Class numbers of quadratic function fields and continued fractions

A function field version of a theorem of F. Hirzebruch relating continued fractions to class numbers of quadratic number fields is established. Our approach is based on Artin's thesis and Zagier's proof of Hirzebruch's theorem. Some of our results seem to be of independent interest, e.g. explicit formulas for Zeta functions of real quadratic function fields.     

1-Universal binary and ternary Hermitian lattices over imaginary quadratic fields

The Ramanujan Journal - A positive definite Hermitian lattice is said to be 1-universal if it represents all positive definite unary Hermitian lattices, including both free and non-free Hermitian...     

Class numbers of complex quadratic fields

Congruence conditions on the class numbers of complex quadratic fields have recently been studied by various investigators, including Barrucand and Cohn, Hasse, and the author. In this paper, we study the class number of Q(√ ? pq), where p ≡ q (mod 4) are distinct primes.     

Class numbers of indefinite binary quadratic forms

We determine the asymptotic average sizes of the class numbers of indefinite binary quadratic forms when ordered by the sizes of their corresponding fundamental units. The proofs make use of the Selberg trace formula.     

Growth of class numbers of positive definite ternary unimodular Hermitian lattices over imaginary number fields

We give a lower bound for class numbers of unimodular ternary Hermitian lattices over imaginary quadratic fields. This shows that class numbers of unimodular Hermitian lattices grow infinitely as the field discriminants grow.     

Class numbers of indefinite binary quadratic forms II

We obtain an asymptotic formula for the averages of class numbers of indefinite binary quadratic forms taken along certain polynomial sequences. It is found that along these sequences the class number is essentially as large as possible.     

2-universal Hermitian lattices over imaginary quadratic fields

A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields.     

Representation of numbers by quadratic forms in algebraic number fields

Let K be the totally real field of algebraic numbers of degree n=[k2e] with the discriminant D=D(K); t=t(x₁, ..., x_s) a totally positive quadratic form of the determinant d>0 over the ring of integers from the field K; S4. Let be the number of representations over of the number m by the form a complete singular series. It is proved that for given s and n, there exists a constant c such that for N(d)>0 it is not true that for all m with m totally positive.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 68–77, 1986.     

On nondecomposable positive definite Hermitian forms over imaginary quadratic fields

Methods are presented for the construction of nondecomposable positive definite integral Hermitian forms over the ring of integers R_m of an imaginary quadratic field ℚ(√−m). Using our methods, one can construct explicitly an n-ary nondecomposable positive definite Hermitian R_m-lattice ( L, h) with given discriminant 2 for every n⩾2 (resp. n⩾13 or odd n⩾3) and square-free m = 12 k + t with k⩾1 and t∈ (1,7) (resp. k⩾1 and t = 2 or k⩾0 and t∈ 5,10,11). We study also the case for discriminant different from 2.     

Polynomial sums over automorphs of a positive definite binary quadratic form

Let P(X) be a homogeneous polynomial in X = (x, y), Q(X) a positive definite integral binary quadratic form, and G the group of integral automorphs of Q(X). Let A(m) = {N ∈ $\text{Z}$ × $\text{Z}$ : Q(N) = m}. It is shown that if Σ_N∈A(m)P(N) = 0 for each m = 1, 2, 3,… then Σ_U∈GP(UX) ≡ 0.     

Class number indivisibility for quadratic function fields

Let M?5. For any odd prime power q and any prime ??q, we show that there are at least pairwise coprime D∈Fq[T] which are square-free and of odd degree ?M, such that ? does not divide the class number of the complex quadratic functions fields .     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

β-Expansions in algebraic function fields over finite fields

The present paper deals with an algebraic function field analogue of β-expansions of real numbers. It completely characterizes the sets with eventually periodic and finite expansions. These characterizations are unknown in the real case.     

Solomon's zeta functions over algebraic function fields

L. Solomon recently introduced a wide-ranging but concrete generalization of the Riemann and Dedkind zeta functions, as well as of Hey's zeta function for a simple algebra over the rationals. The coefficients of Solomon's zeta function give the numbers of certain types of sublattices in a given lattice over an order in a semisimple rational algebra. This paper studies the analogous zeta function and coefficients which arise for an order in a semi-simpleF _q (X) -algebra, whereF _q (X) is a field of rational functions over a finite fieldF _q . Use is made of the analogues for function fields of results on his zeta functions which were first conjectured by Solomon, and later established by C J Bushnell and l Reiner.