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# Séminaire de Théorie Algorithmique des Nombres

Responsable : Damien Robert

Page du séminaire

• Le 26 janvier 2016 à 11:00
• Salle 1
Présentation de WIMS (WWW Interactive Multipurpose Server)

• Le 9 février 2016 à 11:00
• Salle 385
Павел Соломатин (imb)
L-functions of Genus Two Abelian Coverings of Elliptic Curves over Finite Fields
Initially motivated by the relations between Anabelian Geometry andArtin’s L-functions of the associated Galois-representations, here westudy the list of zeta-functions of genus two abelian coverings ofelliptic curves over finite fields. Our goal is to provide a completedescription of such a list.
• Le 1er mars 2016 à 11:00
• Salle 385
Cyril Bouvier (imb)
Nonlinear polynomial selection for the number field sieve: improving Montgomery's method
The number field sieve is the most efficient known algorithm forfactoring large integers that are free of small prime factors. The goalof the polynomial selection, the first stage of this algorithm, is tocompute a pair of integer polynomials. Montgomery proposed a method forgenerating two nonlinear polynomials which relies on the constructionof small modular geometric progressions. In this talk, I will presenttheoretical and practical improvements to Montgomery’s method thatallow us to generate pairs of a quadratic and a cubic polynomials andpairs of two cubic polynomials for larger integer that was previouslypossible.Joint work with Nicholas Coxon.
• Le 8 mars 2016 à 11:00
• Salle 385
Fabien Pazuki (IMB et Université de Copenhague)
Régulateurs de corps de nombres et de variétés abéliennes et propriété de Northcott.
Soit $A$ une variété abélienne définie sur un corps de nombres $K$. On peutdéfinir un régulateur associé au groupe de Mordell-Weil des pointsrationnels $A(K)$, lequel joue un rôle important dans la forme forte dela conjecture de Birch et Swinnerton-Dyer. Si l’on suppose vraie laconjecture de Lang et Silverman, on montre alors que ce régulateurvérifie la propriété de finitude suivante : il n’y a qu’un nombre fini devariétés abéliennes simples de dimension fixée $g$, définie sur $K$, derang non nul et de régulateur borné. On montre de plus (dans le courantde la preuve) une inégalité inconditionnelle entre la hauteur deFaltings de $A$, les premiers de mauvaise réduction de $A$ et le rang deMordell-Weil de $A$. L’exposé commencera par une introduction présentantun résultat similaire et inconditionnel pour les régulateurs defamilles de corps de nombres.
• Le 15 mars 2016 à 11:00
• Salle 385
Bill Allombert (imb)
Survey on computing isogeny between elliptic curves.
We present methods to compute isogenies between elliptic curves, and weapply them to the computation of the isogenies matrix of an ellipticcurve defined over the rational and to the Schoof Elkies Atkinalgorithm for counting point on elliptic curves defined over a finitefield.
• Le 22 mars 2016 à 11:00
• Salle 385
Alexandre Le Meur (Université de Rennes)
Formules de Thomae généralisées aux cas des extensions galoisiennes résolubles de $\mathbb{P}^1$.
D’un point de vue classique, les formules de Thomae relient desrapports de puissances de theta constantes avec les coordonnées affinesdes points de ramification d’une courbe hyperelliptique. A partir desannées 80, plusieurs auteurs, ayant des préoccupations centrés sur laphysique, ont montré des généralisations de ces formules au cas descourbes superelliptiques. Plus récemment, Shau Zemel et Hershel Farkasont écrit un livre en utilisant des arguments essentiellementalgébriques. D’un point de vue arithmétique, ces courbes correspondentà des extensions galoisiennes cycliques d’un corps de fonctions $k(x)$.Nous montrerons comment généraliser ces formules au cas des extensionsrésolubles de $k(x)$ et quelles obstructions peuvent survenir.
• Le 5 avril 2016 à 11:00
• Salle 385
Benjamin Matschke (IMB)
A database of rational elliptic curves with given bad reduction
In this talk we present a database of rational elliptic curves with good reduction outside certain finite sets of primes, including the set {2, 3, 5, 7, 11}, and all sets whose product is at most 1000.

In fact this is a biproduct of a larger project, in which we construct practical algorithms to solve S-unit, Mordell, cubic Thue, cubic Thue–Mahler, as well as generalized Ramanujan–Nagell equations, and to compute S-integral points on rational elliptic curves with given Mordell–Weil basis. Our algorithms rely on new height bounds, which we obtained using the method of Faltings (Arakelov, Parshin, Szpiro) combined with the Shimura–Taniyama conjecture (without relying on linear forms in logarithms), as well as several improved and new sieves. In addition we used the resulting data to motivate several conjectures and questions, such as Baker’s explicit abc-conjecture, and a new conjecture on the number of S-integral points of rational elliptic curves.

This is joint work with Rafael von Känel.

• Le 10 mai 2016 à 11:00
• Salle 385
Nicolas Mascot (University of Warwick)
Calcul de représentations galoisiennes modulaires / Computing modular Galois representations
Nous verrons comment la représentation galoisienne modulo l associée àune forme modulaire classique peut être calculée efficacement, enl’isolant dans la torsion de la jacobienne d’une courbe modulaire. Cecipermet notamment de calculer les coefficients a(p) de la forme en tempspolynomial en log p, ce qui en fait la méthode la plus efficace connueà ce jour.

We will explain how the mod l Galois representation attached to aclassical newform may be efficiently computed, by isolating it amongthe l-torsion of a modular jacobian. This yields a way of computing thecoefficient a(p) of the form in time polynomial in log p, which makesit the most efficient methodknown as of today.

• Le 17 mai 2016 à 11:00
• Salle 385
Nicolas Mascot (University of Warwick)
Certification de représentations galoisiennes modulaires / Certifying modular Galois representations
Nous verrons comment les calculs de représentations galoisiennesprésentés dans l’exposé précédent peuvent être certifiés, en s’appuyantsur la conjecture de modularité de Serre et des calculs explicites decohomologie des groupes.

We will show how the Galois representation computations presented inlast week’s talk may be certified, thanks to Serre’s modularityconjecture and explicit group cohomology computations.

• Le 7 juin 2016 à 10:00
• Salle 385
Jared Asuncion (IMB)
Tower decomposition of Hilbert class fields

• Le 11 octobre 2016 à 14:00
• Salle 385
Enea Milio (Inria Nancy Grand Est)
Une implantation en genre 2 de 'Computing functions on Jacobians and their quotients' de Jean-Marc Couveignes et Tony Ezome
Cet article explique comment définir et évaluer des fonctions sur desJacobiennes de courbes de genre $g$ et sur des quotients de tellesJacobiennes par des sous-groupes isotropes maximaux de la$\ell$-torsion, pour $\ell>2$ premier. Pour le cas spécifique du genre2, il est bien connu qu’à partir d’une courbe hyperelliptique $C$ etd’un sous-groupe isotrope maximal $V$, le quotient $\mathrm{Jac}(C)/V$est la Jacobienne d’une courbe hyperelliptique $C’$,$(\ell,\ell)$-isogène à $C$. L’application de $C$ vers$\mathrm{Jac}(D)$ peut être décrite avec des fractions rationnelles dedegré en $O(\ell)$. L’article donne une méthode pour calculer $C’$ etces fractions. Pour notre exposé, nous nous proposons d’exposer lecontenu de ce papier et de parler de l’implantation que nous avonsfaite en genre 2.
• Le 18 octobre 2016 à 10:00
• Salle 385
Gregor Seiler (ETH Zurich)
Computing ray class fields of imaginary quadratic fields

• Le 8 novembre 2016 à 10:00
• Salle 385
Aurélien Focqué
Algorithmes BMSS et Lercier Sirvent pour SEA dans PARI

• Le 22 novembre 2016 à 10:00
• Salle 385
Razvan Barbulescu
A brief history of pairings
Pairings are a relatively new cryptographic tool which have been theobject of many arithmetic works. In the last few years some of thepairings have become obsolete because of the progress on the underlyingproblem of discrete logarithm in finite fields. We propose ourselves tomake a list of pairings constructions, to explain their advantages butalso their weaknesses. The sporadic curves are vulnerable to the Logjamattack and have never been a popular choice. The small characteristiccurves allow a very good arithmetic but are the target of aquasi-polynomial algorithm. The pairings where the characteristic has alow Hamming weight, which eliminate the cost of modular reductions,have been the object of special attacks. When the embedding degree iscomposite the one can use the tower field arithmetic but there are alsotower field attacks.
• Le 17 janvier 2017 à 10:00
• Salle 385
Damien Stehlé (ENS Lyon)
Tuple lattice sieving
Lattice sieving is asymptotically the fastest approach for solving the shortest vector problem (SVP) on Euclidean lattices. All known sieving algorithms for solving SVP require space which (heuristically) grows as $2^{0.2075n+o(n)}$, where n is the lattice dimension. In high dimensions, the memory requirement becomes a limiting factor for running these algorithms, making them uncompetitive with enumeration algorithms, despite their superior asymptotic time complexity. We generalize sieving algorithms to solve SVP with less memory. We consider reductions of tuples of vectors rather than pairs of vectors as existing sieve algorithms do. For triples, we estimate that the space requirement scales as $2^{0.1887n+o(n)}$. The naive algorithm for this triple sieve runs in time $2^{0.5661n+o(n)}$. With appropriate filtering of pairs, we reduce the time complexity to $2^{0.4812n+o(n)}$ while keeping the same space complexity. We further analyze the effects of using larger tuples for reduction, and conjecture how this provides a continuous tradeoff between the memory-intensive sieving and the asymptotically slower enumeration. Joint work with Shi Bai, Thijs Laarhoven
• Le 14 mars 2017 à 10:00
• Salle 385
Cécile Pierrot (Centrum Wiskunde & Informatica, Amsterdam)
Nearly sparse linear algebra
Linear algebra is a widely used tool both in mathematics and computerscience, and cryptography is no exception to this rule. Yet, itintroduces some particularities, such as dealing with linear systemsthat are often sparse, or, in other words, linear systems inside whicha lot of coefficients are equal to zero. We propose to enlarge thisnotion to nearly sparse matrices, characterized by the concatenationof a sparse matrix and some dense columns, and to design an algorithmto solve this kind of problems. Motivated by discrete logarithmscomputations on medium and high characteristic finite fields, theNearly Sparse Linear Algebra bridges the gap between classical denselinear algebra problems and sparse linear algebra ones, for whichspecific methods have already been established. Our algorithmparticularly applies on one of the three phases of NFS (Number FieldSieve) which precisely consists in finding a non trivial element ofthe kernel of a nearly sparse matrix.

This is a joint work with Antoine Joux.

• Le 23 mai 2017 à 10:00
• Salle 385
Christophe Petit (Oxford)
Post-quantum cryptography from supersingular isogeny problems?
We review existing cryptographic schemes based on the hardness ofcomputing isogenies between supersingular isogenies, and present someattacks against them. In particular, we present new techniques toaccelerate the resolution of isogeny problems when the action of theisogeny on a large torsion subgroup is known, and we discuss the impactof these techniques on the supersingular key exchange protocol ofJao-de Feo.
• Le 30 mai 2017 à 10:00
• Salle 385
Benjamin Wesolowski (EPFL)
Isogeny graphs of ordinary abelian varieties
Fix a prime number $\ell$. Graphs of isogenies of degree a power of$\ell$ are well-understood for elliptic curves, but not forhigher-dimensional abelian varieties. We study the case of absolutelysimple ordinary abelian varieties over a finite field. We analysegraphs of so-called $\mathfrak l$-isogenies, resolving that, inarbitrary dimension, their structure is similar, but not identical, tothe volcanoes’’ occurring as graphs of isogenies of elliptic curves.Specializing to the case of principally polarizable abelian surfaces,we then exploit this structure to describe graphs of a particular classof isogenies known as $(\ell, \ell)$-isogenies. These results lead tonew, provable algorithms to navigate in isogeny graphs, withconsequences for the CM-method in genus 2, for computing explicitisogenies, and for the random self-reducibility of the discretelogarithm problem in genus 2 cryptography.
• Le 6 juin 2017 à 10:00
• Salle 385
Guilhem Castagnos (imb)
Encryption Switching Protocols Revisited: Switching modulo p
Last year, Couteau, Peters and Pointcheval introduced a new primitivecalled encryption switching protocols, allowing to switch ciphertextsbetween two encryption schemes. If such an ESP is built with twoschemes that are respectively additively and multiplicativelyhomomorphic, it naturally gives rise to a secure 2-party computationprotocol. It is thus perfectly suited for evaluating functions, suchas multivariate polynomials, given as arithmetic circuits. Couteau etal. built an ESP to switch between Elgamal and Paillier encryptionswhich do n ot naturally fit well together. Consequently, they had todesign a clever variant of Elgamal over Z/nZ with a costly shareddecryption. In this talk, we first present a conceptually simplegeneric construction for encryption switching protocols. We then givean efficient instantiation of our generic approach that uses twowell-suited protocols, namely a variant of Elgamal in Z/pZ and theCastagnos-Laguillaumie encryption defined over class groups of quadratic fields which is additively homomorphic over Z/pZ. Among otheradvantages, this allows to perform all computations modulo a prime pinstead of an RSA modulus. Overall, our solution leads to significantreductions in the number of rounds as well as the number of bitsexchanged by the parties during the interactive protocols. We also showhow to extend its security to the malici ous setting.

Joint work with Laurent Imbert and Fabien Laguillaumie.

• Le 13 juin 2017 à 10:00
• Salle 385
Bernhard Schmidt (Nanyang Technological University, Singapore)
The Anti-Field-Descent Method
A circulant Hadamard matrix of order $v$ is a matrix of the form$H=\begin{pmatrix}a_1 & a_2 & \cdots & a_v \a_v & a_1 & \cdots & a_{v-1} \\cdots & \cdots & \cdots &\cdots \a_2 & a_3 & \cdots & a_1 \\end{pmatrix}$with $a_i=\pm 1$ such that any two rows of $H$ are orthogonal withrespect to the standard inner product. It is conjectured that there isno circulant Hadamard matrix of order larger than $4$.

One way to study circulant Hadamard matrices is the so-calledfield-descent method’’. The essential fact behind this method is thatcertain cyclotomic integers necessarily are contained in relativelysmall fields and thus must have relatively small complex modulus. Inthis talk, I will present a method which reveals a complementaryphenomenon: certain cyclotomic integers cannot be contained in relativelysmall fields and thus must have relatively large complex modulus. Thismethod provides new necessary conditions for the existence of circulantHadamard matrices.

This is joint work with K. H. Leung.

• Le 17 octobre 2017 à 10:00
• Salle 385
Fredrik Johansson (imb)
Numerics of classical elliptic functions, elliptic integrals and modular forms
We review methods for validated arbitrary-precision numericalcomputation of elliptic functions and their inverses (the complete andincomplete elliptic integrals), as well as the closely related Jacobitheta functions and $\mathrm{SL}_2(\mathbb{Z})$ modular forms. A general strategy consists of two stages:first, using functional equations to reduce the functionarguments to a smaller domain; second, evaluation of a suitable truncatedseries expansion. For elliptic functions and modular forms, one exploitsperiodicity and modular transformations for argument reduction, afterwhich the rapidly convergent series expansions of Jacobi theta functionscan be employed. For elliptic integrals, a comprehensive strategypioneered by B. Carlson consists of using symmetric forms to unify andsimplify both the argument reduction formulas and the series expansions(which involve multivariate hypergeometric functions). Among otheraspects, we discuss error bounds as well as strategies for argumentreduction and series evaluation that reduce the computational complexity.The functions have been implemented in arbitrary-precision complexinterval arithmetic as part of the Arb library.
• Le 24 octobre 2017 à 10:00
• Salle 385
José Manuel Rodriguez Caballero (Labri)
Context-free languages in Algebraic Geometry and Number Theory.
Kassel and Reutenauer computed the zeta function of the Hilbert schemeof n points on a two-dimensional torus and showed it satisfies severalnumber-theoretical properties via modular forms. Classifying thesingularities of this rational function into zeros and poles, we definea word which contains a lot of number-theoretical information about n(the above-mentioned number of points). This nontrivial connectionbetween natural numbers and words can be used to define many classicalsubsets of natural numbers in terms of rational and context-freelanguages (e.g. the set of semi-perimeters of Pythagorean triangles,the set of numbers such that any partition into consecutive parts hasan odd number of parts). Also, some arithmetical functions can bedescribed in way (e.g. the Erdös-Nicolas function, the number of middledivisors). Finally, this approach provides a new technique to provenumber-theoretical results just using relationships among context-freelanguages.
• Le 14 novembre 2017 à 10:00
• Salle 385
Jean Kieffer (ENS Paris)
Accélération du protocole d'échange de clés de Couveignes-Rostovtsev-Stolbunov
Ce protocole d’échange de clés est fondé sur la théorie de lamultiplication complexe: un ordre dans un corps quadratique imaginaireagit sur un ensemble de courbes elliptiques ordinaires isogènes définiessur un corps fini. Pour instancier le protocole, on est amené à calculerdes isogénies de différents degrés entre ces courbes à l’aide desalgorithmes développés pour le comptage de points. Ce cryptosystème peutêtre accéléré par un bon choix de courbe elliptique initiale, notammentpar la présence de points de torsion rationnels, et l’on présente uneméthode de recherche de telles courbes.
• Le 20 novembre 2017 à 14:00
• Salle 385
Christian Klein
Computational approach to compact Riemann surfaces
A purely numerical approach to compact Riemann surfaces starting fromplane algebraic curves is presented. The critical points of the algebraiccurve are computed via a two-dimensional Newton iteration. The startingvalues for this iteration are obtained from the resultants with respect toboth coordinates of the algebraic curve and a suitable pairing of theirzeros. A set of generators of the fundamental group for the complement ofthese critical points in the complex plane is constructed from circlesaround these points and connecting lines obtained from a minimal spanningtree. The monodromies are computed by solving the de ning equation of thealgebraic curve on collocation points along these contours and byanalytically continuing the roots. The collocation points are chosen tocorrespond to Chebychev collocation points for an ensuing Clenshaw–Curtisintegration of the holomorphic differentials which gives the periods ofthe Riemann surface with spectral accuracy. At the singularities of thealgebraic curve, Puiseux expansions computed by contour integration on thecircles around the singularities are used to identify the holomorphicdifferentials. The Abel map is also computed with the Clenshaw–Curtisalgorithm and contour integrals. As an application of the code, solutionsto the Kadomtsev–Petviashvili equation are computed on non-hyperellipticRiemann surfaces.
• Le 28 novembre 2017 à 10:00
• Salle 385
Frank Vallentin
Coloring the Voronoi tessellation of lattices
We define the chromatic number of a lattice: It is the least number ofcolors one needs to color the interiors of the cells of the Voronoitesselation of a lattice so that no two cells sharing a facet are ofthe same color. We compute the chromatic number of the irreducible rootlattices and for this we apply a generalization of the Hoffman bound.
• Le 9 janvier 2018 à 10:00
• Salle 385
Fredrik Johansson (imb)
Numerical integration in complex interval arithmetic
We present a new implementation of validated arbitrary-precisionnumerical evaluation of definite integrals $\int_a^b f(x) dx$,available in the Arb library. The code uses a version of the Petrasalgorithm, which combines adaptive subdivision with Gauss-Legendre (GL)quadrature, evaluating the integrand on complex intervals surroundingthe path of integration to obtain rigorous error bounds. The first partof the talk discusses the general algorithm and its performance forinteresting families of integrals. The second part, which is based onjoint work with Marc Mezzarobba, discusses the fast computation of GLquadrature nodes with rigorous error bounds. It is well known that GLquadrature achieves a nearly optimal rate of convergence for analyticintegrands with singularities well isolated from the path ofintegration, but due to the cost of generating GL quadrature nodes, themore slowly converging Clenshaw-Curtis and double exponentialquadrature rules have often been favored when an accuracy of severalhundreds or thousands of digits is required. We consider the asymptoticand practical aspects of this problem. An order-of-magnitude speedup isobtained over previous code for computing GL nodes with simultaneoushigh degree and precision, which makes GL quadrature viable even atvery high precision.
• Le 22 janvier 2018 à 10:00
• Salle 2
Philippe Moustrou (IMB)
On the Density of Sets Avoiding Parallelohedron Distance 1
Let $\Vert \cdot \Vert$ be a norm on $\mathbb{R}^n$. We consider theso-called unit distance graph $G$ associated with $\Vert \cdot \Vert$:the vertices of $G$ are the points of $\mathbb{R}^n$, and the edgesconnect the pairs $\{x,y\}$ satisfying $\Vert x-y\Vert=1$. We define$m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ as the supremum of thedensities achieved by independent sets of $G$. The number $m_1$ wasintroduced by Larman and Rogers (1972) as a tool to study themeasurable chromatic number $\chi_m(\mathbb{R}^n)$ of $\mathbb{R}^n$for the Euclidean norm.

The best known estimates for $\chi_m(\mathbb{R}^n)$ and$m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ present relations withEuclidean lattices, in particular with the sphere packing problem.

The determination of $m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$for the Euclidean norm is still a difficult question. We study thisproblem for norms whose unit ball is a convex polytope. More precisely,if the unit ball corresponding with $\Vert \cdot \Vert$ tiles$\mathbb{R}^n$ by translation, for instance if it is the Voronoiregion of a lattice, then it is easy to see that$m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)\geq \frac{1}{2^n}$.

C. Bachoc and S. Robins conjectured that equality always holds. We showthat this conjecture is true for $n=2$ and for some families of Voronoiregions of lattices in higher dimensions.

• Le 30 janvier 2018 à 10:00
• Salle 385
Jared Asuncion (IMB)
ECPP in PARI/GP
The elliptic curve primality proving (ECPP) algorithm not only proves(or disproves) the primality of an integer $N$ but also provides, if$N$ is prime, a primality certificate which one can verify quickly. Inthis talk, we recall the steps of ECPP and discuss its implementationin PARI/GP.
• Le 6 mars 2018 à 10:00
• Salle 385
Takashi Fukuda (Nihon University)
Class number calculation for special number fields
I will talk about TC (an interpreter of multiprecision C language whichI developed), Weber’s problem, Coates’ conjecture and an algorithm ofcalculating p-class group of abelian number fields. I also present myproject trying to implement an algorithm mentioned above to PARI/GPduring my stay at IMB.
• Le 20 mars 2018 à 10:00
• Salle 385
Tristan Vaccon (Université de Limoges)
Sur les équations différentielles $p$-adiques à variables séparables
Les trois dernières décennies ont vu le développement de méthodes etalgorithmes $p$-adiques, notamment :
• la factorisation de polynômes rationnels par lemme de Hensel ;
• les algorithmes de comptage de points de Kedlaya et Lauder, reposant surdes résultats avancés de géométrie arithmétique ;
• le calcul d’isogénies entre courbes elliptiques.

Dans toutes ces méthodes et algorithmes, on passe par des calculs surles nombres $p$-adiques, et le problème de la gestion de la précision yest crucial. Avec Xavier Caruso et David Roe, nous avons développé uneméthode, dite de précision différentielle, pour étudier et gérer laprécision $p$-adique.

Dans cet exposé, nous nous intéresserons à l’application de cetteméthode pour l’étude du calcul d’isogénies entre courbes elliptiquesvia la résolution de certaines équations différentielles $p$-adiques àvariables séparables. Il s’agit d’un travail en commun avec PierreLairez de 2016 qui ne traite que du cas $p>2$. Nous présenterons aussidans cet exposé quelques avancées récentes lorsque $p=2$.

• Le 3 avril 2018 à 10:00
• Salle 385
Alex Bartel (Glasgow University)
Cohen-Martinet heuristics revisited
In the early 1990s Henri Cohen and Jacques Martinet offered aprobabilistic model that explains the behaviour of ideal class groupsof number fields in natural families, generalising earlier work byCohen and Lenstra. There is a lot of numerical evidence for thecorrectness of the model, but very few theorems. In joint work withHendrik Lenstra we revisit the Cohen-Martinet heuristics and, amongother things, disprove them in two different ways, but also lendadditional support for the expectation that they are “basically true”.In this talk I will explain one of these disproofs, and will proposepossible corrections.
• Le 24 avril 2018 à 10:00
• Salle 385
Damien Robert (imb)
Huang's proposal for trilinear maps
In a recent paper, Huang proposed atrilinear map involving abelian varieties over finite fields. In this talkwe present his approach.

We will first begin the talk with a review of the standard pairingsconstructions on an abelian variety.

• Le 22 mai 2018 à 10:00
• Salle 385
Jean Kieffer (ENS Paris)
Étude et implémentation de l’algorithme de Schoof–Elkies–Atkin

• Le 12 juin 2018 à 10:00
• Salle 385
Xavier Caruso (Université de Rennes)
Variations autour d'un théorème de Christol
Un célèbre théorème de Christol affirme qu’une série à coefficientsdans $\mathbb{Z}/p\mathbb{Z}$ est algébrique sur$\mathbb{Z}/p\mathbb{Z}(x)$ si et seulement si la suite de sescoefficients est $p$-automatique.

L’objectif de cet exposé sera de raconter de jolies mathématiques enlien de ce théorème. Je commencerai par esquisser deux démonstrations“classiques” de ce résultat, puis montrerai comment les combiner pourobtenir une variante effective du théorème de Christol. Je présenteraiensuite une application de ce résultat à une question de naturealgorithmique.

Si le temps le permet, je discuterai également les liens entre théorèmede Christol et équations différentielles p-adiques et montrerai commentutiliser ce nouvel ingrédient pour accélérer l’algorithme précédent.

(Travail en commun avec A. Bostan, G. Christol et Ph. Dumas.)

• Le 5 juillet 2018 à 10:00
• Salle 385
Jean-François Biasse (University of South Florida)
Fast multiquadratic S-unit computation and application to the calculation of class groups
Let $L=Q(√d_1, … ,√d_n)$ be a real multiquadratic field and S be a setof prime ideals of L that does not contain any divisors of 2. In thispaper, we present a heuristic algorithm for the computation of theS-class group and the S-unit group that runs in time$Poly(log(∆),Size(S)) e^{Õ(√ln d)}$ where $d=max_{i≤n} d_i$ and ∆ is thediscriminant of L. We use this method to compute the ideal class groupof the maximal order $O_L$ of L in time $Poly(log(∆)) e^{Õ(√log d)}$. When$log(d)≤log(log(∆))^c$ for some constant $c < 2$, these methods run inpolynomial time. We implemented our algorithm using Sage 7.5.1.
• Le 18 septembre 2018 à 10:30
• Salle 385
Aurel Page et Pascal Molin
Mini groupe de travail: calcul des caractères de Hecke

• Le 6 novembre 2018 à 10:00
• Salle 385
Elie Eid (Université de Rennes)
Calcul d'isogénies en genre 2
Étant donné une courbe algébrique $C$ de genre $2$ définie sur un corpsfini $K$ de caractéristique impaire et un sous-groupe isotrope maximal$\mathcal{V}$ (pour le couplage de commutateur) de l’ensemble despoints de $l$-torsion où $l$ est un entier (premier) impair, noussavons que le quotient de la jacobienne $J_K(C)$ de $C$ par$\mathcal{V}$ est une variété abélienne de dimension 2 et donc lajacobienne d’une courbe $D$ de genre $2$.

La surjection canonique $\pi_l \: : J_K(C) \longrightarrow J_K(D) =J_K(C) / \mathcal{V}$ est une $(l,l)$-isogénie de noyau $\mathcal{V}$.

Dans cet exposé, on s’intéresse à l’algorithme de Couveignes et Ezomepour trouver l’équation de la courbe $D$ à partir de sa kummerconstruite à l’aide de fonctions définies sur la jacobienne de lacourbe de départ et son quotient, ainsi que les équations quidéfinissent l’isogénie $\pi_l$.

• Le 27 novembre 2018 à 10:00
• Salle 385
Ida Tucker

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