An analogue of the Dedekind–Rademacher sum and certain ray class invariants of real quadratic fields

In this paper, we consider a certain product of double sine functions as an analogue of the Dedekind–Rademacher sum. Its reciprocity formulas are established by decomposition of a certain double zeta function. As their applications, we reconstruct and refine a part of Arakawa?s work on ray class invariants of real quadratic fields, and prove directly explicit relations between various invariants which are defined in terms of the double sine function and are related to the Stark–Shintani conjecture. Moreover, in some examples, new expressions of the invariants are revealed. As two appendices, we give a new proof of Carlitz?s three-term relation for the Dedekind–Rademacher sum and a simple proof of Arakawa?s transformation formula for an analogue of the generalized Eisenstein series originated with Lewittes.     

On the Rubin–Stark Conjecture for a special class of CM extensions of totally real number fields

We use Greithers recent results on Brumers Conjecture to prove Rubins integral version of Starks Conjecture, up to a power of 2, for an infinite class of CM extensions of totally real number fields, called nice extensions under the assumption that a certain Iwasawa –invariant vanishes. As a consequence and under the same assumption, we show that the Brumer–Stark Conjecture is true for nice extensions, up to a power of 2.Mathematical Subject Classification (1991): 11R42, 11R58, 11R27Research on this project was partially supported by NSF grants DMS–9801267 and DMS–0196340.in final form: 16 July 2003     

On Shintani’s ray class invariant for totally real number fields

We introduce a ray class invariant ${X(\mathfrak{C})}$ for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula ${X(\mathfrak{C})=X_1(\mathfrak{C})\cdots X_n(\mathfrak{C})}$ where each ${X_i(\mathfrak{C})}$ corresponds to a real place. Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices. Finally, we describe the behavior of ${X_i(\mathfrak{C})}$ when the signature of ${\mathfrak{C}}$ at a real place is changed. This last result is also interpreted in terms of the derivatives L′(0, χ) of the L-functions and certain Stark units.     

On the dimension of the space of ℝ-places of certain rational function fields

We prove that for every n ∈ ? the space M(K(x ₁, …, x _n ) of ?-places of the field K(x ₁, …, x _n ) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x ₁, …, x _n )) ≤ n. For n = 2 the space M(K(x ₁, x ₂)) has covering and integral dimensions dimM(K(x ₁, x ₂)) = dim_? M(K(x ₁, x ₂)) = 2 and the cohomological dimension dim _G M(K(x ₁, x ₂)) = 1 for any Abelian 2-divisible coefficient group G.     

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000     

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and its ring of integers. Let be a Hermitian vector bundle over . In the first part of this paper we estimate the number of points of bounded height in (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙ ^N _K blown up at a linear subspace of codimension two. Received: 20 February 1998 / Revised version: 9 November 1998     

On the class S0¯ of real sequences

We prove that certain classes of sequences of positive real numbers satisfy some selection principles related to a special kind of convergence.     

On the classification of certain real rank zero C*-algebras

In this paper,we give a classification of real rank zero C^*-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C.     

New identities involving sums of the tails related to real quadratic fields

In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.     

On the divisibility of the class number of the imaginary quadratic field ℚ (\sqrt {a^2 - 4k^n } )

LetD be a positive integer which is square free, and leth (–D) denote the class number of the imaginary quadratic field 2e (–D). In this paper we prove that ifD' exp exp exp 1000,D=4k –a ², wherea, k andn are integers witha>0,k>1,n>1, then

On a certain ideal of Külshammer in the centre of a group algebra

Let G be a finite group and let F be a splitting field of characteristic $ p > 0 $ p > 0 . We show that I² = E₀, where I is a certain ideal of the centre Z of FG, and E₀ is the span of the block idempotents of defect zero.     

TheL <ω-theory of the class of Archimedian real closed fields

For the classA of uncountable Archimedian real closed fields we show that the statement TheL ^<-theory ofA is complete is independent of ZFC. In particular we have the following results:Assuming the Continuum-Hypothesis (CH) is incomplete. Conversely it is possible to build a model of set theory in which is complete and decidable. The latter can also be deduced from the Proper Forcing Axiom (PFA). In this case turns out to be equivalent to the elementary theory of the real numbers (by a quantifier-elimination procedure).Formally: is incomplete. is complete and decidable.     

On the measure of irrationality of the number π

A new estimate of the measure of irrationality of the number π is obtained. The previous result (M. Hata, 1993) is improved by means of another integral construction.     

On transverse Killing fields of metric foliations¶of manifolds with positive curvature

In this paper, we extend a result by H. Takagi on the non-existence of mutually commuting and linearly independent Killing vector fields on positively curved Riemannian manifolds. Further, a kind of “Compact Leaf Theorem” is proved for metric foliations of closed manifolds with positive sectional curvature. Received: 26 May 2000 / Revised version: 28 February 2001     

On the index of contact and multiplicities¶for bigraded rings

We prove that the index of contact p(Z,S)(c) of a set Z with a submanifold S at a point c∈ reg(Z∩S) at which Z is normal pseudo-flat along Z∩S coincides with the Samuel multiplicity of the associated graded ring and give various methods of computing this invariant at such points. Received: 12 July 2001     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

Polynomial behavior of special values of partial zeta functions of real quadratic fields at s = 0

We compute the special values of partial zeta functions at s = 0 for family of real quadratic fields K _n and ray class ideals ${\mathfrak{b}_n}$ such that ${\mathfrak{b}_n^{-1} = [1, \delta(n)]}$ where the continued fraction expansion of δ(n) ? 1 is purely periodic and terms are polynomials in n of degree bounded by d. With additional assumptions, we prove that the special values of the partial zeta functions at s = 0 are given by a quasi-polynomial of degree less than or equal to d as a function of n. We apply this to conclude that the special values of the Hecke’s L-functions at s = 0 for the family ${(K_n, \mathfrak{b}_n, \chi_n:= \chi \circ N_{K_n/\mathbb{Q}})}$ for any Dirichlet character χ behave like quasi-polynomial as well. We compute explicitly the coefficients of the quasi-polynomials. Two examples satisfying the condition are presented, and for these two families, the special values of the partial zeta functions at s = 0 are given.