A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Moduli spaces and multiple polylogarithm motives

In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors A and B in the moduli spaces of n-pointed stable curves of genus 0, and prove that the cohomology of the pair is a framed mixed Tate motive whose period is that integral. It generalizes the results of A. Goncharov and Yu. Manin for multiple ζ-values. Then we apply our construction to the dilogarithm and calculate the period matrix which turns out to be same with the canonical one of Deligne.     

On Polynomially Normal Lattice Configurations

In this paper we extend Champernowne’s construction of a normal sequence in base b to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the constructed configuration with an arbitrary polynomial curve in the plane is a normal sequence in base b.     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

Algebraic Independence Results Related to 〈q, r〉-Number Systems

We define 〈q, r〉-linear arithmetical functions attached to the 〈q, r〉-number systems and give a necessary and sufficient condition for their generating power series to be algebraically independent over . We also deduce algebraic independence of the functions values at a nonzero algebraic number in the circle of convergence.     

Successive Minima and Best Simultaneous Diophantine Approximations

We study the problem of best approximations of a vector by rational vectors of a lattice whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.     

Equidistribution quantitative des points de petite hauteur sur la droite projective

We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over , for each prime p. Le premier auteur tient à remercier chaleureusement le project MECESUP UCN0202, ainsi que l'ACI ``Systèmes Dynamiques Polynomiaux' qui ont permis son séjour à l'Université Catholique d'Antofagasta. Le deuxième auteur est partiellement soutenu par le projet FONDECYT N 1040683. Enfin, nous remercions le rapporteur pour sa lecture détaillée de l'article.     

Note on the rings of integers of certain tame 2-galois extensions over a number field

For a number fieldF, we prove that any tame D4-extensionN/F has a normal integral basis (NIB) if and only if any tame (2, 2)-extensionK/F has a NIB.     

\lambda-presentable Morphisms, Injectivity and (Weak) Factorization Systems

We show that in a locally -presentable category, every -injectivity class (i.e., the class of all the objects injective with respect to some class of -presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of -presentable morphisms. This was known for small-injectivity classes, and referred to as the ‘small object argument.’ An analogous result is obtained for orthogonality classes and factorization systems, where -filtered colimits play the role of the transfinite compositions in the injectivity case. -presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp. pure-injectives). Finally, locally -presentable categories are shown to be cellularly generated by the set of morphisms between -presentable objects.