Integer points on hypersurfaces

Very general hypersurfaces in ⁴ contain r ^2+(4/9) integer points in any ball of radiusr>1. As a consequence, an irreducible algebraic hypersurface in ⁿ (wheren4) which is not a cylinder and is of degreed, contains c(d, n)r ^{n–1–(5/9)} integer points in a ball of radiusr. This improves on the known boundc(d, n)r ^n–(3/2).Meinem verehrten Lehrer Professor E. Hlawka zum siebzigsten Geburtstag gewidmetWritten with partial support from NSF-MCS-8211461.     

S-unit points on analytic hypersurfaces

In analogy with algebraic equations with S-units, we shall deal with S-unit points in an analytic hypersurface, or more generally with values of analytic functions at S-unit points.After proving a general theorem, we shall give diophantine applications to certain problems of integral points on subvarieties of . Also, we shall prove an analogue of a theorem of Masser, important in Mahler's method for transcendence.In the course of the proofs we shall also develop a theory for those algebraic subgroups of whose Zariski closure in An contains the origin. Among others, we shall prove a structure theorem for the family of such subgroups contained in a given analytic hypersurface, obtaining conclusions similar to the case of algebraic varieties.     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

Counting rational points on hypersurfaces of low dimension

Let N(X,B) be the number of rational points of height at most B on a variety X⊂Pn defined over Q. We establish new upper bounds for N(X,B) for hypersurfaces of dimension at most four. We also study N(X^′,B) for the open complements X^′ of all lines or all planes on such hypersurfaces. One of the goals is to show that for smooth hypersurfaces in P⁵ defined by a form F(x₀,…,x₅) of degree d?9. This improves upon a classical upper estimate of Hua for the form .     

Heights of algebraic points lying on curves or hypersurfaces

Our first aim will be to give an explicit version of a generalization of the results of Zhang and Zagier on algebraic points with . Secondly, we will show that distinct algebraic points lying on a given curve of certain type can be distinguished in terms of some height functions. Thirdly, we will derive a bound for the number of points on such a curve whose heights are under a given bound and whose coordinates lie in a multiplicative group of given rank.

     

On smooth hypersurfaces containing a given subvariety

We show when a nonsingular closed subvariety Y of a nonsingular affine real variety X is contained in a nonsingular hypersurface. We also solve this problem in a holomorphic case.     

The density of rational points on non-singular hypersurfaces, II

For any integers d,n 2, let X Pⁿ be a non-singular hypersurfaceof degree d that is defined over the rational numbers. The mainresult in this paper is a proof that the number of rationalpoints on X which have height at most B is O(B^{n – 1} +), for any > 0. The implied constant in this estimate dependsat most upon d, and n. 2000 Mathematics Subject Classification11D45 (primary), 11G35, 14G05 (secondary).     

Counting rational points on smooth cyclic covers

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space ${\mathbb{P}}^{n?1}$. In this paper, we achieve Serre?s conjecture in the special case of smooth cyclic covers of any degree when $n?10$, and surpass it for covers of degree $r?3$ when $n>10$. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput?s method.     

Identifying variable points on a smooth curve

LetX be a compact Riemann surface,n ≥ 2 an integer andx = [x ₁, …,x _n ] an unorderedn-tuple of not necessarily distinct points onX. Byf _x :X →Y _x we denote the normalization which identifies thex ₁, …,x _n and maps them to the only and universal singularity of a complex curveY _x . Thenf _x depends holomorphically onx and is uniquely determined by this parameter. In this context we consider the fine moduli spaceQ _X of all complex-analytic quotients ofX and construct a morphismS ⁿ (X) →Q _X such that each and everyf _x corresponds to the image of the pointx on then-fold symmetric powerS ⁿ (X). For everyn ≥ 2 the mappingS ⁿ (X) →Q _X is a closed embedding; the points of its image have embedding dimensionn(n ? 1) inQ _X . HenceS ²(X) is a smooth connected component ofQ _X . On the other hand, a deformation argument yields thatS ⁿ (X) is part of the singular locus of the complex spaceQ _X provided thatn ≥ 3.     

On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC ^∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC ^∞ boundary such that Ω ? ?².     

Classes of points of cubic hypersurfaces defined over finite fields

The points of a cubic hypersurface are divided into equivalence classes of Ya. I. Manin [1]. It is established that the quasigroup of these equivalence classes over a finite field is isomorphic to either 1 or Z₂ or Z₃ under some weak conditions.