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| Conjectures, consequences, and numerical experiments for p-adic Artin L-functions |
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| Submitted (with R. de Jeu) |
| Abstract |
We conjecture that the \( p \)-adic \( L \)-function of a non-trivial irreducible even Artin character over a totally real field is non-zero at all non-zero integers. This implies that a conjecture formulated by Coates and Lichtenbaum at negative integers extends in a suitable way to all positive integers. We also state a conjecture that for certain characters the Iwasawa series underlying the \( p \)-adic \( L \)-series have no multiple roots except for those corresponding to the zero at \( s = 0 \) of the \(p\)-adic \(L\)-function.
We provide some theoretical evidence for our first conjecture, and prove both conjectures by means of computer calculations for a large set of characters (and integers where appropriate) over the rationals and over real quadratic fields, thus proving many instances of conjectures by Coates and Lichtenbaum and by Schneider. The calculations and the theoretical evidence also prove that certain \( p \)-adic regulators corresponding to \( 1 \)-dimensional characters for the rational numbers are units in many cases. We also verify Gross' conjecture for the order of the zero of the \( p \)-adic \( L \)-function at \( s = 0 \) in many cases. We gather substantial statistical data on the constant term of the underlying Iwasawa series, and propose a model for its behaviour for certain characters. |
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| On the semi-simple case of the Galois Brumer-Stark conjecture for monomial groups |
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| Tokyo J. of Math. 41, 1 (2018), 241-251 |
| Abstract |
In a previous work, we stated a conjecture, called the Galois Brumer-Stark conjecture, that generalizes the (abelian) Brumer-Stark conjecture to Galois extensions. Another generalization of the Brumer-Stark conjecture to non-abelian Galois extensions is due to Nickel. Nomura proved that the Brumer-Stark conjecture implies the non-abelian Brumer-Stark conjecture of Nickel when the group is monomial. In this paper, we use the methods of Nomura to prove that the Brumer-Stark conjecture implies the Galois Brumer-Stark conjecture for monomial groups in the semi-simple case. |
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| The Galois Brumer-Stark conjecture for \(\mathrm{SL}_2(\mathbb{F}_3)\)-extensions |
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| Int. J. Number Theory 12, 165 (2016), 165-188 (with G. Dejou). |
| Abstract |
In A Brumer-Stark Conjecture for non-abelian Galois extensions, we stated a conjecture, called the Galois Brumer-Stark conjecture, that generalizes the (abelian) Brumer-Stark conjecture to Galois extensions. We also proved that, in several cases, the Galois Brumer-Stark conjecture holds or reduces to the abelian Brumer-Stark conjecture. The first open case is the case of extensions with Galois group isomorphic to \(\mathrm{SL}_2(\mathbb{F}_3)\). This is the case studied in this paper. These extensions split naturally into two different types. For the first type, we prove the conjecture outside of \(2\). We also prove the conjecture for \(59\) \(\mathrm{SL}_2(\mathbb{F}_3)\)-extensions of \(\mathbb{Q}\) using computations. The version of the conjecture that we study is a stronger version, called the Refined Galois Brumer-Stark conjecture, that we introduce in the first part of the paper. |
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| Computing \(p\)-adic \(L\)-functions of totally real number fields |
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| Math. Comp. 84 (2015), no. 292, 831-874 (see also ArXiv:1110.0246). |
| Abstract |
We prove explicit formulas for the \(p\)-adic \(L\)-functions of totally real number fields and show how these formulas can be used to compute values and representations of \(p\)-adic \(L\)-functions. |
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| A Brumer-Stark Conjecture for non-abelian Galois extensions |
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| J. Number Theory 142 (2014), 51-88 (with G. Dejou). |
| Abstract |
The Brumer-Stark conjecture deals with abelian extensions of number fields and predicts that a group ring element, called the Brumer-Stickelberger element constructed from special values of \(L\)-functions associated to the extension, annihilates the ideal class group of the extension under consideration. Moreover it specifies that the generators obtained have special properties. The aim of this article is to propose a generalization of this conjecture to non-abelian Galois extensions that is, in spirit, very similar to the original conjecture. |
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| Index formulae for Stark units and their solutions |
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| Pacific J. of Mathematics 266 (2013), n°2, 391-422 (see also ArXiv:1112.2820). |
| Abstract |
Let \(K/k\) be an abelian extension of number fields with a distinguished place of \(k\) that splits totally in \(K\). In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in \(K\), called the Stark unit, constructed from the values of the \(L\)-functions attached to the extension. In this paper, assuming the Stark unit exists, we prove index formulae for it. In a second part, we study the solutions of the index formulae and prove that they admit solutions unconditionally for quadratic, quartic and sextic (with some additional conditions) cyclic extensions. As a result we deduce a weak version of the conjecture ("up to absolute values") in these cases and precise results on when the Stark unit, if it exists, is a square. |
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| Numerical evidence toward a 2-adic equivariant ''Main Conjecture'' |
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| Experiment. Math. 20 (2011), n°2, 169-176 (with A. Weiss). |
| Abstract |
We test a conjectural non abelian refinement of the classical \(2\)-adic Main Conjecture of Iwasawa theory. In the first part, we show how, in the special case that we study, the validity of this refinement is equivalent to a congruence condition on the coefficients of some power series. Then, in the second part, we explain how to compute the first coefficients of this power series and thus numerically check the conjecture in that setting. |
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| Testing the Congruence Conjecture for Rubin-Stark Elements |
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| J. Number Theory 130 (2010), n°6, 1374-1398 (with D. Solomon). |
| Abstract |
The `Congruence Conjecture' was developed by the second author in a previous paper. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence is predicted by earlier conjectures of Rubin and Stark. The first aim of the present paper is to design and apply techniques to investigate the Congruence Conjecture numerically. We then present complete verifications of the conjecture in 48 varied cases with real quadratic base fields.
The Rubin-Stark elements found during this research are available in the table of Rubin-Stark elements over quadratic fields. |
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| On the \(p\)-adic Beilinson conjecture for number fields |
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| Pure Appl. Math. Q. 5 (2009), n°1, 375-434 (with A. Besser, P. Buckingham and R. de Jeu). |
| Abstract |
We formulate a conjectural \(p\)-adic analogue of Borel's theorem relating regulators for higher \(K\)-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and \(p\)-adic \(L\)-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the \(p\)-adic and classical situations. Parts of the conjectures are proved when the number field (or Artin motive) is abelian over the rationals, and all conjectures are verified numerically in some other cases. |
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| Regulators of rank one quadratic twists |
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| J. Théor. Nombres Bordeaux 20 (2008), n°3, 601-624 (with C. Delaunay). |
| Abstract |
We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of an odd quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.
The numerical data obtained during this research is available in the table of Rank one quadratic twists of some elliptic curves. |
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| Calculs et expérimentations en théorie des nombres |
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| Habilitation à diriger des recherches soutenue le 14 juin 2007 à l'Institut Camille Jordan (Université Claude Bernard Lyon 1). |
| Résumé |
Ce mémoire présente l'ensemble de mes travaux en mathématiques après ma thèse. Ces travaux sont dans le domaine de la théorie des nombres, et plus particulièrement sur les aspects explicites de la théorie des nombres, et se partagent en trois parties.
Une première partie concerne l'étude explicite des conjectures de Stark. Dans ce domaine, les travaux présentés portent, d'une part, sur l'utilisation de la conjecture abélienne de rang \(1\) pour la construction du corps de classes de Hilbert de corps quadratiques réels, et, d'autre part, sur la vérification de diverses variations : la conjecture de Brumer-Stark sur des corps quadratiques et cubiques, la conjecture de Stark-Chinburg pour les répresentations icosaèdrales et une conjecture de Solomon portant à la fois sur les valeurs des fonctions \(L\) complexes et des fonctions \(L\) \(p\)-adiques.
Une deuxième partie décrit trois algorithmes sur les nombres \(p\)-adiques. Le premier permet de construire toutes les extensions de degré donné d'un corps \(p\)-adique. Le deuxième factorise les polynômes sur le corps des nombres \(p\)-adiques rationnels. Le troisième calcule la valeur en \(s=1\) de fonctions zêta \(p\)-adiques de corps quadratiques réels.
Une dernière partie est le fruit de collaborations avec mes collègues de Lyon et de Grenoble. Une de ces collaborations porte sur l'utilisation des modules de Drinfeld en cryptographie, les autres sur des aspects explicites de la théorie analytique des nombres. |
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| On Integers of the Form \(p + 2^k\) |
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| Acta Arith. 122 (2006), n°1, 45--50 (avec L. Habsieger). |
| Abstract |
We investigate the density of integers that may be written as \(p + 2^k\), where \(p\) is a prime and \(k\) a non negative integer. |
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| Verifying a \(p\)-adic Abelian Stark Conjecture at \(s=1\) |
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| J. Number Theory 107 (2004), n°1, 168-206 (with D. Solomon). |
| Abstract |
In a previous paper, the second author developed a new approach to the abelian \(p\)-adic Stark conjecture at \(s = 1\) and stated related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these -- the ``Weak Refined Combined Conjecture'' -- in fifteen cases. |
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| Counting Primes in Residue Classes |
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| Math. Comp. 73 (2004), n°247, 1565-1575 (with M. Deléglise and P. Dusart). |
| Abstract |
We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing \(\pi(x)\), the number of primes up to \(x\), can be used for computing efficiently \(\pi(x,k,l)\), the number of primes congruent to \(l\) modulo \(k\) up to \(x\). As an application, we computed the number of prime numbers of the form \(4n \pm 1\) less than \(x\) for several values of \(x\) up to \(10^{20}\) and found a new region where \(\pi(x,4,3)\) is less than \(\pi(x,4,1)\) near \(x = 10^{18}\). |
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| The Brumer-Stark Conjecture in Some Families of Extensions of Specified Degree |
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| Math. Comp. 73 (2004), n°245, 297-315 (with C. Greither and B. Tangedal). |
| Abstract |
As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if \(K/F\) is an abelian extension of relative degree \(2p\), \(p\) an odd prime, we prove the \(\ell\)-part of the Brumer-Stark conjecture for all odd primes \(\ell \not= p\) with \(F\) belonging to a wide class of base fields. In the same setting, we study the \(2\)-part and \(p\)-part of Brumer-Stark with no special restriction on \(F\) and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases. |
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| Corrigendum |
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| Nombre de solutions dans une binade de l'équation \(A^2 + B^2 = C^2 + C\) |
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| Enseign. Math. (2) 50 (2004), n°1-2, 147-182 (avec J.-M. Muller et J.-L. Nicolas). |
| Summary |
Let \(Q(N,\lambda)\) denote the number of integer solutions of the equation \(A^2 + B^2 = C^2 + C\) satisfying \(N\le A\le B\le C\le\lambda N-1/2\). We show that there exists an explicit constant \(\alpha(\lambda)\) such that \(Q(N,\lambda)=c(\lambda)N+O_{\lambda}(N^{7/8} \log N)\). |
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| Factorization Algorithms over Number Fields |
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| J. Symb. Comput. 38 (2004), 1429-1443. |
| Abstract |
The aim of this paper is to describe two new factorization algorithms for polynomials. The first factorizes polynomials modulo the prime ideal of a number field. The second factorizes polynomials over a number field. |
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| Utilisation des modules de Drinfeld en cryptologie |
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| C. R. Math. Acad. Sci. Paris 336 (2003), no. 11, 879-882 (avec R. Gillard, F. Leprévost et A. Panchishkin). |
| Abstract |
We present in this note public-key cryptosystem based on Drinfeld modules. |
| Remark. |
The cryptosystem described in this note has been broken, see Cryptanalysis of a cryptosystem based on Drinfeld modules |
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| Numerical Verification of the Stark-Chinburg Conjecture for Some Icosahedral Representations |
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| Experiment. Math. 12 (2003), n°4, 419-432 (with A. Jehanne and J. Sands). |
| Summary |
We provide some numerical evidence for the Stark-Chinburg conjecture in \(14\) examples of fields cut out by \(2\)-dimensional, odd Galois representations of icosahedral type. |
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| A Fast Algorithm for Polynomial Factorization over \(\mathbb{Q}_p\) |
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| J. Théor. Nombres Bordeaux 14 (2002), n°1, 151-169 (with D. Ford and S. Pauli). |
| Abstract |
We present an algorithm that returns a proper factorization of a polynomial \(\Phi(x)\) over the \(p\)-adic integers \(\mathbb{Z}_p\) (if \(\Phi(x)\) is reducible over \(\mathbb{Q}_p\)) or returns a power basis of the ring of integers of \(\mathbb{Q}_p[x]/\Phi(x)\mathbb{Q}_p[x]\) (if \(\Phi(x)\) is irreducible over \(\mathbb{Q}_p\)). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm. |
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| On the Computation of All Extensions of a \(p\)-Adic Field of a Given Degree |
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| Math. Comp. 70 (2001), n°236, 1641-1659 (avec S. Pauli). |
| Abstract |
Let \(k\) be a \(p\)-adic field. It is well-known that \(k\) has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of \(k\) of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all these extensions. |
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| Computing the Hilbert Class Field of Real Quadratic Fields |
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| Math. Comp. 69 (2000), n°231, 1229-1244 (with H. Cohen). |
| Abstract |
Using the units appearing in Stark's conjectures on the values of \(L\)-functions at \(s=0\), we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. |
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| Stark's Conjectures and Hilbert's Twelfth Problem |
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| Experiment. Math. 9 (2000), n°2, 251-260. |
| Abstract |
We give a constructive proof of a theorem of Tate, which states that (under Stark's Conjecture) the field generated over a totally real field \(K\) by the Stark units contains the maximal real abelian extension of \(K\). As a direct application of this proof, we show how one can compute explicitly real abelian extensions of \(K\). |
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| Numerical Verification of the Brumer-Stark Conjecture |
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| Algorithmic Number Theory (ANTS-IV) (W. Bosma, ed.), Lecture Notes in Comp. Sci. 1838, Springer (2000), 491-504 (avec B. Tangedal). |
| Summary |
We study the Brumer-Stark conjecture computationally in the simplest situation in which it is unproven: an extension \(K/k\) with \(k\) quadratic, \(Gal(K/k)\) isomorphic to \(\mathbb{Z}/4\mathbb{Z}\), the class group of \(K\) non trivial, \(K/\mathbb{Q}\) non Galois. We verify the conjecture in 379 such cases and study the problem of whether the power of 2 dividing the Brumer element can be replaced by a lower 2-power so that the result remains true. |
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| Algorithmes de factorisation dans les extensions relatives et applications de la conjecture de Stark à la construction des corps de classes de rayon |
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| Thèse sous la direction de F. Diaz y Diaz et M. Olivier, soutenue le 27 juin 1997 dans le laboratoire A2X (Université Bordeaux I). |
| Résumé |
Cette thèse suit deux orientations distinctes. D'une part, on expose un nouvel algorithme de factorisation des polynômes à coefficients dans un corps de nombres, ainsi qu'une généralisation de l'algorithme de factorisation modulo un nombre premier dû à Berlekamp au cas des idéaux premiers d'un corps de nombres. D'autre part, on montre comment les conjectures de Stark permettent de construire explicitement certains corps de classes de rayon sur des corps totalement réels. On donne également une table de corps de classes de Hilbert de corps totalement réels de degré 2, 3 et 4 construits par cette méthode. |
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| Unités de Stark et corps de classes de Hilbert |
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| C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), n°11, 1165-1168. |
| Abstract |
Following Stark original idea, we describe how one can use his conjectures to construct the Hilbert Class field of totally real fields by computing accurate approximations of the first derivative of some Artin L-functions. |
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