




Arithmetic Algebraic Geometry / Géométrie algébrique
arithmétique (Kumar Murty and P. Sastry, Organizers)
 D. ARAPURA, Purdue University
Hodge cycles on some moduli spaces

I want to give some criteria for the validity of the Hodge conjecture
for certain moduli spaces of ``vector bundles'' (actually torsion free
sheaves) on curves and surfaces. In the case of curves, the validity of
the conjecture for the Jacobian and its powers implies the validity for
moduli of higher rank vector bundles. In particular, it holds for
generic curves, Fermat curves of prime degree... If time permits,
I'd like to say something about the Tate conjectures as well.
 B. CONRAD, Department of Mathematics, University of Michigan,
Ann Arbor, Michigan 481091109, USA
J_{1}(p) has connected fibers

We prove that for every prime p, the minimal regular proper model of
X_{1}(p) has geometrically integral closed fiber. Consequently, the
component group of J_{1}(p)mod p is trivial. As an application, we
can compute the minimal regular proper model of X_{H}(p) = X_{1}(p)/H for
any subgroup H of (Z/p)^{×}, and the component group of its
Néron model mod p. For example, the map J_{H}(p) ®J_{0}(p) is injective on component groups.
 H. DARMON, McGill
To be announced

 JORDAN ELLENBERG, Princeton University, Princeton, New Jersey, USA
Prop fundamental groups and rational points

A famous circle of conjectures suggests that much of the arithmetic
geometry of a variety X over a number field K is encoded in the
action of the Galois group of K on the etale fundamental group of
X. We discuss some recent results which use the fundamental group of
an algebraic curve C to control rational points on C.
 E. GOREN, Department of Mathematics and Statistics, McGill University,
Montreal, Quebec H3A 2K6
Hilbert modular varieties of low dimension

I shall discuss Hilbert modular surfaces and threefolds in positive
characteristic and their geometry. We use three main tools: local
models, displays and Hecke correspondences at p.
 ERNST KANI, Queen's University, Kingston, Ontario K7L 3N6
Hurwitz spaces of genus 2 covers of elliptic curves

Let E/K be an elliptic curve, where K is a field of characteristic
¹ 2. Then the set of (normalized) genus 2 covers f: C® E of E of degree N ³ 3 can be parametrized by smooth
curve H_{E/K,N}/K which is an open subset of a twist of the usual
modular curve X(N) parametrizing curves with levelNstructure (of
fixed determinant).
By studying the degeneration of the universal cover, one can count the
total number of genus 2 covers of E/K with fixed discriminant (when
K is algebraically closed). This uses the methods of arithmetic
geometry: Neron models, minimal models, modular heights and
intersection theory.
In addition, one can use the above moduli spaces to study the relation
between (certain) modular diagonal quotient surfaces and the Humbert
surfaces which are divisors on M_{2}, the moduli space of genus 2
curves.
 H. KISILEVSKY, Concordia
To be announced

 J. LEWIS, Department of Mathematics, University of Alberta, Edmonton,
Alberta T6G 2G1
K_{2} of curves and the MumfordManin conjecture

We give a formula for a real regulator of the Kgroups of a
projective algebraic manifold, and apply this to K_{2} of curves. We
explain an application of this to points of finite order on a curve.
 YURU LIU, Harvard University, USA
Generalizations of the Turán Theorem and the ErdösKac Theorem

Let m Î N and w(m) the number of distinct prime
divisors of m. The Turán Theorem states

å
m £ x


æ è

w(m)  loglogx 
ö ø

2

<< x loglog x, 

which implies the normal order of w(m) is loglogm. Erdös and
Kac give a refinement of this result. For n Î N, they prove
P_{n} 
ì í
î

m: 
w(m)loglogm

£ y 
ü ý
þ

= 
ó õ

y
¥

e^{[(t2)/2]} dt, 

where P_{n} is the probability measure that takes place mass 1/n at
each m Î N, m £ n. The underlying ideas of these
theorems form the foundations of probabilistic number theory. Indeed,
the setting of these theorems can be generalized. The main purpose of
this talk is to axiomatize the main properties in order to apply the
results in a more general context. We will see applications in the
cases of number fields, polynomials over a finite field, and smooth
projective varieties over a finite field.
 R. MURTY, Queen's University
Effective version of Serre's theorem

Let E be an elliptic curve over Q without complex multiplication.
For all primes q sufficiently large, Serre proved that the field
obtained by adjoing the qdivision points to the rationals has Galois
group isomorphic to GL_{2}((F_{q}) where F_{q} is the finite field of
q elements. We will determine an effective lower bound for how large
q must be. (This is joint work with Alina Carmen Cojocaru.)
 YI OUYANG, University of Toronto, 100 St George St., Toronto,
Ontario M5S 3G3
On the universal norm distribution

We define the universal norm distribution in this talk, which is the
generalization of the universal ordinary distribution and the universal
Euler system introduced by K. Rubin. Furthermore, a Koszul type
resolution (Anderson's resolution) is given to study the universal norm
distribution. By this way, we use spectral sequence method to study the
group cohomology of the universal norm distribution. Namely, let X
be a totally ordered set and let Z be the formal product of X, for
any z Î Z, there is an abelian group G_{z} attached to z. The
universal norm distribution N_{z} is a G_{z}module
satisfying a certain distribution. When we assign properties to X and
G_{z}, N_{z} will have abundant arithmetic structures.
 IVAN SOPROUNOV, University of Toronto
Residues and tame symbols in toric geometry

We construct an analog of Parshin's theory for toroidal varieties. It
turns out to be more explicit than the general theory of Parshin, and
is enriched with the combinatorics inherited from toroidal varieties.
This approach gives a uniform explanation of recent results of
Khovanskii and Gelfond on systems of algebraic equation in
(C^{×})^{n} in terms of symbols and residues on toroidal
varieties, and extend these results to the case of an arbitrary
algebraically closed field.
 Y. ZARHIN, Department of Mathematics, Penn State University,
University Park, Pennsylvania 16802, USA
Jacobians with prescribed cyclotomic endomorphism ring

Let p be an odd prime, K a field of characteristic different from
p, K_{a} its algebraic closure, f(x) a polynomial of degree n > 4
with coefficients in K and without multiple roots, C a smooth
projective model of the affine curve y^{p}=f(x) and J the jacobian.
It is known (Poonen, Schaefer) that the ring End(J) of
K_{a}endomorphisms of J contains a certain subring isomorphic to
the ring of integers Z[m_{p}] in the pth cyclotomic field
Q[m_{p}].
Theorem. Suppose the Galois group of f over K is either the full symmetric group S_{n} or the alternating group A_{n}. Then:
1) If K has characteristic zero then End(J)=Z[m_{p}].
2) If n > 9 and p divides n then Z[m_{p}] is a maximal commutative subring in End(J).

