Titles and
abstracts
Talks
Sara Arias (Universidad de
Barcelona)
Title: Galois representation attached to torsion
points of an abelian variety
Let A be an abelian variety of dimension p defined over a number field
K. The Galois group G of the algebraic closure of K over K, acts on the
torsion points of A, defining, for each prime number p, a
representation R_p from G to Aut(A[p])~GL_2n (Fp ).
When A is an elliptic curve without complex multiplication, the images
of these representations were considered by Serre, who proved that they
are surjective for almost every prime. Later, Serre studies the images
of these representations when A is an abelian variety of dimension n
such that End(A) = Z. He proves that, under certain hypotheses, for
almost every prime p the image of p is the symplectic group GS_p2n
(F_p), using some results of Faltings which are not effective.
When A is an abelian surface, some results have been obtained by
Le Duff and by Dieulefait concerning the determination of primes such
that the image of p is GS_p4 (F_p). In this talk we will discuss some
of these results.
Prof. Antonio Campillo
(Universidad de Valladolid)
Title: Zeta Functions and Poincaré Series for
Local Algebra
Recently a natural concept of Poincaré series was introduced for
multifiltrations on local rings. They allow to recover some data from
the topology of several kind of singularities, in particular the zeta
function of the monodromy is related to the corresponding
Poincaré series. Poincaré series with respect to valuative
multifiltrations are bringing to local algebra, in this way, new
arithmetical methods and thinking which becomes a useful complement of
standard scheme theoretical geometrical tools.
Prof. Graham Everest
(University of East Anglia)
Title: Primitive Divisors in Integral Sequences
Classically, questions have been asked about prime values of terms in
naturally occurring integer sequences. Sequences such as Fibonacci and
Mersenne have proved to be particulary difficult to study in this
respect. In the late 19th century, Zsigmondy and Bang weakened the
sense in which sequences produce primes. They called a prime divisor of
a term of an integer sequence a primitive divisor if has not appeared
as a factor of any earlier term and showed that many interesting
sequences have primitive divisors for all terms from some point on. In
my talk I will recall some of the history of these problems as well as
look at very recent developments, including sequences which arise from
elliptic curves.
Title: Prime and Pure Power Elements of
Diophantine Sets
In a natural way, this talk follows the first one. I will concentrate
mainly upon sets of integers associated to curves of genus 1. Siegel's
Theorem is a key result in the field as is the S-unit Theorem in its
many forms. Baker's Theorem too made a huge impact because it allowed
effective results. I will recall some of the history about the way
these ideas are connected then go on to explain some recent results
which invoke Faltings' Theorem to produce finiteness results about pure
power terms. I will also explain how questions about primality can
sometimes be resolved in genus 1 - and give answers in marked contrast
to the classical cases, thus bringing the end of the second talk round
to the beginning of the first talk.
Irene García-Selfa (Universidad de Sevilla)
Title: Diophantine characterization of torsion
structures on elliptic curves
On this talk we will explain how, given an elliptic curve in short
Weierstrass form, we can determine its torsion structure by the
(non-)existence of solutions for a system of two quasi-homogeneous
diophantine equations. This result generalizes previous work of K. Ono,
D. Qiu and X. Zhang and it might be useful for tackling other problems;
as we will point out.
Dr. Enrique Gozález
(Universidad Autónoma de Madrid)
Title: On the modularity of some jacobian
surfaces and their quotient Q-curves
We consider four families of genus 2 curves defined over the rationals.
We compute their endomorhism algebras and with these results we show
when their jacobians are of GL_2-type and without CM. In this case our
four families split like the square of a Q-curve over a number field.
We have studied when these abelian surfaces are modular. In the modular
case we have computed the corresponding newform for several cases. That
is, level and nebentype of modularity (and the corresponding label).
Finally, we have made some conjectures about the conductor of the
modular Q-curves and the corresponding level.
Prof. Bartolomé López
(Universidad de Cádiz)
Title: Some problems on distinguished Drinfeld
modular forms
The basic results in the theory of Drinfeld modular forms are similar
to those on classical modular forms. However, an interesting part of
that theory, the Fourier expansions of Drinfeld modular forms, seems to
behave quite differently from the classical case. I will present this
part of the Drinfeld theory comparing it with the classical theory.
In particular, one of the facts of the classical theory is that Hecke
action determines eigenforms. To study whether this fact is true or not
for the Drinfeld case seems to be a difficult problem, but it may be
simplified by considering individual eigenforms. The examples that we
have study are the discriminant function and the Eisenstein series of
weight $q-1$; in these cases, we could not give a positive answer to
the previous problem, but at least we were able to propose some
conjectures that would imply that these two eigenforms are determined
by the action of Hecke operators.
Alberto Mínguez (École
Normale Supérieure, Paris)
Title: Howe correspondence
The aim of this talk is to introduce the audience to the theory of
local Howe correspondence. We will also show that for the dual pair of
type (Gl(n), Gl(m)), it can be described explicitely in terms of
Langlands parameters. Moreover, Howe's conjecture, in this case, is
still true for l-modular representations if l is a banal prime but it
is false if l is not banal.
Julio Moyano (Universidad
de Valladolid)
Title: Stöhr Zeta Function and generalized
Poincaré series
Let (O,m) be a Cohen-Macaulay local ring of dimension 1 containing a
finite field F. Assume that the extension degree of O/m over F is
finite. The integral closure of O in its total quotient ring is a
finite intersection of discrete Manis valuation rings V_1, ... V_r,
with associated Manis valuations v_1, ..., v_r, resp. We consider a
fractional ideal b in O and the set S(b):= {(v_1(g}, ... v_r(g)) | g
\in b, g non-zero divisor }. Then:
- we define the Stöhr zeta function associated with O and we put it
in terms of the set S(b), remarking the measure which Stöhr uses for
that.
- we define the generalized Euler characteristic and prove that it
coincides with the Stöhr's measure.
- we express the Stöhr zeta function in terms of an integral wih
respect to the generalized Euler characteristic.
- we compare the Stöhr zeta function with the "generalized Poincaré
series" introduced by Campillo, Delgado y Gussein-Zade.
Prof. Enric Nart
(Universidad Autónoma de Barcelona)
Title: Jacobians in isogeny classes of abelian
surfaces over finite fields
This talk will report on recent work with Everett Howe, Daniel Maisner
and Christophe Ritzenthaler, giving a complete answer to the question
of which polynomials occur as the characteristic polynomials of
Frobenius for genus $2$ curves over finite fields.
The solution of this problem requires the use of several techniques:
- Howe's obstruction for an isogeny class of abelian varieties to
be principally polarizable.
- Kani's construction of split Jacobians by tying two elliptic
curves together along their $n$-torsion groups.
- Counting principally polarized non Jacobians and principally
polarized Deligne modules. Comparison of the two numbers by Brauer
relations in biquadratic fields.
- Mass formulas for quaternion hermitian forms.
- Direct computation of the zeta function of a supersingular curve
in terms of the defining equation.
In the talk these different aspects of the problem will be presented
and some of them will be discussed in more detail.
Prof. Pierre Parent
(Université de Bordeaux I)
Title: On results of uniform big Galois image for
representations arising from
elliptic curves"
Let E be an elliptic curve over Q. Suppose it has no complex
multiplication over the algebraic closure of Q, K. For p a prime
number, consider the representation from Gal(K/Q) to GL_2(F_p) induced
by the Galois action on the group of p-torsion points of E. A milestone
theorem of Serre, published in 1972, asserts that there exists an
integer B_E such that the above representation is surjective if p is
larger than B_E. Serre then asked the following question: can B_E be
chosen independently of E? This boils down to proving the triviality,
for large enough p, of the sets of rational points of four families of
modular curves, namely X_0 (p), X_split (p), X_non-split (p) and X_A_4
(p) (we say that a point of one of these curves is trivial if it is
either a cusp, or the underlying isomorphism class of elliptic curves
has complex multiplication over K. The (so-called exceptional) case of
X_A_4 (p) was ruled out by Serre. The fact that X_0 (p)(Q) is made of
only cusps for p>163 is a well-known theorem of Mazur. In this talk
we will present recent results on the case of X_split (p)(Q) by ourself
and Marusia Rebolledo. If time permits we will also discuss other
applications of our techniques to Shimura curves, which allow to adress
a conjecture of Bruin, Flynn, Gonz\'alez and Rotger on the endomorphism
rings of abelian surfaces over Q.
Prof. Adolfo Quirós
(Universidad Autónoma de Madrid)
Title: Applications of Partial Divided Powers to
Algebraic Groups
In the search for a good p-adic cohomology theory for schemes over a
field k of characteristic p>0, an important role is played by the so
called arithmetic D-modules (see, for example, the work of
Mebkhout-Narváez and of Berthelot).
In particular, Berthelot and his school have shown that rigid
cohomology is a Weil cohomology and that it can (often) be calculated
"de Rham style” using a ring of differential operators build from the
notion of partial divided powers. Moreover, these partial divided
powers can also be used to define crystallline cohomology of higher
(finite) level. Passing to the limit in the level we recover rigid
cohomology modulo torsion.
But we do not want to use crystalline cohomology of higher level only
as a tool to obtain results in rigid cohomology and, in particular, we
do not want to ignore torsion, which is in fact very rich in this
theory.
Our first aim in this talk is to show how partial divided powers can be
used to define the conormal complex of higher level of a group scheme
and to study its relation to invariant differential forms of higher
level. Unlike the classical case (level 0), not all invariant forms are
closed. Actually, the module of closed invariant forms is isomorphic to
the first cohomology group of the conormal complex. We will present
concrete examples and give the relation with de Rham cohomology of
higher level in the case of abelian schemes.
Moreover, we will show that the extension group of the partial divided
power ideal by a smooth group is nothing else but a lifting of the
module of closed invariant differentials of higher level. This is the
first step towards a ``Dieudonne theory of higher level”.
Dr. Antonio Rojas
(University of California, Irvine)
Title: Generalizations of Kloosterman sums
In this talk we will define some families of exponential sums that can
be seen as generalization of the classical Kloosterman sums, and prove
some estimates for them similar to Deligne's result for the classical
ones.
Prof. Norbert Schappacher
(Université de Strasbourg)
Title: Remarks on the abc conjecture
Since the abc-Conjecture is not proved (and we are not going to change
that in this lecture), the talk will discuss:
- various ways to state the conjecture, starting from a medieval
inspiration;
- relations to other theorems and conjectures, at least one of
which is not well-known;
- various proposals that have been made for the error term.
The last item recalls in particular Serge Lang's philosophy of
"radicalized conjectures" in analytic number theory.
Prof. Nùria Vila
(Universidad de Barcelona)
Title: Representaciones de Galois geométricas con
imagen grande
En esta charla consideraré familias compatibles de representaciones de
Galois l-ádicas construidas en la cohomalogía étale de variedades
proyectivas lisas. Presentaré resultados obtenidos en colaboración con
Luis Dieulefait sobre condiciones explícitas para asegurar imagen
genérica grande en familias de representaciones de Galois tres y cuatro
dimensionales. Aplicaremos estos resultados a ejemplos
construidos por van Gemeen y Top en el caso tres
dimensional y por Scholten en el caso cuatro dimensional y
obtendremos la realización de familias grupos lineales y unitarios como
grupos de Galois sobre el cuerpo racional.
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