These are expanded notes prepared for a talk in Arbeitsgemeinschaft: Higher Gross Zagier Formulas held at Oberwolfach, Spring 2017, on Yun-Zhang's beautiful work . This is the last one in a series of 17 talks.
Our goal today is to finish the proof of the main identity for all functions in the spherical Hecke algebra of . For any (unramified everywhere) cuspidal automorphic representation of , the LHS via the analytic spectral decomposition and the RHS via the cohomological spectral decomposition (discussed below) would imply the identity We now have the wonderful opportunity to apply the identity to simplest element in the Hecke algebra, namely the the unit element , and obtain our desired Higher Gross—Zagier formula
Ana's talk has proved the main identity for many 's but we fall short of proving it for the element : in some sense the simplest Hecke function gives the most difficult situation for intersection computation (self-intersection), and considering for sufficiently large allows us to move away from the self-intersection situation and make the computation easier. What we would like to do is to resolve this tension, and deduce the identity for all Hecke functions from sufficiently many 's by just doing commutative algebra. What make the deduction possible are certain key finiteness properties of the Hecke action on the middle cohomology of the moduli of shtukas.
To kill the Eisenstein part, we again make use of the Eisenstein ideal appeared in Ilya's talk on analytic spectral decomposition. Recall we define the Eisenstein ideal to be moreover , which is 1-dimensional as a ring ( is an extension of by the finite group ).
After killing the Eisenstein part, we indeed obtain a finite dimensional vector space.
In particular, it suffices to work with the generic fiber and show is finite dimensional. Let be the geometric generic fiber of , then the map is an isomorphism when and injective when with cokernel .
Let be the finite union of , where instability index is not all . Using the compatibility of the cohomological constant map and the Satake transform, we have a commutative diagram For , by definition and so the bottom row is zero. The cohomological constant map on the right is injective since . It follows that the top row is zero, and hence the image of is contained in , which is finite dimensional as desired. ¡õ
Using a similar argument, one can also prove the following finiteness theorem.
The key point here again is by the compatibility of Satake isomorphism and the cohomological constant term, the -action on the cokernel can be made explicit as -action. One can then choose appropriately so that acts trivially on the cokernels for degree reasons and to make sure that induces a surjection . ¡õ
Let . We have the following immediate consequence:
Now is a finite module over the noetherian ring (by Theorem 2, is a finite module even over ), we obtain the following cohomological spectral decomposition:
Using the key finiteness theorem one can prove the following lemma:
In fact, because is away from the generating set , we have Since the left-hand-side is equal to spans by definition, we know (think: is ``small'' enough).
On the other hand, by the cohomological spectral decomposition (Theorem 3), It remains to rule out the Eisenstein part: i.e., to show that is disjoint from . Suppose is a -point of lies in . Then factors as for some character . It follows that for , we have Hence . Let be the character . Then has finite image and . By class field theory can be identified as a Galois character of and hence for infinitely many by Chebotarev density. This contradicts for all . So is disjoint from and hence is finite dimensional (think: is ``large'' enough). ¡õ
Besides the key finiteness theorems, we need one additional ingredient concerning the local Hecke algebra. It is a bit magical but completely elementary:
Now we can finish the proof of the main identity using the previous two lemmas. For any , look at the commutative diagram Here the vertical arrows are all natural inclusions.
By Lemma 1, is finite dimensional, so is also finite dimensional. Since is quotient of by a nonzero ideal, and , we know the bottom row is surjective by Lemma 2. Now generate as an algebra, so the top row is also surjective. By Lemma 1, , so the map is surjective as desired. ¡õ
Shtukas and the Taylor expansion of $L$-functions, ArXiv e-prints (2015).