ell$). Say the level of $f$ is equal to $N p$, where $p$ is a prime not dividing $N$. One can ask: is there an eigenform $g$ of level $N$ such that $f \equiv g \bmod \ell$. (Here congruence means congruence in $q$-expansions.) This is the question that Ribet solved in his famous Inventiones 100 paper (the paper which reduced FLT to Shimura--Taniyama).

Note that since $p$ is not in the level of $g$, the representation $\rho\_g$ will be unramified locally at $p$. (This comes from knowing that the modular curve of level $N$ has good reduction at $p$, since $p$ does not divide $N$ --- a first application of the theory of reduction of modular curves.) (If $p = \ell$ one must be more careful here, but I will suppress this point.)

Thus if $f \equiv g \bmod \ell,$ so that $\rho\_f$ and $\rho_g$ coincide mod $\ell$, we see that $\rho\_f$, when reduced mod $\ell$, must be unramified at $p$. So this is a necessary condition for the existence of $g$.

It turns out (and Ribet proved) that (under some additional technical hypotheses) this necessary condition is also sufficient. The way the argument goes is the following: the modular curve of level $N p$ has semi-stable singular reduction: it is two smooth curves (coming from level $N$) crossing each other a bunch of times (this is the contribution from the $p$-part of the level). Now the mod $\ell$ Galois representation $\overline{\rho}\_f$ (the reduction of $\rho\_f$ mod $\ell$) is consructed out of the $\ell$-torsion subgroup of the Picard group of this singular curve. Since it is unramified at $p$, it can't be entirely explained by the singularities; some part of it must be arising from the smooth curves, which are of level $N$. (If you like, this is an application of the a certain form of the criterion of Neron--Ogg--Shafarevic.) The Eichler--Shimura relations then show that the system of Hecke eigenvalues attached to $f$, when reduced mod $\ell$, must arise at level $N$: in other words, there is an eigenform $g$ of level $N$ that is congruent to $f$ mod $\ell$.

This is just one typical argument that uses a detailed knowledge of the good and bad reduction of modular curves in various situations. Since the Galois representation attached to modular forms are constructed geometrically from the modular curves, tools like the Neron--Ogg--Shafarevic criterion, and variants thereof, show that there are very close ties between the local properties of the Galois representations at a prime $p$, and the reduction properties of modular curves mod $p$.

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