On the representation of integers by quadratic forms

Let n 4 and let Q [X₁, ..., X_n] be a non-singular quadraticform. When Q is indefinite we provide new upper bounds for theleast non-trivial integral solution to the equation Q = 0, andwhen Q is positive definite we provide improved upper boundsfor the greatest positive integer k for which the equation Q= k is insoluble in integers, despite being soluble modulo everyprime power.     

On representation of numbers by binary forms

An effective method is given for finding all rational points, the denominators of which are formed from a finite number of fixed primes, on the curvef (x, y)=A, wheref (x, y) is a binary form of degree three at least, irreducible over the field of rational numbers, and A is a rational number.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 369–376, April, 1968.     

关于四元二次型表整数个数的均值估计

利用数论函数的可乘性和狄里克雷卷积方法,研究了用四元二次型表整数个数的均值估计,并且得到了相应的渐近公式,推广了相关结果.     

Application of Eichler's reduction formula to the representation of numbers by certain quaternary quadratic forms

It is shown how Eichler's formula can be used to obtain exact formulas for the number of representations of positive integers by certain positive-definite quadratic forms of principal type in four variables, in terms of traces of Frobenius endomorphisms of appropriate elliptic curves over fields of finite characteristic; as examples, some new exact formulas are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 71–76, January, 1971.     

The application of the Shimura lifting to the representation of large numbers of ternary quadratic forms

One obtains an asymptotic formula for the number of the representations of numbers, divisible by a large square, by a positive ternary integral quadratic form. One gives an estimate of the remainder, unimprovable with respect to the quadratic part and uniform with respect to the square-free part of the represented number.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 21–26, 1985.     

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 quadratic forms

The present paper is related to the Jordan—von Neumann characterization of inner product spaces, to the Halperin problem concerning quadratic forms, to some results of the present author on quadratic and sesquilinear forms and to recently obtained results of C. T. Ng and of J. Vukman.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

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.     

Representation of numbers by integral isotropic quadratic forms

Under certain assumptions, we prove that the integral points (x₁,x₂,x₃) ³ with g.c.d. (x₁,x₂,x₃)=1 are asymptotically uniformly distributed on the surface of the two-sheeted hyperboloid t(x₁,x₂,x₃)=m, where f is an isotropic quadratic form over , for given modulus g. This is a generalization of the results of Linnik [3] and Pachev [4].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 66–67, 1986.I would like to thank A. V. Malyshev for suggesting the problem and for his support.     

Orthogonal splitting and class numbers of quadratic forms

Orthogonal splitting for lattices on quadratic spaces over algebraic number fields is studied. It is seen that if the rank of a lattice is sufficiently large, then its spinor genus must contain a decomposable lattice. Also, splitting theory is used to obtain a lower bound for the class number of a lattice (in the definite case) in terms of its rank, via the partition function.     

Representation of large numbers by binary quadratic forms

Sufficient conditions of representability of a given integer by a fixed binary quadratic form are presented. It is shown that almost all integers satisfy these conditions. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 60–64.     

Annihilating polynomials for quadratic forms and Stirling numbers of the second kind

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a non‐vanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2‐adic valuation of Stirling numbers of the second kind. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Indefinite binary quadratic forms with large class numbers

This article describes all discriminants D for which the length of the period of the expansion of D in a continued fraction is equal to 2, 3, 4, or 6. It is proved that such discriminants are small. Similar propositions are proved for prime discriminants D=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 13–21, 1990.