# The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan
Shou-Wu Zhang
Wei Zhang
Pages: 272
https://www.jstor.org/stable/j.cttq94r3

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-x)
4. Chapter One Introduction and Statement of Main Results
(pp. 1-27)

In this chapter, we will state the main result (Theorem 1.2) of this book and describe the main idea of our proof. Let us start with the original work of Gross and Zagier.

LetNbe a positive integer and$f\; \in \;{S_2}({\Gamma _0}(N))$a newform of weight 2. Let$K\; \subset \;\mathbb{C}$be an imaginary quadratic field and$\chi$a character of${\text{Pic(}}{O_K})$. Form the L-series$L(f,\;\chi ,\;s)$as the Rankin–Selberg convolution of the L-series$L(f,\;s)$and the L-series$L(\chi ,\;s).$This L-series$L(f,\;\chi ,\;s)$has a holomorphic continuation to the whole complex plane and satisfies a functional equation relatingsto$2 - s$.

Assume thatK...

5. Chapter Two Weil Representation and Waldspurger Formula
(pp. 28-57)

In this chapter, we will review the theory of Weil representation and its applications to an integral representation of the Rankin-Selberg L-function$L(s,\;\pi ,\;\chi )$and to a proof of Waldspurger’s central value formula. We will mostly follow Waldspurger’s treatment with some modifications including Kudla’s construction of incoherent Eisenstein series.

We will start with the classical theory of Weil representation of$O(F)\; \times \;{\text{S}}{{\text{L}}_2}(F)$on$\mathcal{S}(V)$for an orthogonal spaceVover a local fieldFand its extension to${\text{GO(}}F)\; \times \;{\text{G}}{{\text{L}}_2}(F)$on$\mathcal{S}(V\; \times \;{F^ \times })$by Waldspurger. We then define theta functions, state the Siegel–Weil formula, and define normalized local Shimizu lifting. The main result...

6. Chapter Three Mordell–Weil Groups and Generating Series
(pp. 58-105)

The major goal of this chapter is to introduce Theorem 3.21, an identity between the analytic kernel and the geometric kernel, and describe how it is equivalent to Theorem 1.2. We first define the generating series, and then use it to define the geometric kernel. The analytic kernel is the same as that in the Waldspurger formula, except that we take derivative here. As a bridge between these two theorems, we also introduce Theorem 3.15, an identity formulated in terms of projectors.

In §3.1, we review some basic notations and results on Shimura curves.

In §3.2, we will review the...

7. Chapter Four Trace of the Generating Series
(pp. 106-170)

The goal of this chapter is to prove Theorem 3.17 and Theorem 3.22 in the last chapter.

Before going to the proofs, in §4.1 we give more details on the new space$\overline \mathcal{S} (V \times {\mathbb{R}^ \times })$of Schwartz functions including the formation of theta series and Eisenstein series by them.

Theorem 3.17 asserts the modularity of the generating series. The major part of its proof is in §4.2, where we reduce the problem to the results in [YZZ]. In this way, the modularity is proved on the open Shimura variety. To extend to the compactification, it suffices to prove the degree (as a...

8. Chapter Five Assumptions on the Schwartz Function
(pp. 171-183)

In this chapter and the rest of this book, we assume all the geometric assumptions in §3.6.5.

In this chapter, we impose some assumptions on the Schwartz function$\phi \: \in \:\overline \mathcal{S} (\mathbb{V}\: \times \:{\mathbb{A}^ \times })$, which we will keep from the rest of this book. These assumptions greatly simplify the computations, but imply the kernel identity for all$\phi$.

In §5.1, we restate the kernel identity in terms of un-normalized kernel functions$Z{(g,\;\phi ,\;\chi )_U}$and$I'{(0,\;g,\;\phi ,\;\chi )_U}$. It depends onU, but we always fix aUfrom now on. The rest of this book is to work on this version.

In §5.2, we state the assumptions. It...

9. Chapter Six Derivative of the Analytic Kernel
(pp. 184-205)

Let$\phi \: = \:{\phi _f}\: \otimes \:{\phi _\infty }\: \in \:\overline \mathcal{S} (\mathbb{V}\: \times \:{\mathbb{A}^ \times })$be a Schwartz function. Assume that the archimedean part${\phi _\infty }$is standard, and that the finite part${\phi _f}$is invariant under the action of$K\; = \;U\; \times \;U$for some open compact subgroupUof$\mathbb{B}_f^ \times$.

Recall that in §5.1 we have introduced the series

$I{(s,\;g,\;\chi ,\;\phi )_U} = \int_{T(F)\backslash T(\mathbb{A})/Z(\mathbb{A})}^* {I{{(s,\;g,\;r(t,\;1)\phi )}_U}\;\chi (t)\,dt,}$

where

$I{(s,\;g,\;\phi )_U} = \sum\limits_{u \in \mu _U^2\backslash {F^ \times }} {\sum\limits_{\gamma \in {P^1}(F)\backslash {\text{S}}{{\text{L}}_2}(F)} {\delta {{(\gamma g)}^s}} } \sum\limits_{{x_1}\; \in \;E} {r(\gamma g)} \phi ({x_1},\;u)$.

In this chapter, we compute the derivative$I'{(0,\;g,\;\chi ,\;\phi )_U}$and its holomorphic projection$\mathcal{P}r\,I'{(0,\;g,\;\chi ,\;\phi )_U}$. We assume all the assumptions in §5.2, which significantly simplify the results. The main content of this section is various local formulae. We usually fixUand abbreviate$I{(s,\;g,\;\phi )_U}$and$I{(s,\;g,\;\chi ,\;\phi )_U}$as$I(s,\;g,\;\phi )$and$I(s,\;g,\;\chi ,\;\phi )$, or even as$I(s,\;g)$and...

10. Chapter Seven Decomposition of the Geometric Kernel
(pp. 206-229)

Let$\phi {\text{ }} = {\text{ }}{\phi _f} \otimes {\phi _\infty } \in {\text{ }}\overline \mathcal{S} {(\mathbb{V}\, \times {\text{ }}\mathbb{A}{\mathbb{Q}^ \times })^{U \times {\text{ }}U}}$be a Schwartz function with standard${\phi _\infty }.$Assume that$- 1\; \notin \;U$to simply notations. Recall that in §5.1 we have introduced the generating series

$Z{(g,\;\phi )_U} = {Z_0}{(g,\;\phi )_U} + {Z_*}{(g,\;\phi )_U},\quad \;g\; \in \;{\text{G}}{{\text{L}}_2}(\mathbb{A})$.

Here the non-constant part

${Z_*}{(g,\;\phi )_U} = \sum\limits_{a \in {F^ \times }} {\sum\limits_{x \in K\backslash \mathbb{B}_f^ \times } {r(g)\phi {{(x)}_a}} } \;Z{(x)_U}$.

We further have the height series as follows:

$Z{(g,\;({h_1},\;{h_2}),\;\phi )_U} = \;{\langle Z{(g,\;\phi )_U}\;[{h_1}]_U^ \circ ,\;[{h_2}]_U^ \circ \rangle _{{\text{NT}}}},{\text{ }}{h_1},\;{h_2}\; \in \;{\mathbb{B}^ \times }$;

$Z{(g,\;\chi ,\;\phi )_U} = \;\int_{T(F)\backslash T(\mathbb{A})/Z(\mathbb{A})}^* {Z{{(g,\;(t,\;1),\;\phi )}_U}\;\chi (t)\,dt}$.

By Lemma 3.19,$Z{(g,\;({h_1},\;{h_2}),\;\phi )_U}$is cuspidal ing. So we can replace$Z{(g,\;\phi )_U}$by${Z_*}{(g,\;\phi )_U}$in the definition of$Z{(g,\;({h_1},\;{h_2}),\;\phi )_U}$. The constant term${Z_0}{(g,\;\phi )_U}$will be ignored in the rest of this book.

The goal of this chapter is to decompose the height series

$Z{(g,\;({t_1},\;{t_2}),\;\phi )_U} = \;{\langle {Z_*}{(g,\;\phi )_U}\;[{t_1}]_U^ \circ ,\;\;[{t_2}]_U^ \circ \rangle _{{\text{NT}}}},\quad \;{t_1},\;{t_2}\; \in \;{\mathbb{B}^ \times }$.

We presume the assumptions in §5.2. Then there is not horizontal self intersection in...

11. Chapter Eight Local Heights of CM Points
(pp. 230-250)

The goal of this chapter is to prove Theorem 7.8, namely, to compute the local heights and compare them with the derivatives computed before. We check the theorem place by place. We assume all the assumptions in §5.2 throughout this chapter.

According to the reduction of the Shimura curve, we divide the situation to the following four cases:

archimedean case:$\upsilon$is archimedean;

supersingular case:$\upsilon$isnonsplit inEbut split in$\mathbb{B}$;

superspecial case:$\upsilon$isnonsplit in bothEand$\mathbb{B}$;

ordinary case:$\upsilon$is split in bothEand$\mathbb{B}$.

The treatments in different cases are similar in...

12. Bibliography
(pp. 251-254)
13. Index
(pp. 255-256)