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 imaginary quadratic fields

The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number . The first complete results were for by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any . Indeed, after Oesterlé handled , in 1985 Serre wrote, ``No doubt the same method will work for other small class numbers, up to 100, say.' However, more than ten years later, after doing , Wagner remarked that the case seemed impregnable. We complete the classification for all , an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterlé work, which used an elliptic curve -function with an order 3 zero at the central critical point, to instead consider Dirichlet -functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large ``exceptional modulus' of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.

     

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.     

On class numbers of quadratic extensions¶over function fields

We consider class numbers of quadratic extensions over a fixed function field. We will show that there exist infinitely many quadratic extensions which have class numbers not being divisible by 3 and satisfy prescribed ramification conditions. Received: 24 October 1997 / Revised version: 26 February 1998     

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.     

Class numbers and Iwasawa invariants of quadratic fields

Let and be quadratic fields with 2 (mod 3) a positive integer. Let be the respective Iwasawa -invariants of the cyclotomic -extension of these fields. We show that if , then 3 does not divide the class number of and .

     

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.     

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.     

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.     

Class numbers of cyclotomic function fields

Let be a prime power and let be the finite field with elements. For each polynomial in , one could use the Carlitz module to construct an abelian extension of , called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of , similar to the role played by cyclotomic number fields for abelian extensions of . We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in . Two types of properties are obtained for the -parts of the class numbers of the fields in this tower, for a fixed prime number . One gives congruence relations between the -parts of these class numbers. The other gives lower bound for the -parts of these class numbers.

     

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 .