The density of zeros of forms for which weak approximation fails

The weak approximation principal fails for the forms x3 + y3 + z3 = kw3, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these forms. Evidence, both numerical and theoretical, is presented, which suggests that, for forms of the above type, the product of the local densities still gives the correct global density. Let f(x1,..., xn) ∈ Q[x1,..., xn] be a rational form. We say that f satisfies the weak approximation principle if the following condition holds. (WA): Given an ε> 0 and a finite set S of places of Q, and zeros (xν1,..., x ν n) ∈ Qnν of the form f, we can find a rational zero (x1,..., xn) of f such that, |xi − xνi |ν < ε, for 1 ≤ i ≤ n and ν ∈ S. Alternatively, we may write X(K) for the points on the hypersurface f = 0 whose coordinates lie in the field K, and consider the produc

Artin's Conjecture on Zeros of pp-Adic Forms

This is an exposition of work on Artin's Conjecture on the zeros of $p$-adic forms. A variety of lines of attack are described, going back to 1945. However there is particular emphasis on recent developments concerning quartic forms on the one hand, and systems of quadratic forms on the other.Comment: Submitted for publication as part of ICM 201

Lattice points in the sphere

Our goal in this paper is to give a new estimate for the number of integer lattice points lying in a sphere of radius R centred at the origin. Thus we define S(R) = #{x ∈ ZZ3: ||x| | ≤ R}

A mean value estimate for real character sums

There are a number of well known estimates for averages of Dirichlet polynomi-als. For example one has ∫

The distribution and moments of the error term in the Dirichlet divisor problem

This paper will consider results about the distribution and moments of some of the well known error terms in analytic number theory. To focus attention we begin by considering the error term ∆(x) in the Dirichlet divisor problem, which is defined a

The density of rational points on curves and surfaces

Let $C$ be an irreducible projective curve of degree $d$ in $\mathbb{P}^3$, defined over $\overline{\mathbb{Q}}$. It is shown that $C$ has $O_{\varepsilon,d}(B^{2/d+\varepsilon})$ rational points of height at most $B$, for any $\varepsilon>0$, uniformly for all curves $C$. This result extends an estimate of Bombieri and Pila [Duke Math. J., 59 (1989), 337-357] to projective curves. For a projective surface $S$ in $\mathbb{P}^3$ of degree $d\ge 3$ it is shown that there are $O_{\varepsilon,d}(B^{2+\varepsilon})$ rational points of height at most $B$, of which at most $O_{\varepsilon,d}(B^{52/27+\varepsilon})$ do not lie on a rational line in $S$. For non-singular surfaces one may reduce the exponent to $4/3+16/9d$ (for $d=4$ or 5) or $\max\{1,3/\sqrt{d}+2/(d-1)\}$ (for $d\ge 6$). Even for the surface $x_1^d+x_2^d=x_3^d+x_4^d$ this last result improves on the previous best known. As a further application it is shown that almost all integers represented by an irreducible binary form $F(x,y)\in\mathbb{Z}[x,y]$ have essentially only one such representation. This extends a result of Hooley [J. Reine Angew. Math., 226 (1967), 30-87] which concerned cubic forms only. The results are not restricted to projective surfaces, and as an application of other results in the paper it is shown that $\#\{(x_1,x_2,x_3)\in\mathbb{N}^3:x_1^d+x_2^d+x_3^d=N\} \ll_{\varepsilon,d} N^{\theta/d+\varepsilon}$ with $\theta=\frac{2}{\sqrt{d}}+\frac{2}{d-1}.$ When $d\ge 8$ this provides the first non-trivial bound for the number of representations as a sum of three $d$-th powers