# Modular Forms and Special Cycles on Shimura Curves. (AM-161)

Stephen S. Kudla
Michael Rapoport
Tonghai Yang
Pages: 384
https://www.jstor.org/stable/j.ctt32bc0s

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Acknowledgments
(pp. ix-x)
4. Chapter One Introduction
(pp. 1-26)

In this monograph we study the arithmetic geometry of cycles on an arithmetic surface $M$ associated to a Shimura curve over the field of rational numbers and the modularity of certain generating series constructed from them. We consider two types of generating series, one for divisors and one for 0-cycles, valued in ${\widehat{{\rm{CH}}}^1}(M)$ and ${\widehat{{\rm{CH}}}^2}(M)$ , the first and second arithmetic Chow groups of $M$ , respectively. We prove that the first type is a nonholomorphic elliptic modular form of weight $\frac{3}{2}$ and that the second type is a nonholomorphic Siegel modular form of genus two and weight $\frac{3}{2}$ . In fact we...

5. Chapter Two Arithmetic intersection theory on stacks
(pp. 27-44)

The aim of the present chapter is to outline the (arithmetic) intersection theory on Deligne-Mumford (DM) stacks that will be relevant to us. The stacks $M$ we consider will satisfy the following conditions:>

$M$ is regular of dimension 2 and is proper and flat over $S{\text{ = Spec }}\mathbb{Z}$ , and is a relative complete intersection over Spec $\mathbb{Z}$ . Also we assume $M$ to be connected (and later even geometrically connected).

Let $M = {M_\mathbb{C}} = M{ \times _{{\text{Spec }}\mathbb{Z}}}{\text{Spec }}\mathbb{C}$ be the complex fiber of $M$ . Then M is given by an orbifold presentation,

$M = [\Gamma \backslash X]$ ,

where...

6. Chapter Three Cycles on Shimura curves
(pp. 45-70)

In this chapter we review the objects which will be our main concern. We introduce the moduli problem attached to a quaternion algebra $B$ over $\mathbb{Q}$ solved by the DM-stack $M$ , which has relative dimension 1 over Spec $\mathbb{Z}$ , has semistable reduction everywhere, and is smooth outside the ramification locus of $B$ . This stack has a complex uniformization by $\mathbb{C}\backslash \mathbb{R}{\text{ = }}\mathfrak{H} + \cup {\mathfrak{H}^ - }$ and, for every prime $p$ in the ramification locus of $B$ , a $p$ -adic uniformization by the Drinfeld upper half space.

We then recall from [12] the construction of classes $\widehat{{\text{Z}}}(t,\upsilon )$ in ${\widehat{{\text{CH}}}^1}(M)$ for every $t \in {\text{ }}\mathbb{Z}$ and every $\upsilon \in {\text{ }}{\mathbb{R}_{ > 0}}$ . This is...

7. Chapter Four An arithmetic theta function
(pp. 71-104)

In this chapter, we consider the generating function ${\widehat\phi _1}$ whose coefficients are the special divisors $\widehat{{\text{Z}}}(t,\upsilon ) \in {\widehat{{\text{CH}}}^1}(M)$ defined in the previous chapter. We prove that ${\widehat\phi _1}$ is a modular form of weight $\frac{3}{2}$ (Theorem A). In the first section we introduce the decomposition

(4.0.1) ${\widehat{{\text{CH}}}^1}(M){\text{ = }}\widetilde{{\text{MW}}} \oplus {\text{ }}\mathbb{R}\widehat{\text{\omega }}{\text{ }} \oplus {\text{ Vert }} \oplus {\text{ }}a({A^0}{({M_\mathbb{R}})_0})$

of ${\widehat{{\text{CH}}}^1}(M)$ into its ‘Mordell-Weil’ component, its ‘Hodge bundle’ component, its ‘vertical’ component, and its ‘ ${C^\infty }$ ’-component. We then reduce the proof of Theorem A to an assertion about the various components of ${\widehat\phi _1}$ with respect to this decomposition. The modularity of the Hodge component follows from [16]. In Section 4.3, we prove the modularity of the...

8. Chapter Five The central derivative of a genus two Eisenstein series
(pp. 105-166)

In this chapter, we study an incoherent Eisenstein series of genus two in detail and, in particular, compute its derivative at $s{\text{ = 0}}$ , the central point for the functional equation. This Eisenstein series was first introduced in [4]. In the first few sections, we consider the Fourier coefficients associated to $T \in {\text{Sy}}{{\text{m}}_2}(\mathbb{Q})$ with det $(T) \ne 0$ . These coefficients, which are given by a product of local factors, were studied in [4] and [8]. In Sections 5.4 through 5.8, we deal with the Fourier coefficients for $T$ ’s with rank $(T) = 1$ . These coefficients, which are not given as a product of local factors, are of...

9. Chapter Six The generating function for 0-cycles
(pp. 167-180)

In this chapter, we give the definition of a generating function for 0-cycles on the arithmetic surface $M$ . More precisely, we consider a generating series of the form

(6.0.1) ${\widehat\phi _2}(\tau ) = \mathop \sum \limits_{T \in {\text{Sy}}{{\text{m}}_2}(\mathbb{Z}){\text{V}}} {\text{ }}\widehat{\text{Z}}(T,\upsilon ){q^T}$ ,

where $\tau = u + i\upsilon \in {\mathfrak{H}_2}$ , the Siegel space of genus two, ${q^T} = e({\text{tr(}}T\tau {\text{)}})$ , and $\widehat{\text{Z}}(T,\upsilon ) \in {\widehat{{\text{CH}}}^2}(M)$ . Since we are working with arithmetic Chow groups with real coefficients, as explained in Chapter 2, there is an isomorphism

(6.0.2) $\widehat{{\text{deg}}}:{\widehat{{\text{CH}}}^2}(M)\tilde \to \mathbb{R}$ .

For example, if $Z$ is a 0-cycle on $M$ with $Z \simeq {\text{Spec(}}R{\text{)}}$ for an Artin ring $R$ , then

(6.0.3) $\widehat{{\text{deg}}}:(Z,0) \mapsto \log \left| R \right|$

is the usual arithmetic degree of $Z$ . The first step is to define the terms for ‘good’ positive...

10. Chapter Six: Appendix The case $p = 2,p{\text{ }}\left| {D(B)} \right.$
(pp. 181-204)

In this appendix, we extend the results of [3] concerning intersections of the special cycles on the Drinfeld space to the case $p = 2$ . We will denote by $B' = {M_2}({\mathbb{Q}_p})$ the matrix algebra over ${\mathbb{Q}_p}$ and by ${V'}$ its subspace of traceless elements. To any $j \in V'$ there is associated a special cycle on the Drinfeld space. In the appendix to section 11 of [4] the geometry of an individual special cycle $Z(j)$ is described in the case $p = 2$ . What we have to deal with, then, is the relative position of two such special cycles.

We remark that elsewhere in this book we...

11. Chapter Seven An inner product formula
(pp. 205-264)

In Chapter 4 we defined the ${\widehat{{\text{CH}}}^1}(M)$ -valued generating function ${\widehat\phi _1}(\tau ) = {\sum _{t \in \mathbb{Z}}}\widehatZ(t,\upsilon ){q^t}$ and showed its modularity. It follows that the height pairing $\langle {\widehat\phi _1}({\tau _1}),{\widehat\phi _1}({\tau _2})\rangle$ of two of these generating functions is then a series with coefficients in $\mathbb{R}$ which is a modular form of two variables, of weight $\frac{3}{2}$ in each. Here we are identifying, as in Chapter 2, ${\widehat{{\text{CH}}}^2}(M)$ with $\mathbb{R}$ via the arithmetic degree map. On the other hand, in the previous chapter, we defined a generating series for 0-cycles with coefficients in ${\widehat{{\text{CH}}}^2}(M)$ and proved that it is a Siegel modular form of genus two and weight $\frac{3}{2}$ . This was...

12. Chapter Eight On the doubling integral
(pp. 265-350)

In this chapter, we obtain some information about the doubling zeta integral for the metaplectic cover ${{G'}_A}$ of ${\text{S}}{{\text{L}}_2}(\mathbb{A})$ . In the standard doubling integral, as considered in [5], one integrates the pullback to ${\text{S}}{{\text{p}}_n}(\mathbb{A}) \times {\text{S}}{{\text{p}}_n}(\mathbb{A})$ of a Siegel Eisenstein series on ${\text{S}}{{\text{p}}_{2n}}(\mathbb{A})$ against a pair of cusp forms ${f_1}$ and ${f_2}$ on ${\text{S}}{{\text{p}}_n}(\mathbb{A})$ . If ${f_1}$ and ${f_2}$ lie in an irreducible cuspidal automorphic representation $\sigma$ , the result is a product of a partial L-function ${L^S}(s + \tfrac{1}{2},\sigma )$ of $\sigma$ associated to the standard degree $2n + 1$ representation of the L-group ${}^L{\text{S}}{{\text{P}}_n} = S{O_{2n + 1}}$ and certain ‘bad’ local zeta integrals at the places in the finite set...

13. Chapter Nine Central derivatives of L-functions
(pp. 351-370)

In this chapter, we first use the Borcherds generating function, ${\phi _{{\text{Bor}}}}(\tau ,\varphi )$ , to define an arithmetic analogue of the classical Shimura-Waldspurger lift described in Section 8.2. We show that this lift, whose target is the Mordell-Weil group of a Shimura curve over $\mathbb{Q}$ , is compatible with the local theta correspondence, and hence there are local obstructions to nonvanishing, just as in the classical case. We then formulate a conjectural analogue of the result of Waldspurger, Theorem 8.2.5, and characterize the nonvanishing of the arithmetic theta lift in terms of theta dichotomy (local obstructions) and the nonvanishing of the central derivative $L'(\frac{1}{2},{\text{Wald(}}\sigma {\text{,}}\psi {\text{)}})$ ....

14. Index
(pp. 371-373)