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?



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


