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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 |
| Review |
The author uses one of Stark's conjectures for constructing Hilbert class fields H over totally real algebraic number fields k. Provided that the conjecture is true he demonstrates that H can be obtained from sufficiently good approximations of the values L'(0,χ) for suitable characters χ. (Once a candidate for H is found, it can of course be shown to be the Hilbert class field of k independently from Stark's conjecture.) Two examples of Hilbert class fields of cubic fields k of class numbers 3, 5 are presented. (Reviewed by M. Pohst.) |
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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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| Computing the Hilbert Class Field of Real Quadratic Fields |
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| Math. Comp. 69 (2000), n°231, 1229-1244 (with H. Cohen) |
| Review |
This paper presents an efficient technique for computing the Hilbert class field Hk of a real quadratic number field k. The method uses a special case of Stark's conjectures on the values of L-functions at s = 0; however, the field obtained by the algorithm can be proved or disproved to be the Hilbert class field independently of the conjecture. The authors implemented this method as part of the software package PARI/GP.
The algorithm proceeds essentially in four stages. The first step determines a quadratic extension K of Hk which is abelian over k so that one of the real embeddings of k stays real in K and the other becomes complex. (The abelian rank-one Stark conjecture is used in connection with this field K.) Next, approximations of the conjugates over k of a generator of the extension Hk /k are computed; this results in approximations of the coefficients of a generating polynomial of Hk /k. The third step finds these coefficients, which are algebraic integers, from the approximations. The final stage checks whether the polynomial thus obtained is in fact a generating polynomial for Hk /k.
The paper concludes with an example and tables of Hilbert class fields of all real quadratic number fields of discriminant less than 2000. (Reviewed by R. Scheidler.) |
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| Stark's Conjectures and Hilbert's Twelfth Problem |
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| Experiment. Math. 9 (2000), n°2, 251-260 |
| Review |
Let K be a totally real number field and let v be an infinite place of K. Consider a finite abelian extension N/K in which v is totally split. Let S denote the set of infinite places of K together with the finite places ramified in N/K. A conjecture by Stark predicts the existence of an S-unit e in N having a certain natural connection to the derivatives at s=0 of the Artin L-functions LS (s, χ) attached to the characters χ of Gal(N/K). The author denotes by KStark the extension of K generated by all such units e with varying N (and varying places above v of N). By a theorem of Tate, the Stark conjecture implies that the maximal real abelian extension of K is contained in KStark. The author gives a new proof for this by constructing, for any finite real abelian extension L/K, Stark units e1 ,...,er such that L=Q(e1 + e1 -1,..., er + er -1). He in fact computes LS '(0, χ) from a formula in which the main factor (from the computational point of view) is the value at s=1 of the function Λ(s, χ) defined in conjunction with the functional equation of Dirichlet's L-function L(s, χ). For the computation of Λ(1, χ) he uses a result of E. C. Friedman [in Séminaire de Théorie des Nombres, 1987--1988 (Talence, 1987--1988), Exp. No. 5, 23 pp., Univ. Bordeaux I, Talence, 1988; MR 90i:11136]. Since the whole procedure is based on a conjecture, a method is presented to check the correctness of the result. In an example, the author constructs the Hilbert class field L=Q(e1 + e1 -1) of K=Q(√(82)); another example shows the construction of a ray class field of a real cubic field. (Reviewed by T. Metsänkylä.) |
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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) |
| Review |
Let K/k be an abelian extension of number fields, with Galois group G=G(K/k), where k is totally real and K totally complex. For each σ in G and for the set S consisting of the Archimedean primes of k and the prime ideals in k that ramify in K/k, denote by ζS (s,σ) the partial zeta function of K/k. Let γ =w Σσ ∈ G ζS (0,σ)σ-1, where w is the number of roots of 1 in K. Then γ is an element of the group ring Z[G], called the Brumer element of K/k. The Brumer-Stark conjecture asserts that γ annihilates the class group of K; more precisely, γ maps each fractional ideal to a principal ideal whose generator α can be chosen so that K(α1/w)/k is abelian and the absolute value of α is 1 at all Archimedean primes of K. The authors study this conjecture computationally in the simplest situation in which it is unproven: k quadratic, G isomorphic to Z/4Z, the class group of K nontrivial, K/Q non-Galois. They verify the conjecture in 379 such cases. They are also concerned with the problem of whether the exact power of 2 dividing γ can be replaced by a lower 2-power so that the result remains true.
The article contains one detailed example and two tables listing all the 379 settled extensions K/k and the class groups of the corresponding fields K. (Reviewed by T. Metsänkylä.) |
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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) |
| Review |
Let p be a fixed prime number, Qp denote the field of p-adic rational numbers, Q-p an algebraic closure of Qp , k an intermediate field, finite over Qp , and P the prime ideal of k. Using his lemma, Krasner gave formulae for the number of finite extensions of a given degree m > 1 and discriminant Pd , d non-negative. The aim of the present paper is, following Krasner's methods, to get a set of polynomials defining all of these extensions.
The general case is reduced in Section 2 to the computation of totally ramified extensions. An ultrametric distance on the set of Eisenstein polynomials of degree m is introduced in Section 4 and used in Section 5 in the construction of a set of polynomials defining all totally ramified extensions of degree m. In Section 8 are given algorithms for the computation of a minimal set of polynomials generating all the extensions of degree m and discriminant Pd. This interesting paper is completed by two examples (Section 9) and some discussions on future developments (Section 10). (Reviewed by Victor Alexandru.) |
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| A Fast Algorithm for Polynomial Factorization over Qp |
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| J. Théor. Nombres Bordeaux 14 (2002), n°1, 151-169 (with D. Ford and S. Pauli) |
| Review |
An algorithm is presented for factorization of a polynomial with integral p-adic coefficients into irreducible factors. It is based on the observation that if Φ ∈ Zp [X] is monic and square-free then it is reducible over Qp if and only if for some polynomial θ ∈ Qp [X] the resultant of Φ(X) and t-θ(X) lies in Zp [t] and has at least two distinct irreducible factors mod p. The authors do not discuss the running time of this algorithm, but provide experimental data showing that it is faster than the "Round Four" algorithm of H. Zassenhaus. They state that this algorithm will be implemented in the forthcoming version of the package PARI. (Reviewed by W. Narkiewicz.) |
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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) |
| Review |
This interesting work provides the first numerical evidence for the Stark-Chinburg conjecture for some non-solvable Galois extensions of the rational numbers.
More precisely, the authors discuss 14 examples of fields cut out by 2-dimensional, odd Galois representations of icosahedral type. Thus, the Galois groups of these fields embed into GL2 (C) such that the canonical image in PGL2 (C) is isomorphic to the alternating group on 5 symbols. Such Galois representations can be constructed by picking candidates (from existing tables) for the A5 -field cut out by the attached projective representation. One then lifts this projective representation to a linear 2-dimensional representation (which is always possible). Computing the coefficients of the Artin L-series of the lift is, however, not a trivial task. The choice of the above 14 examples is made so as to make this computation manageable; more precisely, the examples were chosen so that a lift with odd, quadratic determinant exists.
Odd, icosahedral representations over Q are strongly expected---and in many cases proved to be---modular of weight 1, i.e., are such that the coefficients of the L-series coincide with the Fourier coefficients of some newform of weight 1.
Assuming this for a given representation ρ, one can meaningfully speak of its L-function at the point s=0. Roughly speaking, the Stark-Chinburg conjecture predicts the existence of a real unit of the field cut out by ρ with special properties related to the value of the derivative of the L-function at s=0.
The first section of the paper gives a clearly written account of the statement of the Stark-Chinburg conjecture, especially in the 2-dimensional case at hand. (Reviewed by Ian Kiming.) |
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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) |
| Review |
A new approach to the p-adic abelian Stark conjecture at s=1 was developed in the previous papers of D. R. Solomon [see J. Number Theory 94 (2002), no. 1, 10--48; MR1904961 (2003d:11169); Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 379--417; MR1906480 (2003d:11168)]. The conjectures of Harold Stark were made in the 1970's and 1980's [Advances in Math. 7 (1971), 301--343 (1971); MR0289429 (44 #6620); Advances in Math. 17 (1975), no. 1, 60--92; MR0382194 (52 #3082); Advances in Math. 22 (1976), no. 1, 64--84; MR0437501 (55 #10427); Adv. in Math. 35 (1980), no. 3, 197--235; MR0563924 (81f:10054)] concerning the values at s=1 and s=0 of complex Artin L-series attached to Galois extensions of number fields K/k. A systematic approach to the Stark conjectures was presented in the book of J. T. Tate [Les conjectures de Stark sur les fonctions L d'Artin en s=0, Lecture notes by Dominique Bernardi and Norbert Schappacher, Progr. Math., 47, Birkhäuser Boston, Boston, MA, 1984; MR0782485 (86e:11112)], and recent developments on the Stark conjectures were discussed in a paper of D. R. Hayes [J. Reine Angew. Math. 497 (1998), 83--89; MR1617427 (99h:11131)]. In the most general terms these conjectures concern the special values of Artin L-functions of number fields and their analogies, relating them to certain "regulators of S-units" and analogous objects.
In the present paper a numerical investigation is developed and applied to one of the analogous p-adic conjectures---the "weak refined combined conjecture"---in 15 cases. This conjecture (which is Conjecture 3.16 of [D. R. Solomon, op. cit.; MR1906480 (2003d:11168)]) was motivated by K. Rubin's refined, higher order Stark conjecture at s=0 for an abelian extension of a number field [K. Rubin, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 33--62; MR1385509 (97d:11174)].
A p-adic version of Shintani's method is used for a totally real finite extension K of Q of degree n, and its maximal order OK .
If Clf (K) is the group of ray classes of fractional ideals I of K, modulo f, N(I)=[OK :I] for I subset of OK then according to Shintani, the special values at non-positive integers of the partial zeta function ζK (a, f, s) = ∑I ∈ a N(I)-s are rational numbers which can be expressed in terms of certain generating functions which generalize the generating function of Bernoulli numbers [see T. Shintani, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393--417; MR0427231 (55 #266)].
These generating functions are associated to "cone decompositions" in F⊗ R which are defined non-canonically [see also H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge Univ. Press, Cambridge, 1993; MR1216135 (94j:11044) (Chapter I)].
Extensive numerical examples are given on 7 pages for 15 real quadratic fields and for small primes p=3,11,13, 17, 19, 41, using PARI/GP. (Reviewed by Alexey A. Panchishkin.) |
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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) |
| Review |
Let π(x) denote the numbers of primes up to x. In the 1870's, Meissel designed a method for computing π(x) that is more efficient than the sieve of Eratosthenes. J. C. Lagarias, V. S. Miller and A. M. Odlyzko [Math. Comp. 44 (1985), no. 170, 537-560] improved this to a method requiring O(x2/3/log x) time. M. Deléglise and J. Rivat [Math. Comp. 65 (1996), no. 213, 235-245] made further improvements and obtained an algorithm requiring O(x2/3/log2 x) time.
Let π(x,k,l) denote the number of primes p ≤ x with p congruent to k mod l. In this article, the authors show how the above-mentioned algorithm of Deléglise and Rivat may be modified for computing π(x,k,l). The resulting algorithm also requires O(x2/3/log2 x) time. As an application, they compute π(x,4,±1) for several values of x up to 1020, and they find a previously unknown region where π(x,4,3) < π(x,4,1). The new region is near x=1018. (Reviewed by S. W. Graham.) |
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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) |
| Review |
Let K/F be a finite abelian extension of number fields with Galois group G, K totally complex, and F totally real. The Brumer conjecture (B) and the Brumer-Stark conjecture (BS) predict that certain elements of the group ring Z[G] annihilate the ideal class group CK of K. In this paper the authors first establish an important link between (B) and (BS). This link allows one to translate progress on (B) into new results on (BS).
The paper is divided into two parts. The first one presents the theoretical results and the second one presents the computational work, where certain situations not covered in full in the theoretical part are studied. As a starting point the equivalence between (BS) and (BS)l , the l-primary analog of (BS), for all l and the link between (B)l and (BS)l are proved.
One of the main results is that if K/F is an abelian extension of degree 2p, p an odd prime, then (BS)p holds in general with the exception of two well-defined specific classes and if K/F is a sextic abelian extension with K totally complex and F either a real quadratic or a real cubic field, then (BS) holds in full with some exceptions. Some cases of these exceptions are solved in the computational part of the paper. (Reviewed by Gabriel D. Villa-Salvador.) |
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| Nombre de solutions dans une binade de l'équation A2 + B2 = C2 + C |
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| Enseign. Math. (2) 50 (2004), n°1-2, 147-182 (avec J.-M. Muller et J.-L. Nicolas) |
| Review |
Let Q(N,λ) denote the number of integer solutions of the equation in the title, subject to the conditions N≤ A≤ B≤ C≤λ N-1/2. This quantity arises from a question in computer science.
It is shown that Q(N,λ)=c(λ)N+Oλ (N7/8 log N) with an explicitly given constant c(λ). When λ=2 a numerically explicit estimate for the error term is presented. Extensive numerical data for this case are also given. The proof of the main result uses a simple linear transform to put the equation into the shape X2+1=Z2-Y2, and then employs ideas from the work of C. Hooley [Acta Math. 117 (1967), 281--299; MR0204383 (34 #4225)] on large prime factors of x2+1. The argument ultimately depends on estimates for the Kloosterman sum. (Reviewed by D. R. Heath-Brown.) |
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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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| On Integers of the Form p + 2k |
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| Acta Arith. 122 (2006), n°1, 45--50 (avec L. Habsieger) |
| Review |
Let ρ(x) denote the proportion of odd integers n≤ x which can be written as the sum of a prime and a power of 2. It was shown by N. P. Romanov [Math. Ann. 109 (1934), no. 1, 668--678; MR1512916; Zbl 0009.00801] that if d0 =liminf ρ(x), then d0 must be strictly positive. In the other direction, P. Erdös [Summa Brasil. Math. 2 (1950), 113--123; MR0044558 (13,437i)] showed that there is an infinite arithmetic progression of odd numbers not representable as described, whence d1 = limsup ρ(x)<1. The aim of the present paper is to estimate d0 and d1 , and it is indeed shown that 0.1866≤ d0 ≤ d1 ≤0.9819. The lower bound uses a refined version of Romanov's method, due to J. Pintz and I. Z. Ruzsa [Acta Arith. 109 (2003), no. 2, 169--194; MR1980645 (2004c:11185)], while the upper bound comes from a numerical search involving powers of 2 to a suitable large highly composite modulus. (Reviewed by D. R. Heath-Brown.) |
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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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| Numerical evidence toward a 2-adic equivariant ''main conjecture'' |
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| To appear in Experimental Mathematics (with A. Weiss) |
| Abstract |
Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that for the prime 2 and a dihedral extension of order 8 over Q, this conjecture is equivalent to a congruence condition of the coefficients of a power series with 2-adic integral coefficients constructed using the 2-adic L-series associated to the extension. We then verify that this congruence condition holds in a large number of examples. |
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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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| Testing the Congruence Conjecture for Rubin-Stark Elements |
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| Preprint 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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