Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the O-constants and Ω-constants. We implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results. Our algorithms are based on a modification of Shanks ’ baby-step giantstep strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order. 1.

Let O be a real quadratic order of discriminant \Delta. For elements ff in O we develop a compact representation whose binary length is polynomially bounded in log log H(ff), log N(ff) and log \Delta where H(ff) is the height of ff and N(ff) is the norm of ff. We show that using compact representations we can in polynomial time compute norms, signs, products, and inverses of numbers in O and principal ideals generated by numbers in O. We also show how to compare numbers given in compact represention in polynomial time.

Abstract. We prove more special cases of the Fontaine-Mazur conjecture regarding p-adic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], Golod and Shafarevich showed in 1964 [8] that there exists a number field with an infinite, everywhere unramified pro-p extension. In fact it is easy to obtain many examples [8], [13], [23]. Very little is known, however, regarding the structure

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Abstract. We present a technique to recover f ∈ Q(ζp) where ζp is a primitive pth root of unity for a prime p, given its norm g = f ∗ ¯ f in the totally real field Q(ζp + ζ −1 p). The classical method of solving this problem involves finding generators of principal ideals by enumerating the whole class group associated with Q(ζp), but this approach quickly becomes infeasible as p increases. The apparent hardness of this problem has led several authors to suggest the problem as one suitable for cryptography. We describe a technique which avoids enumerating the class group, and instead recovers f by factoring Nf, the absolute norm of f, (for example with a subexponential sieve algorithm), and then running the Gentry-Szydlo polynomial time algorithm for a number of candidates. The algorithm has been tested with an implementation in PARI. 1

Abstract. Let K = Q ( √ −d) be an imaginary quadratic field and let Q ( √ 3d) be the associated real quadratic field. Starting from the Cohen-Lenstra heuristics and Scholz’s theorem, we make predictions for the behaviors of the 3-parts of the class groups of these two fields as d varies. We deduce heuristic predictions for the behavior of the Iwasawa λ-invariant for the cyclotomic Z3extension of K and test them computationally. The Cohen-Lenstra heuristics [1] give predictions for frequencies of class numbers and class groups of number fields. In the following, we investigate a related situation and a more specific question: I. Are there heuristics for the Iwasawa lambda invariants, similar to those of Cohen and Lenstra for class groups of number fields? The λ2-invariants of imaginary quadratic fields are given by a simple formula of Ferrero [4] and Kida [5] and are correspondingly not suitable for a heuristic analysis. We therefore consider the first nontrivial case, namely the λ-invariant for the cyclotomic Z3-extension of an imaginary quadratic field K as K varies. When 3 does not split in K, the frequency

Abstract. In this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes < 10000 not dividing the degree that divide the class numbers of fields of conductor ≤ 2000. Cohen–Lenstra heuristics allow us to conjecture that no larger prime divisors should exist. Previous computations have been largely limited to prime power conductors. 1.

Under the assumption of the well-known heuristics of Cohen and Lenstra (and the new extensions we propose) we give proofs of several new properties of class numbers of imaginary quadratic number fields, including theorems on smoothness and normality of their divisors. Some applications in cryptography are also discussed. 1

Résumé On explore le spectre d’un peigne de Dirac pondéré supporté par le quasicristal de Thue-Morse, et on le caractérise à un ensemble de mesure nulle près, au moyen de la Conjecture de Bombieri-Taylor, pour les pics de Bragg, et d’une autre conjecture que l’on appelle Conjecture de Aubry-Godrèche-Luck, pour la composante singulière continue. La décomposition de la transformée de Fourier du peigne de Dirac pondéré est obtenue dans le cadre de la théorie des distributions tempérées. Nous montrons que l’asymptotique de l’arithmetique des sommes p-raréfiées de Thue-Morse (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), précisément les fonctions fractales des sommes de chiffres, jouent un rôle fondamental dans la description de la composante singulière continue du spectre, combinées à des résultats classiques sur les produits de Riesz de Peyrière et de M. Queffélec. Les lois d’échelle dominantes des suites de mesures approximantes sont contrôlées sur une partie de la composante singulière continue par certaines inégalités dans lesquelles le nombre de classes de diviseurs et le régulateur de corps quadratiques

. Let D be a positive integer such that D; D \Gamma 1 are not perfect squares; denote by X 0 ; Y 0 ; X 1 ; Y 1 the least positive integers such that X 2 0 \Gamma (D \Gamma 1)Y 2 0 = 1; X 2 1 \Gamma DY 2 1 = 1; and put ae(D) = log X 1 = log X 0 : We prove here that ae(D) can be arbitrarily large. Indeed, we exhibit an infinite family of values of D for which ae(D) AE D 1=6 = log D: We also provide some heuristic reasoning which suggests that there exists an infinitude of values of D for which ae(D) AE p D log log D= log D; and that this is the best possible result under the Extended Riemann Hypothesis. Finally, we present some numerical evidence in support of this heuristic. 1. Introduction In his very entertaining book [17] (see pp. 260--263) on the properties of particular integers Roberts remarks (as did Carmichael [3] (p.33 footnote) and Beiler [2] (p. 255)) on the two Pellian equations x 2 \Gamma 1620y 2 and x 2 \Gamma 1621y 2 = 1 : "The first of these has smalle...