Reduction theory over quadratic imaginary fields

Hurwitz developed a reduction theory for real binary quadratic forms of positive discriminant based on least-remainder continued fractions. For each quadratic imaginary field k, we develop a similar theory for complex binary quadratic forms of nonzero discriminant. This uses a Markov partition for the geodesic flow over the quotient of hyperbolic 3-space by the Bianchi group B_k. When k has a Euclidean algorithm, our theory is based on least-remainder continued fractions.     

Special values of generalized \mathbf{\lambda} functions at imaginary quadratic points

We study a modular function Λ _k,? that is one of generalized λ functions. We show that Λ _k,? and the modular invariant function j generate the modular function field with respect to the modular subgroup Γ ₁(N). Further, we prove that Λ _k,? is integral over Z[j]. From this result we obtain that a value of Λ _k,? at an imaginary quadratic point is an algebraic integer and generates a ray class field over a Hilbert class field.     

Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.     

A note on Pall partitions over finite fields

A Pall partition for a quadratic space V is a collection of disjoint (except for {0}) maximal totally isotropic subspaces whose union contains all of the isotropic vectors in V. In this paper it is shown that no non-degenerate quadratic space of dimension 4k+1, k?1, over a finite field of odd characteristic can have a Pall partition. The method of proof consists of assuming such a partition exists and showing by various counting arguments that this leads to the existence of an impossible array of ordered pairs.     

A note on isometries over local or finite fields

Let K   be a finite or a local field of characteristic ≠2$\ne 2$. We give a new proof, in a slightly more general case, for the following classical theorem of Milnor. If two unitary operators of a quadratic space over K have the same irreducible minimal polynomial, then they are conjugate via a unitary operator. Our arguments are short and elementary.     

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.      

Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique

In this paper, we propose two exact algorithms for the GQAP (generalized quadratic assignment problem). In this problem, given M facilities and N locations, the facility space requirements, the location available space, the facility installation costs, the flows between facilities, and the distance costs between locations, one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized. This problem generalizes the well-known quadratic assignment problem (QAP). Both exact algorithms combine a previously proposed branch-and-bound scheme with a new Lagrangean relaxation procedure over a known RLT (Reformulation-Linearization Technique) formulation. We also apply transformational lower bounding techniques to improve the performance of the new procedure. We report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time. Six of these instances were open.     

Hyperelliptic curves and homomorphisms to ideal class groups of quadratic number fields

For the function field K of hyperelliptic curves over Q we define a subgroup of the ideal class group called the group of Z-primitive ideals. We then show that there are homomorphisms from this subgroup to ideal class groups of certain quadratic number fields.     

A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randi? index χ = χ(G) of a graph G   is defined by χ(G)=_∑(uv)(_δuδv)^-1/2$\chi (G)={\sum }_{(\mathit{uv})}({\delta }_u{\delta }_v{)}^{-1/2}$, where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number n_k of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randi? index attained its minimum value with given k (k ? n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number n_k of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.     

Pairwise commuting derivations of polynomial rings

We prove that n pairwise commuting derivations of the polynomial ring (or the power series ring) in n variables over a field k of characteristic 0 form a commutative basis of derivations if and only if they are k-linearly independent and have no common Darboux polynomials. This result generalizes a recent result due to Petravchuk and is an analogue of a well-known fact that a set of pairwise commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector.     

APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

Local densities of 2-adic quadratic forms

In this paper, we give an explicit from formula for the local density number of representing a two by two 2-integral matrix T by a quadratic 2-integral lattice L over . The non-dyadic case was dealt in a previous paper. The special case when L is a (maximal) lattice in the space of trace zero elements in a quaternion algebra over yields a clean and interesting formula, which matches up perfectly with the non-dyadic case in terms of the Gross-Keating invariants. This work is used to compare the central derivative of a genus two Eisenstein series with certain generating function of arithmetic 0-cycles on certain Shimura curve, in a joint work with Kudla and Rapoport.     

On generic and maximal k-ranks of binary forms

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k in any number of variables. The second one (by the fourth author) deals with the maximal k-rank of binary forms. We settle the first conjecture in the cases of two variables and the second in the first non-trivial case of the 3-rd powers of quadratic binary forms.     

A characterization of linear polynomials

We give a characterization of linear polynomials over a number field k as the only non-constant polynomials f for which the equation f(x)=g(y) has a k-rational solution for each polynomial g over k.     

Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula

Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y)     

Cyclicity of the unramified Iwasawa module

For a number field k and a prime number p, let k _?? be the cyclotomic Z _p -extension of k with finite layers k _n . We study the finiteness of the Galois group X _?? over k _?? of the maximal abelian unramified p-extension of k _?? when it is assumed to be cyclic. We then focus our attention to the case where p?=?2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of k _n . As a consequence, we determine the complete list of real quadratic number fields for which X _?? is cyclic non trivial. We then apply these results to the study of Greenberg??s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya.     

Quadratic function fields with invariant class group

Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case.     

Hasse principle for classical groups over function fields of curves over number fields

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thélène conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map has trivial kernel, denoting the set of places of k.The conjecture is true if G is of type 1A∗, i.e., isomorphic to SL₁(A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thélène, being an immediate consequence of the theorems of Merkurjev-Suslin [S1] and Kato [K]. Gille [G] proves the conjecture if G is defined over k and F=k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types 2A∗, B_n, C_n, D_n (D₄ nontrialitarian), G₂ or F₄; a group is said to be of type 2A∗, if it is isomorphic to SU(B,τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′|k involution τ.     

Toroidal automorphic forms for some function fields

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL₂ is toroidal if all its right translates integrate to zero over all non-split tori in GL₂, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g?1, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an “automorphic” proof for the Riemann hypothesis for the zeta function of those curves.     

Imaginary quadratic function fields

In Bautista-Ancona and Diaz-Vargas (2006) [B-D] a characterization and complete listing is given of the imaginary quadratic extensions K of k(x), where k is a finite field, in which the ideal class group has exponent two and the infinite prime of k(x) ramifies. The objective of this work is to give a characterization and list of these kind of extensions but now considering the case in which the infinite prime of k(x) is inert in K. Thus, we get all the imaginary quadratic extensions of k(x), in which the ideal class group has exponent two.