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.     

Distribution of integral points on some hyperboloids of one sheet

Results of B. F. Skubenko (Izv. Akad. Nauk SSSR, Ser. Mat.,26, 721–752 (1962)) are generalized to indefinite ternary quadratic formsf(x)=f ₀(Cx), which are contained in the simplest formf ₀(x)=x ₁ x ₃–x _r ^e We prove that the integral points on the hyperboloid of one sheetf(x)=m,m<0, are uniformly distributed over area (in the sense of hyperbolic metric) and over residue classes for given modulus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 83–93, 1983.     

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.     

Weierstrass points and multiple intersection points of closed geodesics

We classify smooth complex projective surfaces of degreed and class , satisfying either (i) –d16, or (ii) 25. All these surfaces are rational or ruled. Indeed, we prove that the smallest value of the class of a non-ruled surface is 30 and in fact there are at least two surfacesS, both of degreed=10 and sectional genusg=6, with Kodaira dimension (S)=0 and class =30. Finally, we classify the smoothk-folds (k3) whose sectional surface has class 23.     

Fixed points and uniqueness of complex geodesics

In this work we examine the conditions which guarantee the uniqueness of a complex geodesic whose range contains two fixed points of a holomorphic mapf of a bounded convex circular domain in itself and is contained in the fixed points set off.     

Homogeneous geodesics and the critical points of the restricted Finsler function

In this paperwe study the set of homogeneous geodesics of a left-invariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. We extend J. Szenthe’s result on homogeneous geodesics to left-invariant Finsler metrics. This result gives a relation between geodesic vectors and restricted Minkowski norm in Finsler setting. We show that if a compact connected and semisimple Lie group has rank greater than 1, then for every left-invariant Finsler metric there are infinitely many homogeneous geodesics through the identity element.     

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.     

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.     

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.     

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.     

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).     

Fixed points of an integral operator

A characterization of the functions fixed by a class of integral operators associated with hyperbolically-harmonic functions is given.     

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 $$x^3 + y^3 = u^2 + v^2 .$$