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 [1]. This is the last one in a series of 17 talks.

You may also want to check out Tony Feng's excellent notes of all talks and my earlier course notes.

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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.

Remark 1
is infinite dimensional caused by the fact that is only *locally* of finite type. This infinite dimensionality can already be seen when , where we recover the classical Hecke action on the space of automorphic forms of level 1: Here the space of cusp forms is finite dimensional, but the space of Eisenstein series is infinite dimensional.

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 ).

Remark 2
One can describe the Hecke action on the Eisenstein part explicitly: a -point of factors as for some character . In particular , . When , we get . This is completely analogous to classical Eisenstein ideal (= -th Fourier coefficient of the ``weight 2'' Eisenstein series), which was invented by B. Mazur to study rational points on modular curves and torsion points on elliptic curves.

After killing the Eisenstein part, we indeed obtain a finite dimensional vector space.

Proof (Sketch)
Recall that is a union of open substacks of finite type with instability index bounded by . The key point here is that one can understand the difference between the cohomology of using horocycles discussed in Lizao's talk. More precisely, when , the cone of the natural map is equal to , which is a local system concentrated in degree . Here , , and is the quotient torus of the Borel of .

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.

Proof (Sketch)
For a sequence of open substacks of finite type Define the images Suppose the cohomological correspondence induced by sends to , then we have the induced map on associated graded It turns out one can construct such sequence such that these induced maps are surjective for all for some . The result then follows since is finite dimensional.

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:

Proof
We have an embedding By Theorem 2, is a finite -module. Also is finite -module (due to the finiteness of ). It follows that RHS is a finite -module and hence is a finite -module. Because is a polynomial algebra, it follows that is a finitely generated -algebra.
¡õ

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:

Theorem 3([1, Theorem 7.14])

- There is a decomposition Here is a finite set of closed points.
- There is a unique decomposition of -modules such that , and is finite dimensional over .

Remark 3
The fact that is a finite set of closed points and is finite dimensional follows from Theorem 1.

Theorem 4 ([1, Theorem 9.2])
For any , we have the identity of rational numbers

Proof
Define to be the image of in . Then both sides of the identify only depend on the image of in . Define to be the linear subspace spanned by 's, where effective divisor of degree . By Ana's Talk, we already proved the identity for . So it remains to show that the composition is surjective.

Using the key finiteness theorem one can prove the following lemma:

Proof (Sketch)
is a finitely generated -algebra, so there exists a finite set such that the images of () generate as -algebra. Enlarge so that also contains all for . Let be the ideal generated by the images of (), then works.

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:

Proof
Let . Then under the Satake transform we may identify , where . Note that , thus it suffices to show that for any nonzero ideal in the ring of symmetric polynomials in , and any , we have For any , write (). We define the lowest degree of to be the smallest such that . Then it suffices to show that for any given , we can find an element with same the lowest degree as that of . Since is a PID, we may assume for some . Notice that for any , there is no cancellation in the product , and so one can always choose some such that has a prescribed lowest degree (in particular, the lowest degree of ) as desired.
¡õ

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

[1]Shtukas and the Taylor expansion of $L$-functions, ArXiv e-prints (2015).