Distribution of integral points on certain two-sheeted hyperboloids

Yu. V. Linnik's investigations [Vestn. Leningr. Univ., No. 2, 3–23; No. 5, 3–32; No. 8, 15–27 (1955)] are refined and generalized to indefinite ternary quadratic forms of a sufficiently general form (to forms contained in the form x₁x₃—x ₂ ² ]. The method of investigation is improved. The presentation is substantially simplified.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 93, pp. 87–141, 1980.     

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).     

A connection of the sum of Salié sums with the distribution of integral points on hyperboloids

Based on the uniform distribution of integral points on hyperboloids, we obtain nontrivial estimates for the sum of Salié sums.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 56–62, 1982.     

Distribution of integral points on the determinant surface

One obtains an asymptotic formula with remainder term for the number of second-order integral matrices with an increasing determinant, belonging to a given region of the discriminant surface and to a given residue class. The results are more accurate than in A. M. Istamov's paper (this issue, pp. 14–17) and are obtained in a somewhat different manner. The presentation is more detailed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 93, pp. 30–40, 1980.     

Distribution of integral points on second-order hyperbolic surfaces

One gives generalizations of the fundamental results of A. V. Malyshev (Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,1, 6–83 (1966)) and of B. Z. Moroz (ibid. pp. 84–113) to arbitrary second-order surfaces in R^S of hyperbolic type (s ≥ 4).     

Distribution of integral points on second-order and certain third-order surfaces

Under certain assumptions regarding the bounds of the zeros of the Dirichlet L -functions, one obtains results on the asymptotics of the number of integral points in arbitrary domains on second-order surfaces of an arbitrary form. The method is based on reduction to the case of the simplest hyperboloids. As an application, one has obtained results on the distributions of the integral points on surfaces of the form $$x^3 + y^3 = u^2 + v^2 .$$      

Canonical representations on two-sheeted hyperboloids

The two-sheeted hyperboloid in ℝⁿ can be identified with the unit sphere Ω in ℝⁿ with the equator removed. Canonical representations of the group G = SO ₀(n − 1, 1) on are defined as the restrictions to G of the representations of the overgroup = SO ₀(n, 1) associated with a cone. They act on functions and distributions on the sphere Ω. We decompose these canonical representations into irreducible constituents and decompose the Berezin form. Bibliography: 12 titles. To the memory of my teacher F. A. Berezin __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 91–124.     

Harmonic analysis on hyperboloids

The regular representation of O(n, N) acting on $\text{L}{}^2\text{(}\text{O(n, N)}\text{O(n, N ? 1)}\text{)}$ is decomposed into a direct integral of irreducible representations. The homogeneous space $\text{O(n, N)}\text{O(n, N ? 1)}$ is realized as the Hyperboloid $\text{H = {(x, t) ? R}{}^{\mathrm{n\; +\; N}}\text{: ¦ t ¦}{}^2\text{? ¦ x ¦}{}^2\text{= 1}}$. The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □_h on H, which is the tangential part of the operator □ = Δ_x ? Δ_t on R^{n + N}. The spectrum of □_h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □u = 0 which vanish in the region $\text{¦ t ¦ < ¦ x ¦}$, and a continuous part $\text{H}$_?. As a representation of O(n, N), $\text{H}$_? ⊕ $\text{H}$_? is unitarily equivalent to the regular representation on L² of the cone $\text{{(x, t) : ¦ x ¦}{}^2\text{= ¦ t ¦}{}^2\text{}}$, and the intertwining operator is obtained by solving the equation □u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL(2, R).     

Multiples of integral points on elliptic curves

If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P∈E(Q), nP is integral for at most one value of n>C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E^′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ⊆E^′(Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.     

Distribution of some functionals of the integral of a random walk

The asymptotic behavior of the probabilities associated with the first crossing of a straight line or parabola by the integral of a Brownian curve is studied.Dedicated to the memory of Mikhail Konstantinovich Polivanov.L. D. Landau Institute of Theoretical Physics, USSR Academy of Sciences Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 323–3353, March, 1992.     

On the distribution of integral points on cones

New results on the distribution of integral points on the cones

Uniform distribution of integral points on multidimensional ellipsoids

Let Q(X), X^T=(x₁,...,x_l), be a positive definite, integral-valued, primitive, quadratic form of l4 variables, let () be the number of solutions of Eq. Q(X)=n, let (,) be the number of the solution of the equation Q(X)=n such that X/, where is an arbitrary convex domain on Q(X)=1 with a piecewise smooth boundary. One investigates the asymptotic behavior of the quantity (,) (n). In the case of an even l4 the result is formulated in the following manner: for (n,N)=1 and n one has, >o, where() is the measure of the domain , normalized by the condition(E)=1, where E is the ellipsoid Q(X)=1. Weaker results have been obtained earlier by various authors. In the case of the simplest domains (belt, cap) the remainder in (1) can be brought to the form. The last estimate for large l is close to an unimprovable one. The proof makes use of the theory of modular forms and of Deligne's estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 144–153, 1986.     

Effective analysis of integral points on algebraic curves

LetK be an algebraic number field,S?S _\t8 a finite set of valuations andC a non-singular algebraic curve overK. Letx∈K(C) be non-constant. A pointP∈C(K) isS-integral if it is not a pole ofx and |x(P)| _v >1 impliesv∈S. It is proved that allS-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if

1. x:C→P ¹ is a Galois covering andg(C)≥1;
2. the integral closure of $\bar Q$ [x] in $\bar Q$ (C) has at least two units multiplicatively independent mod $\bar Q$ *.

This generalizes famous results of A. Baker and other authors on the effective solution of Diophantine equations.     

Concerning one integral inequality

The inequality of Zinger [1] (Theorem 1) is proved. The proof uses a monotone mapping x?F(x).