Workshop on:
Intersection of Arithmetic Cycles and Automorphic Forms
December 12-16, 2005.

Organizers: Eyal Goren and Henri Darmon (McGill).



J. Bruinier

Borcherds products and Green functions

This lecture gives an introduction to the theory of Borcherds products on Shimura varieties of type $\operatorname{O}(2,n)$.  Borcherds products are particular meromorphic modular forms which have a striking infinite product expansion. They are obtained from elliptic modular forms by means of a regularized theta lifting. We explain some of their geometric and arithmetic properties. We show how the lifting can be generalized to lift weak Maass forms to certain automorphic Green functions for Heegner divisors. We discuss some fundamental properties of these Green functions, for instance, their behavior at the boundary. It is shown that they define Green objects in the extended arithmetic intersection theory due to Burgos, Kramer and K\"uhn. Moreover, the relationship to the Green functions constructed by Kudla is briefly discussed.

Arithmetic intersection theory and Hilbert modular surfaces

We report on joint work with J.~Burgos and U.~K\"uhn. We consider certain arithmetic Hirzebruch-Zagier divisors on Hilbert modular surfaces. The properties of Borcherds products and automorphic Green functions of the previous talk are used to show that the generating series of these Hirzebruch-Zagier divisors is an elliptic modular form of weight $2$ with values in an arithmetic Chow group. Moreover, we consider the intersection of the generating series with the line bundle of modular forms equipped with the Petersson metric. In particular, the arithmetic self intersection number of the line bundle of modular forms is determined. These results confirm conjectures of Kudla, Maillot and Roessler in the special case of Hilbert modular surfaces.

CM values of Hilbert modular functions

Gross and Zagier found an explicit formula for the values of the $j$-function at CM points as a special case of their famous work on the Gross-Zagier formula. We report on joint work with T.~Yang, in which we extend some of their arguments to obtain exact formulas for the CM-values of rational functions on Hilbert modular surfaces associated to certain non-biquadratic CM fields. If time permits, we also discuss a natural family of rational functions on a Hilbert modular surface which is obtained by a ``twisted'' Borcherds lifting. For instance, on any Hilbert modular surface, there is a rational function, defined over the underlying real quadratic field, which is the lifting of the classical $j$-invariant.

B. Conrad

Deformation theory and local intersection numbers

In this talk and the sequel I wish to explain to a "general" audience some of the basic ideas and arguments underlining a purely deformation-theoretic approach to the arithmetic intersection theory calculations in the original work of Gross and Zagier.  Roughly speaking, the goal is to eliminate all appeals to the "numerology" of modular curves (e.g., uniform arguments for all j-values and all CM fields) and to provide a conceptual interpretation for various formulas.

To this end, in the first talk I will first explain some basic background in deformation theory on moduli spaces, especially the Serre-Tate lifting theorem and complete local rings on modular varieties.  These notions are applied to compute some intersection numbers among Heegner cycles on some modular curves in terms of isomorphism groups between pairs of elliptic curves.  (This corresponds to Theorems 4.1, 5.1, and 6.4 in my paper in the MSRI "Gross-Zagier" book.)

Deformation theory and self-intersection numbers

We explain the conceptual relationship between deformation rings in characteristic 0 and characteristic p, and combine this with Kodaira-Spencer theory and some group-scheme arguments to compute some local self-intersection numbers for Heegner points on modular curves.  This gives the local terms whose sum is the non-archimedean contribution to the global height-pairing formula of Gross and Zagier. (This roughly corresponds to Lemma 9.1, Theorem 9.2, and Lemma 10.1 in my paper in the MSRI proceedings.)

S. Kudla

Generating series for arithmetic cycles


In the first two lectures, I will report on some of the results contained in the forthcoming book,  Modular Forms and Special Cycles on Shimura Curves, joint with M. Rapoport and T. Yang.

In the first lecture I will describe the generating function for divisors on the arithmetic surface $\mathcal M$ associated to a Shimura curve $M$ over $\Q$.  The fact that this series is the $q$--expansion of a modular form of weight $3/2$ is proved by analyzing its components in various subspaces of the arithmetic Chow group $\CH^1(\mathcal M)$. For example, the modularity of the Mordell-Weil component is derived using Borcherds results on the generators and relations for certain spaces of modular forms and his construction of meromorphic modular forms on $\mathcal M$.

The modular generating function for 0-cycles on an arithmetic surface

One of the main results of [KRY] is the identification of a certain generating series for $0$--cycles on the arithmetic surface $\mathcal M$ associated to a Shimura curve with the central derivative of a Siegel-Eisenstein series of genus $2$. I will try to explain what goes into the proof of  this result:

(i) non-singular coefficients

(ii) rank 1 coefficients

(iii) the constant term

If there is time, I will discuss the `arithmetic inner product formula' which relates the height pairing of two generating functions for divisors to the restriction of the generating function for $0$--cycles.

Special values of Borcherds forms and integrals of Green functions

In this lecture, I will discuss some results of J. Schofer's thesis on the CM values of Borcherds forms. In the case of Hilbert modular surfaces, these results complement those of Bruinier and Yang, explained in the third lecture of Bruinier.  I will then explain how a similar technique can be used to relate the integrals of Green functions to certain terms in the derivatives of genus $1$ Eisenstein series.

S.-W. Zhang

Generic Abelian Varieties are not Jacobians

I will explain a proof of the following statement:

any Hilbert modular  variety of dimension g>4 can't be included in the modular space of curves of genus g. Combined with my work on equidistribution on CM-points, this result implies the finiteness of certain curves with CM Jacobians. This is a report of a joint work in progress with Johan de Jong.

Period integrals and special values of L-functions

Let G be a classical group and H a reductive subgroup of "Stiefel type". The purpose of this talk is to outline a procedure to evaluate period integrals over H of cusp forms on G. The result is roughly expressed as a special value of a standard L-function. This is a report of joint work in progress with Jian-Shu Li.

J. Cogdell

L-functions, modularity, and functoriality

(Colloquium Talk)

There is a very interesting, and still very mysterious, complex analytic invariant attached to an arithmetic object -- its L-function. (The Riemann zeta function is an example.) There is also a family of more analytic objects that have similar complex analytic invariants -- modular forms or automorphic forms. In this talk I would like to discuss both arithmetic and automorphic L-functions. I will pay particular attention to the the nature of the ``Converse Theorem for GL(n)'', which morally says: any object with a nice L-function should be modular. I will explain how this leads naturally to both Langlands' conjectures on the modularity of Galois representations and Langlands' Functoriality conjecture. Finally I will discuss the Converse Theorem as a practical tool for establishing functoriality, concentrating on the cases of the lifting of automorphic forms from the classical groups to GL(n). The hope is that the talk will be expository, self contained, and understandable to a general mathematical audience.




B. Howard

Interpolation of Heegner points in Hida families

I will describe the construction of a family of big Heegner points for Hida's universal ordinary deformation of a modular Galois representation, and explain how this construction suggests two possible generalizations of the Gross-Zagier theorem: one to the Iwasawa theory of Hida families and one to modular forms of nontrivial nebentype.

K. Bringmann

Arithmetic properties of coefficients of Maass Poincare series of half-integral weight.

We generalize a result of Zagier, describing the duality of weakly holomorphic modular forms of weight 1/2 and 3/2. We show that Zagier's result is a special case of a generic duality for general weight. For this we consider Maass Poincare series and conclude the duality by comparing Fourier coefficients. Moreover we write these coefficients as traces of singular moduli.  As a plus we obtain exact formulas for these traces. This is joint work with Ken Ono.

J. Burgos Gil

Mini course on Arakelov Geometry II

The Arakelov theory developed by Gillet and Sou\'e is well suited to work with smooth metrics. However, the hermitian metrics that appear naturally when studying modular varieties are not smooth at the boundary. For instance the Petersson metric of the line bundle of modular forms has logarithmic singularities along the cusps. Analogously, the line bundle that defines the Faltings' modular height are  log-singular near the boundary of the modular variety. Higher rank examples can be found in the proof by Mumford of the Hirzebruch  proportionality principle in the non compact case. Nevertheless, in all  these examples, the singularities that appear are so mild that  they share many properties of smooth metrics.   

In this lecture we will review these examples and we will show how to extend the formalism of Arakelov geometry in order to be able to study with log-singular hermitian metrics. This is a particular case of the cohomological arithmetic Chow groups introduced in the previous lecture.    

J. Funke

A singular theta lift and the construction of Green currents for unitary groups

Borcherds introduced a singular theta lift from SL_2 to O(n,2), which on one hand gave rise to remarkable product expansions of automorphic forms for O(n,2), and on the other hand was used by Bruinier to construct Green functions for "special" divisors on the associated locally symmetric space. In previous work jointly with Bruinier, we established a duality statement between the Borcherds lift and a theta lift introduced by Kudla and Millson. From this result, we can also deduce a relationship between Bruinier's Green function and the one constructed by Kudla in the context of his joint program with Rapoport and Yang on generating series in arithmetic geometry. In this talk, we discuss these ideas and present results of work in progress on the extension to unitary groups.

J. Getz

Hilbert modular generating functions with coefficients in intersection homology

In a famous paper, Hirzebruch and Zagier considered families of homology classes $\{Z_m\}_{m \in \ZZ_{\geq 0}}$ on certain Hilbert modular surfaces and showed that the generating series $\sum_{n=0}^{\infty} Z_m \cdot Z_n q^n$ are elliptic modular forms with nebentypus.  This work can be seen as giving a geometric interpretation of the Doi-Naganuma lifting.

We prove the modularity of analogous generating series in the context of intersection homology classes on the product of two Hilbert modular varieties of arbitrary dimension or a single Hilbert modular variety of arbitrary dimension.  The later case is a work in progress with M. Goresky.

P. Jenkins

p-adic properties of traces of singular moduli

Zagier initiated the study of traces of singular moduli $\textnormal{Tr}(d)$ and their generalizations as coefficients of certain weakly holomorphic half integral weight modular forms.  We discuss the $p$-adic properties of these traces and consequent congruences.  In the case where $p$ splits in $\mathbb{Q}(\sqrt{-d})$, we recover Edixhoven's observation that $\textnormal{Tr}(p^{2n} d) \equiv 0 \pmod{p^n}$.

J. Kramer

Mini course on Arakelov Geometry I

We start by recalling the definition of the (naive) Weil height for rational points in projective space measuring the arithmetic complexity of these points. Then, we give a survey on the theory of cohomological arithmetic Chow groups developed jointly with J. Burgos and U. K\"uhn. As a special case, we will recover Arakelov Geometry as developed by H. Gillet and C. Soul\'e. In particular we find an interpretation of the classical Weil height by means of arithmetic intersections. This, in turn, gives rise to a definition of a height for higher dimensional cycles.

U. Kuehn

Mini course on Arakelov Geometry III

We present calculations of arithmetic intersection numbers on Hilbert modular surfaces by means of the cohomological arithmetic Chow rings with pre-log-log forms. These rely on the interplay of Borcherds products and arithmetic intersection theory.  We show how to transform in certain cases the abstract formula for the star product of Green objects associated with Hirzebruch-Zagier divisors into quantities we can actually calculate. To complete our calculations at the finite places the arithmetic properties of the  input space for the Borcherds lift to Hilbert modular forms are used  in a vital way.

K. Lauter

Intersection of CM points with the reducible locus on the Siegel moduli space

Given a primitive quartic CM field, $K$, one can study the values at CM points associated to $K$ of certain Siegel modular functions studied by Igusa.  The values are algebraic numbers which generate unramified abelian extensions of the reflex field of $K$.  When computing their minimal polynomials over ${\bf Q}$, rational primes in the denominators of the coefficients correspond to primes where at least one of the abelian varieties with CM by $K$ reduces to a product of supersingular elliptic curves with the product polarization.  We call such primes \emph{evil primes for} $K$.   In joint work with Eyal Goren, we showed that for fixed $K$, such primes are bounded by a quantity related to the discriminant of the field $K$.  As a consequence, we showed that certain analogues of elliptic units defined by DeShalit-Goren were actually $S$-units for an explicit set $S$.

In some sense, there are few evil primes, since if we fix $K$, then there are a finite number of evil primes for $K$.  But in subsequent work with Goren, we show that evil primes are ubiquitous in the sense that, for any rational prime $p$, there are an infinite number of quartic CM fields $K$ for which $p$ is evil for $K$.  This generalizes recent work of Elkies, Ono, and Yang.

J. Rouse

Traces of Singular Moduli on Hilbert Modular Surfaces

Suppose that $p \equiv 1 \pmod{4}$ is prime and let $K = Q(\sqrt{p})$. Hirzebruch and Zagier proved that generating functions for the intersection numbers of Hirzebruch-Zagier divisors on the Hilbert modular surface $(H \times H)/SL_{2}(O_K)$ are weight 2 modular forms. Using work of Bruinier and Funke, we show that generating functions for traces of singular moduli over these intersections are weakly holomorphic weight 2 modular forms. For the singular moduli of $j(z) - 744$ we compute these generating functions explicitly, and factorize their ``norms'' as products of Hilbert class polynomials.

R. Livne

Examples of universal Kummer families over Shimura curves

Shimura curves over Q parameterize certain abelian surfaces, but it is hard to describe the universal families explicitly. In joint work with A. Besser we describe explicitly the associated Kummer families in some examples as elliptic fibration.

A. Raghuram

Special values of symmetric power L-functions.

The first half of the talk will be an introduction to Deligne's conjectures on the special values of symmetric power L-functions associated to a holomorphic cuspform. The latter part of the talk will be a report on some work in progress, which is joint work with Freydoon Shahidi, on the special values of the fourth symmetric power L-functions.

B. Klingler

On the Andre-Oort conjecture

This is joint work with A. Yafaev . We will indicate the proof of the following

Theorem: Let $Sh_{K}(G, X)$ be a Shimura variety and let $S$ be a set of special subvarieties in $Sh_{K}(G,X)(C)$. We make one of the following assumptions:

1. assume the Generalized Riemann Hypothesis for CM fields or

2. assume there is a faithful representation V of G such that the corresponding generic Hodge structures attached to the special subvarieties in $S$ lie in one isomorphism class.

Then every irreducible component of the Zariski-closure of $S$ in $Sh_{K}(G,X)(C)$ is a special subvariety.

In the special case where S is a set of special points the theorem above proves the Andre-Oort conjecture under one of the two assumptions.

S. Dasgupta

Shintani zeta-functions and Gross-Stark units for totally real fields


Let $F$ be a totally real number field and let $p$ be a finite prime of $F$, such that $p$ splits completely in the finite abelian extension $H$ of $F$. Stark has proposed a conjecture stating the existence of a $p$-unit in $F$ with absolute values at the places above $p$ specified in terms of the values at zero of the partial zeta functions associated to $H/F$.  Gross proposed a refinement of Stark's conjecture which gives a conjectural formula for the image of Stark's unit in $F_p^\times/E$, where $F_p$ denotes the completion of $F$ at $p$ and $E$ denotes the topological closure of the group of totally positive units of $F$.  We propose a further refinement of Gross' conjecture by proposing a conjectural formula for the exact value of Stark's unit in $F_p^\times$.

D. Helm

A Deligne-Rapoport model for U(2) Shimura varieties

The Deligne-Rapoport model of the reduction of a modular curve at a prime with "\Gamma_0(p)"-level structure is a key tool in the study of the arithmetic of modular forms.  We construct an analogous model for U(2) Shimura varieties, a particular class of higher-dimensional analogues of modular curves.  As a consequence of this model, we are able to give a completely geometric proof of certain cases of the Jacquet-Langlands correspondence.

C. Consani

Noncommutative geometry and motives

In the recent years noncommutative geometry, number-theory and the theory of motives have come to an interesting cross-point. NCG gives through the noncommutative space of adele classes a set-up in which Weil explicit formulas acquire a geometric meaning.

In the talk I will describe a cohomological interpretation of the spectral realization of the zeroes of the zeta-function as the cyclic homology of a `suitable' non-commutative space. A central role in this construction is played by few ideas that naturally lead to the definition of a theory of noncommutative motives. [joint work with A. Connes and M. Marcolli]

H. Darmon

A Gross-Kohnen-Zagier Theorem for Stark-Heegner points

This is a preliminary report on work in progress with Gonzalo Tornaria.

I will discuss relations between ``Stark-Heegner points" and the fourier coefficients of modular forms of weight 3/2, in the spirit of the formula of Gross, Kohnen and Zagier for classical Heegner points. The proof suggests a connection between Hida's theory of p-adic families of modular forms and the ``arithmetic theta-lifts" arising in the work of Kudla and others which are the main focus of this workshop.