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.

[-] Contents
Overview
Key finiteness theorems
Cohomological spectral decomposition
Finish of the proof of the main identity

TopOverview

Our goal today is to finish the proof of the main identity $$(\log q)^{-r} \mathbb{J}_r(f)=\mathbb{I}_r(f)$$ for all functions in the spherical Hecke algebra $f\in \mathcal{H}=C_c(K\backslash G(\mathbb{A})/K)$ of $G=\PGL_2$. For any $\pi$ (unramified everywhere) cuspidal automorphic representation of $G(\mathbb{A})$, the LHS via the analytic spectral decomposition and the RHS via the cohomological spectral decomposition (discussed below) would imply the identity $$\lambda_\pi(f)\cdot \mathcal{L}^{(r)}(\pi_{F'}, 1/2)\sim \langle [\Sht_T]_\pi, f*[\Sht_T]_\pi\rangle.$$ We now have the wonderful opportunity to apply the identity to simplest element in the Hecke algebra, namely the the unit element $\mathbf{1}_K\in \mathcal{H}$, and obtain our desired Higher Gross—Zagier formula $$\mathcal{L}^{(r)}(\pi_{F'}, 1/2)\sim\langle [\Sht_T]_\pi, [\Sht_T]_\pi\rangle.$$

Ana's talk has proved the main identity for many $h_D$'s but we fall short of proving it for the element $\mathbf{1}_K$: in some sense the simplest Hecke function gives the most difficult situation for intersection computation (self-intersection), and considering $h_D$ for sufficiently large $D$ 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 $h_D$'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.

TopKey finiteness theorems

Recall the middle cohomology $$V=H^{2r}_c(\Sht_{G}^{r}, \mathbb{Q}_\ell),$$ which admits an action of the Hecke algebra $$\mathcal{H}=C_c(K\backslash G(\mathbb{A})/K, \mathbb{Q}_\ell)= \bigotimes_{x\in |X|}\nolimits' \mathcal{H}_x.$$
Remark 1 $V$ is infinite dimensional caused by the fact that $\Sht_G^r$ is only locally of finite type. This infinite dimensionality can already be seen when $r=0$, where we recover the classical Hecke action on the space of automorphic forms of level 1: $$\mathcal{A}=C_c(G(F)\backslash G(\mathbb{A})/K, \mathbb{Q}_\ell)=\mathcal{A}_{\mathrm{Eis}}\oplus\mathcal{A}_\mathrm{cusp}.$$ Here the space of cusp forms $\mathcal{A}_\mathrm{cusp}$ 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 $$I_\mathrm{Eis}:=\ker(\mathcal{H}\xrightarrow{\mathrm{Sat}} \mathcal{H}_{A}\cong \mathbb{Q}_\ell[\Div_X(k)]\rightarrow \mathbb{Q}_\ell[\Pic_X(k)]),$$ moreover $\mathcal{H}/I_\mathrm{Eis}\cong \mathbb{Q}_\ell[\Pic_X(k)]^{\iota}$, which is 1-dimensional as a ring ($\Pic_X(k)$ is an extension of $\mathbb{Z}$ by the finite group $\Jac_X(k)$).

Definition 1 Define $Z_\mathrm{Eis}:=\Spec \mathcal{H}/I_\mathrm{Eis}$ a closed subscheme of $\Spec \mathcal{H}$, which is reduced and 1-dimensional.
Remark 2 One can describe the Hecke action on the Eisenstein part explicitly: a $\overline{\mathbb{Q}_\ell}$-point of $Z_\mathrm{Eis}$ factors as $$\mathcal{H}\rightarrow \mathbb{Q}_\ell[\Pic_X(k)]\xrightarrow{\chi} \overline{\mathbb{Q}_\ell}$$ for some character $\chi: \Pic_X(k)\rightarrow \overline{\mathbb{Q}_\ell}^\times$. In particular , $h_x\mapsto \chi(t_x)+q_x \chi(t_x^{-1})$. When $\chi=1$, we get $h_x\mapsto 1+q_x$. This is completely analogous to classical Eisenstein ideal $T_p\mapsto 1+p$ (= $p$-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.

Theorem 1 $I_\mathrm{Eis}\cdot V$ is a finite dimensional $\mathbb{Q}_\ell$-vector space.
Proof (Sketch) Recall that $\Sht_G=\cup_{d} \Sht_G^{\le d}$ is a union of open substacks of finite type $\Sht_G^{\le d}$ with instability index bounded by $d$. The key point here is that one can understand the difference between the cohomology of $$H^i_c(\Sht_G^{<d}, \mathbb{Q}_\ell)\text{ and } H^i_c(\Sht_G^{\le d}, \mathbb{Q}_\ell)$$ using horocycles discussed in Lizao's talk. More precisely, when $d>2g-2$, the cone of the natural map $$\mathbf{R}\pi_{G,!}^{<d}\mathbb{Q}_\ell\rightarrow \mathbf{R}\pi_{G, !}^{\le d}\mathbb{Q}_\ell$$ is equal to $\mathbf{R}\pi_{H,!}^d\mathbb{Q}_\ell[-r](-r/2)$, which is a local system concentrated in degree $r$. Here $\pi_G: \Sht_G\rightarrow X^r$, $\pi_H: \Sht_H\rightarrow X^r$, and $H\cong \mathbb{G}_m$ is the quotient torus of the Borel of $G$.

In particular, it suffices to work with the generic fiber and show $I_\mathrm{Eis}\cdot H^r_c(\Sht_{G,\bar\eta})$ is finite dimensional. Let $\bar\eta$ be the geometric generic fiber of $X^r$, then the map $$H^i_c(\Sht_{G,\bar\eta}^{<d})\rightarrow H^i_c(\Sht_{G,\bar\eta}^{\le d})$$ is an isomorphism when $i<r$ and injective when $i=r$ with cokernel $\cong H_0(\Sht_{H,\bar\eta}^{d})$.

Let $U$ be the finite union of $\Sht_G^{\le d}$, where instability index $d$ is not all $> 2g-2$. Using the compatibility of the cohomological constant map and the Satake transform, we have a commutative diagram $$\xymatrix{H^r_c(\Sht_{G,\bar\eta}) \ar[r]^{f*} \ar[d]^{\prod \gamma_d}& H^r_c(\Sht_{G,\bar\eta}) \ar[r] \ar[d]^{\prod \gamma_d} & H^r_c(\Sht_{G,\bar \eta})/H^r_c(U_{\bar \eta}) \ar[d]^{\prod\limits_{d>2g-2} \gamma_d} \\ \prod_{d} H_0(\Sht_{H,\bar \eta}^d) \ar[r]^{\mathrm{Sat}(f)*} & \prod_{d} H_0(\Sht_{H,\bar \eta}^d) \ar[r] & \prod_{d>2g-2} H_0(\Sht^d_{H,\bar \eta})}$$ For $f\in I_\mathrm{Eis}$, by definition $\mathrm{Sat}(f)*=0$ and so the bottom row is zero. The cohomological constant map on the right is injective since $d>2g-2$. It follows that the top row is zero, and hence the image of $f*$ is contained in $H^r_c(U_{\bar\eta})$, which is finite dimensional as desired. ¡õ

Using a similar argument, one can also prove the following finiteness theorem.

Theorem 2 $V$ is a finitely generated $\mathcal{H}_x$-module for any $x\in |X|$.
Proof (Sketch) For a sequence of open substacks of finite type $$U_0\subseteq U_1\subseteq \cdots U_n\subseteq\cdots\subseteq \Sht_G,$$ Define the images $$F_{\le n}:=\im(H^{2r}(U_n, \mathbb{Q}_\ell)\rightarrow V).$$ Suppose the cohomological correspondence $C(h_x)$ induced by $h_x$ sends $F_{\le n}$ to $F_{\le n+1}$, then we have the induced map on associated graded $$\Gr_{n}^FV\xrightarrow{\Gr_n C(h_x)} \Gr_{n+1}^FV.$$ It turns out one can construct such sequence $U_i$ such that these induced maps $\Gr_nC(h_x)$ are surjective for all $n\ge n_0$ for some $n_0$. The result then follows since $F_{\le n_0}$ is finite dimensional.

The key point here again is by the compatibility of Satake isomorphism and the cohomological constant term, the $C(h_x)$-action on the cokernel can be made explicit as $t_x+q_x t_x^{-1}$-action. One can then choose $U_i$ appropriately so that $q_x t_x^{-1}$ acts trivially on the cokernels for degree reasons and to make sure that $C(h_x)$ induces a surjection $\Gr_n^FV\rightarrow \Gr_{n+1}^FV$. ¡õ

TopCohomological spectral decomposition

Let $\overline{\mathcal{H}}=\im (\mathcal{H}\rightarrow\End_{\mathbb{Q}_\ell}(V)\times \mathbb{Q}_\ell[\Pic_X(k)])$. We have the following immediate consequence:

Corollary 1 $\overline{\mathcal{H}}$ is a finitely generated $\mathbb{Q}_\ell$-algebra (in particular, a noetherian ring).
Proof We have an embedding $$\overline{\mathcal{H}}\hookrightarrow \End_{\mathcal{H}_x}(V \oplus \mathbb{Q}_\ell[\Pic_X(k)]^\iota).$$ By Theorem 2, $V$ is a finite $\mathcal{H}_x$-module. Also $\mathbb{Q}_\ell[\Pic_X(k)]^\iota$ is finite $\mathcal{H}_x$-module (due to the finiteness of $\Jac_X(k)$). It follows that RHS is a finite $\mathcal{H}_x$-module and hence $\overline{\mathcal{H}}$ is a finite $\mathcal{H}_x$-module. Because $\mathcal{H}_x\cong \mathbb{Q}_\ell[h_x]$ is a polynomial algebra, it follows that $\overline{\mathcal{H}}$ is a finitely generated $\mathbb{Q}_\ell$-algebra. ¡õ

Now $V$ is a finite module over the noetherian ring $\overline{\mathcal{H}}$ (by Theorem 2, $V$ is a finite module even over $\mathcal{H}_x$), we obtain the following cohomological spectral decomposition:

Theorem 3([1, Theorem 7.14])
  • There is a decomposition $$\Spec \overline{\mathcal{H}}^\mathrm{red}=Z_{\mathrm{Eis}}\coprod Z_{0}^r.$$ Here $Z_0^r$ is a finite set of closed points.
  • There is a unique decomposition of $\mathcal{H}$-modules $$V=V_\mathrm{Eis} \oplus V_0$$ such that $\supp V_\mathrm{Eis}\subseteq Z_{\mathrm{Eis}}$, $\supp V_0= Z_{0}^r$ and $V_0$ is finite dimensional over $\mathbb{Q}_\ell$.
Remark 3 The fact that $Z_0^r$ is a finite set of closed points and $V_0$ is finite dimensional follows from Theorem 1.

Remark 4 When $r=0$, we exactly recover the decomposition $V_\mathrm{Eis}=\mathcal{A}_\mathrm{Eis}$ and $V_0=\mathcal{A}_\mathrm{cusp}$.

TopFinish of the proof of the main identity

Theorem 4 ([1, Theorem 9.2]) For any $f\in \mathcal{H}$, we have the identity of rational numbers $$(\log q)^{-r}\mathbb{J}_r(f)=\mathbb{I}_r(f).$$
Proof Define $\tilde{\mathcal{H}}$ to be the image of $\mathcal{H}$ in $\End(V) \times \End(\mathcal{A})$. Then both sides of the identify only depend on the image of $f$ in $\tilde{\mathcal{H}}$. Define $\mathcal{H}'\subseteq \mathcal{H}$ to be the linear subspace spanned by $h_D$'s, where $D$ effective divisor of degree $\ge d_0=\max\{2g'-1,2g\}$. By Ana's Talk, we already proved the identity for $f\in \mathcal{H}'$. So it remains to show that the composition $\mathcal{H}'\hookrightarrow \mathcal{H}\twoheadrightarrow\tilde{\mathcal{H}}$ is surjective.

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

Lemma 1 There is an ideal $I\subseteq \tilde{\mathcal{H}}$ such that $\tilde{\mathcal{H}}/I$ is finite dimensional and is generated by the image of $\mathcal{H}'$.
Proof (Sketch) $\tilde{\mathcal{H}}$ is a finitely generated $\mathbb{Q}_\ell$-algebra, so there exists a finite set $S\subseteq |X|$ such that the images of $h_x$ ($x\in S$) generate $\tilde{\mathcal{H}}$ as $\mathbb{Q}_\ell$-algebra. Enlarge $S$ so that $S$ also contains all $h_x$ for $\deg x\le d_0$. Let $I\subseteq \tilde{\mathcal{H}}$ be the ideal generated by the images of $h_y$ ($y\not\in S$), then $I$ works.

In fact, because $h_y$ is away from the generating set $S$, we have $$\im(\bigotimes_{x\in S} \mathcal{H}_x)\cdot \im(h_y)\subseteq \im(\mathcal{H}').$$ Since the left-hand-side is equal to $\tilde{\mathcal{H}}\cdot \im(h_y)$ spans $I$ by definition, we know $I\subseteq \im(\mathcal{H}')$ (think: $I$ is ``small'' enough).

On the other hand, by the cohomological spectral decomposition (Theorem 3), $$\Spec \tilde{\mathcal{H}}\subseteq Z_\mathrm{Eis}\cup Z_{0,\ell}^r\cup Z_{0,\ell }^0.$$ It remains to rule out the Eisenstein part: i.e., to show that $\Spec\tilde{\mathcal{H}}/I$ is disjoint from $Z_\mathrm{Eis}$. Suppose $\sigma: \tilde{\mathcal{H}}/I\rightarrow \overline{\mathbb{Q}}_\ell$ is a $\overline{\mathbb{Q}}_\ell$-point of $\Spec\tilde{\mathcal{H}}/I$ lies in $Z_\mathrm{Eis}$. Then $\mathcal{H}\rightarrow \tilde{\mathcal{H}}/I\rightarrow \overline{\mathbb{Q}}_\ell$ factors as $$\mathcal{H}\rightarrow \mathbb{Q}[\Pic_X(k)]\xrightarrow{\chi} \overline{\mathbb{Q}}_\ell,$$ for some character $\chi$. It follows that for $y\not\in S$, we have $$0=\chi(\mathrm{Sat}(h_y))=\chi(t_y)+q_y \chi(t_y^{-1}).$$ Hence $\chi(t_y)^2=-q_y$. Let $\chi': \Pic_X(k)\rightarrow \overline{\mathbb{Q}_\ell}^\times$ be the character $\chi'{}=\chi\cdot q^{-\deg/2}$. Then $\chi'$ has finite image and $\chi'(t_y)^2=-1$. By class field theory $\chi'$ can be identified as a Galois character of $F$ and hence $\chi'(t_y)=1$ for infinitely many $y\in|X|$ by Chebotarev density. This contradicts $\chi'(t_y)\ne1$ for all $y\not \in |S|$. So $\Spec \tilde{\mathcal{H}}/I$ is disjoint from $Z_\mathrm{Eis}$ and hence $\tilde{\mathcal{H}}/I$ is finite dimensional (think: $I$ 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:

Lemma 2 For any nonzero ideal $I\subseteq \mathcal{H}_x$, and for any $m\ge1$, we have $$I+\mathrm{span}\{h_{nx}\}_{n\ge m}= \mathcal{H}_x.$$
Proof Let $t=q_x^{-1/2}t_x$. Then under the Satake transform we may identify $h_{nx}=q_x^{n/2}T_n$, where $T_n=t^n+t^{n-2}+\ldots +t^{-n}$. Note that $T_n-T_{n-2}=t^n+t^{-n}$, thus it suffices to show that for any nonzero ideal $I$ in the ring $\mathbb{Q}[(t+t^{-1})]$ of symmetric polynomials in $t^\pm$ , and any $m\ge1$, we have $$I+\mathrm{span}\{t^n+t^{-n}\}_{n\ge m}= \mathbb{Q}[(t+t^{-1})].$$ For any $g\in \mathbb{Q}[(t+t^{-1})]$, write $g=\sum_{k\ge 0} c_k (t^k +t^{-k})$ ($c_k\in \mathbb{Q}$). We define the lowest degree of $g$ to be the smallest $k\ge0$ such that $c_k\ne0$. Then it suffices to show that for any given $g$, we can find an element $h\in I$ with same the lowest degree as that of $g$. Since $\mathbb{Q}[(t+t^{-1})]$ is a PID, we may assume $I=(f)$ for some $f\in \mathbb{Q}[(t+t^{-1})]$. Notice that for any $l\ge0$, there is no cancellation in the product $f\cdot (t^l+t^{-l})$, and so one can always choose some $l\ge0$ such that $f\cdot (t^{l}+t^{-l})$ has a prescribed lowest degree (in particular, the lowest degree of $g$) as desired. ¡õ

Now we can finish the proof of the main identity using the previous two lemmas. For any $x\in |X|$, look at the commutative diagram $$\xymatrix{\mathcal{H}' \ar[r] &\mathcal{H} \ar[r] &\tilde{\mathcal{H}} \ar[r] & \tilde{\mathcal{H}}/I\\ \mathcal{H}'\cap \mathcal{H}_x \ar[u] \ar[r] & \mathcal{H}_x \ar[u]\ar[rr] & & \ar[u] \im(\mathcal{H}_x) }$$ Here the vertical arrows are all natural inclusions.

By Lemma 1, $\tilde{\mathcal{H}}/I$ is finite dimensional, so $\im(\mathcal{H}_x)$ is also finite dimensional. Since $\im(\mathcal{H}_x)$ is quotient of $\mathcal{H}_x$ by a nonzero ideal, and $\mathcal{H}'\cap \mathcal{H}_x=\{h_{nx}: \deg nx\ge d_0\}$, we know the bottom row is surjective by Lemma 2. Now $\{\mathcal{H}_x,x\in |X|\}$ generate $\tilde{\mathcal{H}}$ as an algebra, so the top row is also surjective. By Lemma 1, $I\subseteq\im( \mathcal{H}'\rightarrow \tilde{\mathcal{H}})$, so the map $\mathcal{H}'\rightarrow \tilde{\mathcal{H}}$ is surjective as desired. ¡õ

Last Update: 04/16/2017. Copyright © 2015 - 2017, Chao Li.

References

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