A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field

Let k be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic forms over k. We have two main results. The first result is on the principal part of the global zeta function associated with the prehomogeneous vector space. The second result is on a mean value theorem for degree zero divisor class groups of quadratic extensions over k, which is a consequence of the first one.     

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.     

Worst-case complexity bounds for algorithms in the theory of integral quadratic forms

The problem of factoring an integer and many other number-theoretic problems can be formulated in terms of binary quadratic Diophantine equations. This class of equations is also significant in complexity theory, subclasses of it having provided most of the natural examples of problems apparently intermediate in difficulty between P and NP-complete problems, as well as NP-complete problems [2, 3, 22, 26]. The theory of integral quadratic forms developed by Gauss gives some of the deepest known insights into the structure of classes of binary quadratic Diophantine equations. This paper establishes explicit polynomial worst-case running time bounds for algorithms to solve certain of the problems in this theory. These include algorithms to do the following: (1) reduce a given integral binary quadratic form; (2) quasi-reduce a given integral ternary quadratic form; (3) produce a form composed of two given integral binary quadratic forms; (4) calculate genus characters of a given integral binary quadratic form, when a complete prime factorization of its determinant D is given as input; (5) produce a form that is the square root under composition of a given form (when it exists), when a complete factorization of D and a quadratic nonresidue for each prime dividing D is given as input.     

Cusp forms in S 6(Γ 0(23)), S 8(Γ 0(23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables

In this study we find bases of S ₆(Γ ₀(23)), S ₈(Γ ₀(23)) and obtain explicit formulae for the number of representations of numbers by some quadratic forms in 12 and 16 variables that are direct sums of binary quadratic forms $F_{1}=x_{1}^{2}+x_{1}x_{2}+6x_{2}^{2}$ and $\varPhi_{1}=2x_{1}^{2}+x_{1}x_{2}+3x_{2}^{2}$ (or its inverse) with discriminant ?23.     

Exact arithmetic on the Stern–Brocot tree

In this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials.     

On character sums and class numbers

A mean-value estimate for character sums with real characters is proved in an elementary way. The application is a proof of a conjecture on the error term in the asymptotic expression of the sum Σ_d≤xh^k(?d), where h(?d) is the class number of the quadratic field with the negative discriminant ?d, and k is an integer ≥ 2.     

Two problems of the theory of quadratic maps

Given quadratic forms q ₁, …, q _k , two questions are studied: Under what conditions does the set of common zeros of these quadratic forms consist of the only point x = 0? When is the maximum of these quadratic forms nonnegative or positive for any x ≠ 0? Criteria for each of these conditions to hold are obtained. These criteria are stated in terms of matrices determining the quadratic forms under consideration.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Dihedral Congruence Primes and Class Fields of Real Quadratic Fields

We show that for a real quadratic field F the dihedral congruence primes with respect to F for cusp forms of weight k and quadratic nebentypus are essentially the primes dividing expressions of the form ε^k−1₊±1 where ε₊ is a totally positive fundamental unit of F. This extends work of Hida. Our results allows us to identify a family of (ray) class fields of F which are generated by torsion points on modular abelian varieties.     

Arithmetics of binary quadratic forms,symmetry of their continued fractions and geometry of their de Sitter world

This article concerns the arithmetics of binary quadratic forms with integer coefficients, the De Sitters world and the continued fractions.Given a binary quadratic forms with integer coefficients, the set of values attaint at integer points is always a multiplicative tri-group. Sometimes it is a semigroup (in such case the form is said to be perfect). The diagonal forms are specially studied providing sufficient conditions for their perfectness. This led to consider hyperbolic reflection groups and to find that the continued fraction of the square root of a rational number is palindromic.The relation of these arithmetics with the geometry of the modular group action on the Lobachevski plane (for elliptic forms) and on the relativistic De Sitters world (for the hyperbolic forms) is discussed. Finally, several estimates of the growth rate of the number of equivalence classes versus the discriminant of the form are given.Partially supported by RFBR, grant 02-01-00655.     

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229-247], it was proved that the Selberg zeta function for SL₂(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL₂(Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.     

On the rational equivalence of full decomposable forms

Let k be any finite normal extension of the rational field Q and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. Consider the set F of the full decomposable forms which correspond to the invertible fractional ideals of D. In a recent paper the author has given arithmetic criteria to determine which classes of improperly equivalent forms in F integrally represent a given positive rational integer m. These criteria are formulated in terms of certain integer sequences which satisfy a linear recursion and need only be considered modulo the primes dividing m. Here, for the most part, we consider partitioning F under rational equivalence. It is a found that the set of rationally equivalent classes in F is a group under composition of forms analogous to Gauss' and Dirichlet's classical results for binary quadratic forms. This leads us to given criteria as before to determine which classes of rationally equivalent forms in F rationally represent m. Moreover, by applying the genus theory of number fields, we find arithmetic criteria to determine when everywhere local norms are global norms if the Hasse norm principle fails to hold in $\text{k}\text{Q}$.     

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.     

On a formal analogue of the Bernoulli numbers

Using properties of the modular forms G_k, it is shown that G_K(z) = α_kω^2k where z is an integer in an imaginary quadratic field and α_k is algebraic and involves analogues of the Bernoulli numbers. A recursion formula (3) is given for these numbers.     

La classe invariante d'une forme binaire

We explain how to associate to any irreducible binary form an element of the class group in the corresponding ring. This class does not depend on the choice of the form modulo the action of SL₂. The question is to generalize the classical theory of quadratic forms. To cite this article: D. Simon, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids

For any non-uniform lattice Γ in SL₂(?), we describe the limit distribution of orthogonal translates of a divergent geodesic in Γ\SL₂(?). As an application, for a quadratic form Q of signature (2, 1), a lattice Γ in its isometry group, and v ₀ ∈ ?³ with Q(v ₀) > 0, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit v ₀Γ of norm at most T, when the stabilizer of v ₀ in Γ is finite. Our result in particular implies that for any non-zero integer d, the smoothed count for the number of integral binary quadratic forms with discriminant d ² and with coefficients bounded by T is asymptotic to c · T log T + O(T).     

Trace formulae and applications to class numbers

In this paper we compute the trace formula for Hecke operators acting on automorphic forms on the hyperbolic 3-space for the group PSL₂( $\mathcal{O}_K $ ) with $\mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field K of class number H _K > 1. Furthermore, as a corollary we obtain an asymptotic result for class numbers of binary quadratic forms.     

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.     

Rings of integral quaternions

A quaternion order derived from an integral ternary quadratic form f = Σa_ijx_ix_j of discriminant d = 4 det (a_ij) is m-maximal if m is not divisible by any prime p such that p² | d, or p 6; d and c_p = 1. If R is m-maximal and m is a product p₁ … p_r of primes, then any primitive element α of R has unique right-divisor ideals of each norm p₁ … p_k (k = 1, …, r). This generalizes Lipschitz's ninety-year-old theorem. We characterize m-maximal orders, study their ideals, and show how the preceding result yields formulas for the number of representations of integers by certain quaternary quadratic forms.     

Red numbers and quadratic field

A natural number is said red if the period of the continued fraction of its square root has odd length. For any quadratic field \mathbbQ(?D)\mathbb{Q}(\sqrt{D}), we show how the parity of the periods length of the continued fractions of its irrationalities depends on the redness of their discriminant.