Workshop on:
Intersection of
Arithmetic Cycles and Automorphic Forms
December 12-16, 2005.
Organizers: Eyal Goren and Henri Darmon (McGill).
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Abstracts |
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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 |
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Arithmetic
intersection theory and Hilbert modular surfaces |
We report on joint work
with J.~ |
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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. |
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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.) |
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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.) |
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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 $\mathcal
M$. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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J. |
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. |
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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. |
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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. |
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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 $\textnormal{Tr}(p^{2n} d) \equiv 0 \pmod{p^n}$. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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 $F_p^\times$. |
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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. |
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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] |
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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. |