A cohomological characterization of Alexander schemes

Vistoli defined Alexander schemes in [19], which behave like smooth varieties from the viewpoint of intersection theory with Q-coefficients. In this paper, we will affirmatively answer Vistoli’s conjecture that Alexander property is Zariski local. The main tool is the abelian category of bivariant sheaves, and we will spend most of our time for proving basic properties of this category. We show that a scheme is Alexander if and only if all the first cohomology groups of bivariant sheaves vanish, which is an analogy of Serre’s theorem, which says that a scheme is affine if and only if all the first cohomology groups of quasi-coherent sheaves vanish. Serre’s theorem implies that the union of affine closed subschemes is again affine. Mimicking the proof line by line, we will prove that the union of Alexander open subschemes is again Alexander. Oblatum 1-XII-1997 & 14-XII-1998 / Published online: 10 May 1999     

On the Shimura variety having Mumford’s fake projective plane as a connected component

We give an explicit construction of a unitary Shimura surface that has Mumford’s fake projective plane as one of its connected components. Moreover, as a byproduct of the construction, we show that Mumford’s fake projective place has a model defined over the 7th cyclotomic field.     

ABC implies the radicalized Vojta height inequality for curves

The truncated or radicalized counting function of a meromorphic function counts the number of times that f takes a value a, but without multiplicity. By analogy, one also defines this function for numbers. In this sequel to [M. van Frankenhuijsen, The ABC conjecture implies Vojta's height inequality for curves, J. Number Theory 95 (2002) 289-302], we prove the radicalized version of Vojta's height inequality, using the ABC conjecture. We explain the connection with a conjecture of Serge Lang about the different error terms associated with Vojta's height inequality and with the radicalized Vojta height inequality.     

Gauss curvature flow: the fate of the rolling stones

We prove Firey’s 1974 conjecture that convex surfaces moving by their Gauss curvature become spherical as they contract to points. Oblatum 20-VII-1998 & 19-III-1999 / Published online: 6 July 1999     

Oriented Chow groups, Hermitian K-theory and the Gersten conjecture

We show that the oriented Chow groups of Barge–Morel appear in the E ₂-term of the coniveau spectral sequence for Hermitian K-theory. This includes a localization theorem and the Gersten conjecture (over infinite base fields) for Hermitian K-theory. We also discuss the conjectural relationship between oriented and higher oriented Chow groups and Levine’s homotopy coniveau spectral sequence when applied to Hermitian K-theory.     

Densité de points et minoration de hauteur

We obtain a lower bound for the normalised height of a non-torsion subvariety V of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of V, up to a power of a “log”. We thus extend the results of Amoroso and David on the same problem on a multiplicative group . We prove furthermore that the optimal lower bound (conjectured by David and Philippon) is a corollary of the conjecture of David and Hindry on the abelian Lehmer's problem. We deduce these results from a density theorem on the non-torsion points of V.     

A lower bound for the <Emphasis Type="Italic">r</Emphasis>-order of a matrix modulo <Emphasis Type="Italic">N</Emphasis>

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity. Author’s address: Dipartimento di Matematica e Informatica, Via Delle Scienze 206, 33100 Udine, Italy     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field. The author was partly supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

Cyclic orders: Equivalence and duality

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassé’s recent proof of Gallai’s conjecture. We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders are equivalent. We then derive short proofs of Gallai’s conjecture and a theorem “polar to” the main result of Bessy and Thomassé, using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

Dyson's theorem for curves

Let K be a number field and X₁ and X₂ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X₁×X₂. If Xi=P₁ we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell-Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.     

Sur la reéduction modulo p des polynoômes absolument irreéductibles

 On the reduction modulo p of absolutely irreducible polynomials. Let K be a number field and F(X,Y) be an absolutely irreducible polynomial of K[X,Y]. In this note, using an effective version of Riemann-Roch theorem and a version of the implicit functions theorem, we calculate a positive number A such that if ℘ is prime ideal of the ring of integers of K with norm , then the reduction of F(X,Y) modulo ℘ is an absolutely irreducible polynomial. (Réu le 1 Février 1999; en forme finale 21 Septembre 1999)     

A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K ₀-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K ₀-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart in [BaFQ]. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut (cf. [B2, Par. (l), p. 353] and also Ch. Soulé’s question in [SABK, 1.5, p. 162]). Oblatum 22-I-1999 & 20-II-2001?Published online: 4 May 2001     

Elliptic units and sign functions

In the first part of this paper we give a new definition of the elliptic analogue of Sinnott’s group of circular units. In this we essentially use the ideas discussed in Oukhaba (in Ann Inst Fourier, 55(33):753–772, 2005). In the second part of the paper we are interested in computing the index of this group of elliptic units. This question is closely related to the behaviour of the universal signed ordinary distributions introduced in loc. cit. Such distributions have a natural resolution discovered by Anderson. Consequently, we can apply Ouyang’s general index formula and the powerful Anderson’s theory of double complex to make the computations     

Divisibility of zeta functions of curves in a covering

We prove, as an analogy of a conjecture of Artin, that if is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator of the zeta function of X divides that of Y in . Then, we give some interpretations of this result in terms of semi-abelian varieties. Received: 23 July 2001     

Generalized Kneser coloring theorems with combinatorial proofs

The Kneser conjecture (1955) was proved by Lovász (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matoušek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol’nikov, Alon-Frankl-Lovász, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver’s theorem. Oblatum 17-IV-2001 & 12-IX-2001?Published online: 19 November 2001 An erratum to this article is available at .     

Re-extending Chebyshev’s theorem about Bertrand’s conjecture

In this paper, Chebyshev’s theorem (1850) about Bertrand’s conjecture is re-extended using a theorem about Sierpinski’s conjecture (1958). The theorem had been extended before several times, but this extension is a major extension far beyond the previous ones. At the beginning of the proof, maximal gaps table is used to verify initial states. The extended theorem contains a constant r, which can be reduced if more initial states can be checked. Therefore, the theorem can be even more extended when maximal gaps table is extended. The main extension idea is not based on r, though. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1701–1706, December, 2007.     

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     

Rational classes and divisors on curves of genus 2

We describe a method of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel’s morphism (X − T) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.V. G. L. Neumann supported by CNPq, Brazil     

The minimal height of quadratic forms of given dimension

Given an arbitrary n, we consider anisotropic quadratic forms of dimension n over all fields of characteristic ≠ 2 and prove that the height of an n-dimensional excellent form (depending on n only) is the (precise) lower bound of the heights of all forms of dimension n. The second and third authors were partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS. The James D.Wolfensohn Fund and The Ellentuck Fund support is acknowledged by the second author. Received: 9 December 2005