RÉSUMÉS / ABSTRACTS

Abstract: When Henri Cohen and his coworkers set out to write PARI twenty years ago, GP was an afterthought. While GP has become by far the most commonly used interface to the PARI library, both the gp interpretor and the GP language are rather primitive in design. Paradoxically, while gp allows to handle very high-level objects, GP itself is a low-level language coming straight from the seventies. We rewrote gp as a compiler/evaluator pair and are ready to implement some high-level features that should move GP into the nineties. We give a mathematician-oriented account of how the compiler/evaluator works and of the planned changes to the language.

Abstract: A paper of H. Cohen and the author dated 1985 introduced the Dirichlet series whose n-th coefficient is +1 or -1 according to the parity of the sum of the binary digits of the integer n.

We will follow the direct and indirect descent of this paper, from the original computation of infinite products to the proof of a conjecture of Shallit, from a new proof of the existence of logarithmic density for automatic sequences to the consideration of generalized Dirichlet series with automatic or automatic-like coefficients.

We will make use of works of H. Cohen, M. Mendès France, J. Peyrière, J. Shallit, and the author, with mention of a paper by D. Essouabri.

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Abstract: In 1873 H.A. Schwarz gave a classification of Gauss hypergeometric functions that are at the same time algebraic over the rational functions. Later, this Schwarz list has been extended to higher order hypergeometric functions and several types of Appell functions. In this lecture we discuss the generalisation to general hypergeometric functions in several variables, known as GKZ-hypergeometric functions.

Abstract: I will speak on the work of Murty, Luca and myself proving that fields with class number divisible by a given integer have positive logarithmic density, and on the subsequent development due to Levin.

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Abstract: Digital signatures play an important role in many computer security solutions. However, no provably secure digital signature algorithm is known. Also, the digital signature schemes used in practice are threatened by quantum computers. Therefore, it is necessary to search for alternative digital signature algorithms. In this talk we describe a generic construction of digital signatures that only require a cryptographic hash function. We explain the construction of such hash functions based on shortest vector problems in certain classes of lattices.

Abstract: Let (u_n) be a linear recurrent sequence of integers. We survey recent results on the Diophantine equation u_n = y^p in the three unknowns n, y and p. We focus on two particular equations, namely on (10^n - 1)/9 = y^p and on F_n = y^p, where F_n is the n-th Fibonacci number.

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Abstract: We show how the classical theory of reduction of real binary forms with respect to the action of SL(2,Z) may be extended to a reduction theory for binary forms with complex coefficients under the action of certain discrete groups. In particular, we give some explicit results concerning the reduction of binary cubics and quartics with coefficients in the ring of integers of an imaginary quadratic field of class number one (such as Z[i]), and mention applications to the enumeration of cubic fields and two-descent on elliptic curves.

Abstract: Soit E une courbe elliptique définie sur Q de conducteur p premier. Si C est un sous-groupe d'ordre p de E[p], on note P_C dans E l'image du point (E,C) de X_0(p) par le revêtement modulaire associé à E. Le point P_C est appelé un self-point. On démontre que ce point est d'ordre infini et on en déduit des minorations explicites (dans les sens où on construit effectivement des points linéairement indépendants) du rang des groupes de Mordell-Weil E(K_n) avec K_n=Q(E[p^n]). (Avec Christian Wuthrich.)

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Abstract: The classical Raabe formula computes a definite integral of the logarithm of Euler's Gamma-function. We compute analogous p-adic integrals of the p-adic log Gamma functions (both Diamond's and Morita's) and prove a p-adic Raabe formula. We show that the difference equation together with the p-adic Raabe formula uniquely characterize the p-adic log Gamma functions. If time permits we will also describe all solutions to the Raabe formula, which is an integro-differential equation in the p-adic setting. (Joint work with Henri Cohen.)

Abstract: Combining conjectures of Lichtenbaum and Zagier, we obtain experimental values for higher regulator lattices, which suggest - or corroborate - higher Stark-type conjectures and a curious symmetry.

Abstract: Finding very short vectors in lattices is an algorithmic problem of great importance, and of great difficulty as the dimension tends to infinity; many tasks in computational number theory and cryptology can be reduced to it. The classical algorithm, described by Kannan and independently by Fincke-Pohst, was analyzed by several authors, who obtained a complexity of d^{d/2(1+o(1))} when d tends to infinity, up to polynomial factors. We shall prove that the complexity is actually d^{d/2e (1 + o(1))}. The main tools are the estimation of the number of integral points inside an ellipsoid, and a sharp estimate on the geometry of strongly reduced lattice bases.

Abstract: I will decribe how to compute Galois groups of rational polynomials. These methods are implemented in the computer algebra system Magma. They work independently of the degree of the polynomial and we have been able to compute Galois groups of high degree, say 64 or 128. This is a joint project with Claus Fieker.

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Abstract: The original Cohen-Lenstra heuristics provide a conjectural description of the distribution of class groups of quadratic fields. The lecture, which is intended for the novice, will explain the model behind the conjecture and the intuition on which it is based.

Abstract: Primality proving is like a mystery book in which the detective has to prove the innocence of a suspect by finding a lot of witnesses who can testify it. Information has to be gathered concerning a given integer N suspected to be prime (innocent?). After an easy beginning, say 2^(N-1) = 1 mod N, then more scrutiny is needed. In the ancient times where no general purpose algorithms were known, properties obtained via Fermat like tests would be assembled to help. Nowadays, we have several algorithms that can prove a number to be prime. Combining several ingredients can still be done and even different algorithms can be merged to reach the goal. This talk will survey old results and sketch some attempts at combining recent algorithms.

Abstract: This is a joint work with Marc Deléglise and Paul Zimmermann. Let S_n denote the symmetric group with n letters, and g(n) the maximal order of an element of S_n. If the standard factorization of M into primes is M = q_1^{a_1}q_2^{a_2} ... q_k^{a_k}, we define ell(M) to be q_1^{a_1} + q_2^{a_2} + ... + q_k^{a_k}. One century ago, E. Landau proved that g(n) = max_{ell(M) <= n} M and that, when n goes to infinity, log g(n) ~ sqrt(n log(n)). There exists a basic algorithm to compute g(n) for 1 < n < N ; its running time is O(N^{3/2} / sqrt(log N)) and the needed memory is O(N); it allows computing g(n) up to, say, one million. In this talk, we describe an algorithm to calculate g(n) for n up to 10^{15}. The main idea is to use the so-called ell-superchampion numbers. Similar numbers, the superior highly composite numbers, were introduced by S. Ramanujan to study large values of the divisor function tau(n).

Abstract: We develop effective methods for solving unit equations over global function fields. In case the number of variables is 2 or 3 we also present applications.

Abstract: We prove that the sum of the digits of prime numbers and squares is equidistributed in residue classes (joint work with Christian Mauduit).

Abstract: Cet exposé sera consacré à des résultats obtenus dans un travail en commun avec Christian Krattenthaler. J'expliquerai pourquoi les coefficients de Taylor de fonctions de la forme q(z)=zexp(G(z)/F(z)) sont des entiers, où F(z) et G(z)+log(z) F(z) sont des solutions spécifiques de certaines équations différentielles hypergéométriques ayant une monodromie unipotente maximale en z=0. Ceci implique de nombreux résultats similaires pour les coefficients de Taylor des 'mirror maps' de familles d'intersections complètes de type Calabi-Yau dans des espaces projectifs à poids.

Abstract: For every squarefree natural number n, the Jacobian J0(n) of the modular curve X0(n) is a semistable abelian variety that is defined over Q. It has good reduction outside n . We show, conversely, that for every odd squarefree n < 30, any semistable abelian variety over Q with good reduction outside n is isogenous to a power of J0(n).

Abstract: For many curves of higher genus, Baker's Theory provides astronomical bounds for the size of integral points. I explain how to combine Baker's bounds with the Mordell-Weil sieve to compute all the integral points on some curves of higher genus. This is based on joint work with Bugeaud, Mignotte, Stoll and Tengely.

Abstract: The integers Z and hence the rationals Q may be thought of as containing all finite fields F_p by way of reduction mod p. I explain how this thought deals with reduction of the continued fraction expansion of a formal Laurent series in 1/X with coefficients in Z. I discuss the quadratic irrational case, that of hyperelliptic curves, in some detail and mention links to its numerical analogue.

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Abstract: An analogue of Cremona's and Mazur's theory of visibility of Shafarevich-Tate groups is the capitulation of ideal classes in cyclotomic fields. We discuss some computations (using PARI, of course) and relations with Cohen-Lenstra heuristics.

Abstract: We consider a rank 1 elliptic curve E that satisfies a Heegner hypothesis for an imaginary quadratic field of small class number (3 or 5 perhaps). We can then consider the lattice generated by the conjugates of the Heegner point in the Mordell-Weil group of E over the Hilbert class field. We can investigate this in two ways, either by computing the Heegner point in the Hilbert class field directly, or by computing L'(E,1) and using Gross-Zagier, as all we really need is the height of the point in the HCF. We discuss how we might expect the obtained lattices to be distributed, and give data that we have accumulated (joint work with S. R. Donnelly).

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Abstract: In 1979, Henri Cohen and François Dress implement a small arithmetic interpretor named ISABELLE on a TI 980, using assembly language. A few years later, using faster computers and higher-level languages, Henri starts a new project, the PARI system. I will recall some of the big successes of GP/PARI (and a few failures) during the last two decades.