Titres et resumés

 

 

Conférences Comunications
A. Campillo
 S. Arias
G. Everest
 I. García-Selfa
B. López  E. González
E. Nart
  A. Mínguez
P. Parent
  J. Moyano
A. Quirós
  
A. Rojas
N. Schappacher
N. Vila

 

 

 

Conférences



Sara Arias (Universidad de Barcelona)

Titre: 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.

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Prof. Antonio Campillo (Universidad de Valladolid)

Titre: 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.

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Prof. Graham Everest (University of East Anglia)

Titre: 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.

Titre: 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.

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Irene García-Selfa (Universidad de Sevilla)

Titre: 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.

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Dr. Enrique Gozález (Universidad Autónoma de Madrid)

Titre: 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.

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Prof. Bartolomé López (Universidad de Cádiz)

Titre: 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.

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Alberto Mínguez (École Normale Supérieure, Paris)

Titre: 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.

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Julio Moyano (Universidad de Valladolid)

Titre: 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:

  1. 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.
  2. we define the generalized Euler characteristic and prove that it coincides with the Stöhr's measure.
  3. we express the Stöhr zeta function in terms of an integral wih respect to the generalized Euler characteristic.
  4. we compare the Stöhr zeta function with the "generalized Poincaré series" introduced by Campillo, Delgado y Gussein-Zade.

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Prof. Enric Nart (Universidad Autónoma de Barcelona)

Titre: 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:
  1. Howe's obstruction for an isogeny class of abelian varieties to be principally polarizable.
  2. Kani's construction of split Jacobians by tying two elliptic curves together along their $n$-torsion groups.
  3. Counting principally polarized non Jacobians and principally polarized Deligne modules. Comparison of the two numbers by Brauer relations in biquadratic fields.
  4. Mass formulas for quaternion hermitian forms.
  5. 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.

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Prof. Pierre Parent (Université de Bordeaux I)

Titre: 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.

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Prof. Adolfo Quirós (Universidad Autónoma de Madrid)

Titre: 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”.

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Dr. Antonio Rojas (University of California, Irvine)

Titre: 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.

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Prof. Norbert Schappacher (Université de Strasbourg)

Titre: 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:
  1. various ways to state the conjecture, starting from a medieval inspiration;
  2. relations to other theorems and conjectures, at least one of which is not well-known;
  3. 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.

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Prof. Nùria Vila (Universidad de Barcelona)

Titre: 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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