Distribution of Values of Rational Maps on the F p -Points on an Affine Curve

 Let p be a prime number, let be the algebraic closure of , let be an irreducible curve in and a rational map defined on the curve . We investigate the distribution on the torus of the images through h of the -points of . Received 9 May 2001     

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

Sur la torsion de la distribution ordinaire universelle attachée à un corps de nombres

We study the torsion subgroup of the universal ordinary distribution related to a general number field. We describe a way to control this subgroup. We apply this method to the special case of an imaginary quadratic field, and we give examples of such fields where these torsion subgroups are non-trivial. Received: 4 January 2001 / Revised version: 17 May 2001     

Deux théorèmes de non-annulation de valeurs spéciales de fonctions L

In answer to questions recently raised by Merel [Mer], we prove two non-vanishing theorems for the central value of automorphic L-functions: let p be prime and let χ be a primitive character modulo p. Then for all p large enough 1. If χ is not quadratic and even, there exists a primitive weight 2 form f of level p with . 2. If χ is quadratic and even, then there exists a primitive weight 2 form f of level p with . Received: 12 March 2000 / Revised version: 26 September 2000     

On the genus of a maximal curve

The upper limit and the first gap in the spectrum of genera of -maximal curves are known, see [34], [16], [35]. In this paper we determine the second gap. Both the first and second gaps are approximately constant times , but this does not hold true for the third gap which is just 1 for while (at most) constant times q for This suggests that the problem of determining the third gap which is the object of current work on -maximal curves could be intricate. Here, we investigate a relevant related problem namely that of characterising those -maximal curves whose genus is equal to the third (or possible the forth) largest value in the spectrum. Our results also provide some new evidence on -maximal curves in connection with Castelnuovo's genus bound, Halphen's theorem, and extremal curves. Received: 1 January 2001 / Revised version: 30 July 2001 / Published online: 23 May 2002     

The Nash sheaf of a complete resolution

Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang–Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. Received: 9 April 2001 / Revised version: 26 July 2001     

On p-quaternionic pairings of non-elementary type

We show that a p-analogue to the elementary type conjecture is not true. In fact, p-quaternionic pairings of non-elementary type are constructed for every prime p≥ 3. Received: 17 March 1999 / Revised version: 19 August 1999     

The First Power Mean of the Inversion of L-Functions and General Kloostermann Sums

 The main purpose of this paper is using the estimate for classical Kloostermann sums and E. Bombieri–A. I. Vinogradov’s important work to study the first power mean of the inversion of Dirichlet L-functions with the weight of general Kloostermann sums, and give an interesting asymptotic formula. Received 30 March 2001; in revised form 12 November 2001     

Siegel modular forms and representations

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ _n =Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F _<4, is then used to establish meromorphic continuation for the spinor L-function when n=3. Received: 28 March 2000 / Revised version: 25 October 2000     

Cycles for Rational Maps of Good Reduction Outside a Prescribed Set

Let K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let R _S be the ring of S-integers of K. In the present paper we study the cycles in for rational maps of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P ₀, P ₁,… ,P _n−1) and (Q ₀, Q ₁,… ,Q _n−1) of points of are equivalent if there exists an automorphism A ∈ PGL₂(R _S ) such that P _i = A(Q _i ) for every index i∈{0,1,… ,n−1}. We prove that if we fix two points , then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P ₀, P ₁ as consecutive points is finite and depends only on S and K. We also prove that this result is in a sense best possible.     

On Fractional Part Sums: A Mean-Square Asymptotics over Short Intervals

 This article is concerned with sums 𝒮(t) = ∑ _n  ψ(tf(n/t)) where ψ denotes, essentially, the fractional part minus ?, f is a C ⁴-function with f″ ≠ 0 throughout, summation being extended over an interval of order t. We establish an asymptotic formula for ∫ _T−Λ ^T+Λ (𝒮(t))²dt for any Λ = Λ(T) growing faster than log T. Received April 30, 2001; in revised form February 15, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Pseudo-Eisenstein forms and cohomology¶of arithmetic groups. I

Let ψ be a compactly supported closed differential form on the e[P] of the Borel–Serre boundary of an arithmetically defined locally symmetric space S. A closed compactly supported differential form E (ψ) on S is defined by a pseudo-Eisenstein series attached to ψ. Its degree is the degree of ψ shifted by the codimension of e[P] in S. Non-vanishing results for the cohomology class [E(ψ)] represented by E(ψ) are obtained by use of Poincaré duality and results on cohomology classes represented by ordinary Eisenstein series. Received: 21 July 2001 / Revised version: 17 September 2001     

L-functions for Jacobi forms and the basis problem

This paper deals with Jacobi forms Φ on ?×ℂ. The Rankin–Selberg doubling method is employed to study properties of the standard L-function of Hecke–Jacobi eigenforms. It is shown that every analytic Klingen–Jacobi Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. Hecke–Jacobi cusp eigenforms of weight k > 4 and k≡ 0 mod 4 can written explicitly as a linear combination of theta series. Finally the basis problem of Jacobi forms of square-free index is solved. Received: 12 March 2000 / Revised version: 17 September 2001     

The high temperature case for the random K-sat problem

We give a completely rigorous proof that the replica-symmetric solution holds at high enough temperature for the random K-sat problem. The most notable feature of this problem is that the order parameter of the system is a function and not a number. Received: 21 April 1998 / Revised version: 24 April 2000 / Published online: 21 December 2000     

On a Diophantine Inequality Over Primes (II)

In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality is solvable, where p ₁,···,p ₅ are primes.     

Invariant subsets of rank 1 manifolds

It is proved that for a Riemannian manifold M with nonpositive sectional curvature and finite volume the space of directions at each point in which geodesic rays avoid a sufficiently small neighborhood of a fixed rank 1 vector v∈UM looks very much like a generalized Sierpinski carpet. We also show for nonpositively curved manifolds M with dim M≥ 3 the existence of proper closed flow invariant subsets of the unit tangent bundle UM whose footpoint projection is the whole of M. Received: 6 July 2000 / Revised version: 11 October 2001     

A mean value theorem on the binary Goldbach problem and its application

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p ₁ + p ₂ = n is solvable in prime variables p ₁, p ₂ such that p ₁ + 2 = P ₃, and for every sufficiently large odd integer satisfying ≢ 1(mod 6), the equation p ₁ + p ₂ + p ₃ = is solvable in prime variables p ₁, p ₂, p ₃ such that p ₁ + 2 = P ₂, p ₂ + 2 = P ₃. Here P _k denotes any integer with no more than k prime factors, counted according to multiplicity.     

Efficient separation routines for the symmetric traveling salesman problem II: separating multi handle inequalities

This paper is the second in a series of two papers dedicated to the separation problem in the symmetric traveling salesman polytope. The first one gave the basic ideas behind the separation procedures and applied them to the separation of Comb inequalities. We here address the problem of separating inequalities which are all, in one way or another, a generalization of Comb inequalities. These are namely clique trees, path, and ladder inequalities. Computational results are reported for the solution of instances of the TSPLib using the branch and cut framework ABACUS. Received: September 1999 / Accepted: August 2001?Published online February 14, 2002     

Efficient separation routines for the symmetric traveling salesman problem I: general tools and comb separation

This is the first of a series of two papers dedicated to efficient separation heuristics for violated inequalities in the context of solving to optimality instances of the Traveling Salesman Problem via branch-and-cut. In this paper we give the basic ideas behind these heuristics and design heuristics for comb separation. The second paper will deal with more complex inequalities such as Clique Tree, Star or Path and Ladder inequalities and give computational results. Received: September 1999 / Accepted: August 2001?Published online February 14, 2002     

Metric properties and exceptional sets of β-expansions over formal Laurent series

This paper is concerned with the metric properties of β-expansions over the field of formal Laurent series. We will see that there are essential differences between β-expansions of the formal Laurent series case and the classical real case. Also the Hausdorff dimensions of some exceptional sets, with respect to the Haar measure, are determined. Authors’ addresses: Bing Li and Jian Xu, School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P.R. China; Jun Wu, Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China