The density of rational points on curves and surfaces. (With an appendix by J.-L. Colliot-Thélène). (English) Zbl 1039.11044

Heath-Brown, D. R.

Ann. Math. (2) 155, No. 2, 553-598 (2002).

Let \(\vec x:= (x_1,\dots, x_n)\), suppose that \(n\geq 3\), and let \(F(\vec x)\) be a homogeneous polynomial of degree \(d\) with algebraic coefficients. Let \(Z_n\) be the set of those \(\vec a\) in \(\mathbb Z^n\) with h.c.f.\((a_1,\dots, a_n)= 1\) and such that if \(i\) is the smallest index for which \(a_i\neq 0\), then \(a_i> 0\). For \(\vec B= (B_1,\dots, B_n)\) with each \(B_i\geq 1\), let \(N(F;\vec B)\) stand for the number of \(\vec a\) in \(\mathbb Z_n\) such that \(F(\vec a)= 0\) and \(| a_i|\leq B_i\) for \(1\leq i\leq n\); we write \(N(F; B):= N(F;\vec B)\) if \(B_i= B\) for \(1\leq i\leq n\). In this remarkable work, the author obtains several upper estimates for the quantity \(N(F;\vec B)\). We shall cite 4 of the 14 theorems proved in the paper. Let \(\varepsilon> 0\); let \(V:= \prod^n_{i=1} B_i\), let \(T:= \max\{\prod^n_{i=1} B^{f_i}_i\}\), with the maximum taken over all integer \(n\)-tuples \(\vec f\) for which the corresponding monomial \(\prod^n_{i=1} x^{f_i}_i\) occurs in \(F(\vec x)\) with nonzero coefficient, and let \(\| F\|\) be defined as the maximum modulus of the coefficients of \(F\). According to the author, the following theorem is the fundamental result in this paper.
Theorem 14. Let \(F(\vec x)\in \mathbb Z[\vec x]\) and suppose that \(F(\vec x)\) is irreducible over \(\mathbb Q\). Then there exists \(D\) depending only on \(n\), \(d\) and \(\varepsilon\), and an integer \(k\) satisfying the following conditions:
1) \(k\ll_\varepsilon(V^d/T)^\delta V^\varepsilon(\log\| F\|)^{2n-3}\), where \(\delta:= d^{-(n-1)/(n-2)}\);
2) for each \(j\leq k\) there is a form \(F_j(\vec x)\) in \(\mathbb Z^n[\vec x]\) having degree at most \(D\) and not divisible by \(F(\vec x)\);
3) \(\prod^k_{j=1} F_j(\vec a)= 0\) for each \(\vec a\) counted by \(N(F;\vec B)\).
Theorem 2. Let \(F(\vec x)\) be an integral quadratic form of rank at least 3. Then \(N(F; B)\ll_{\varepsilon,n} B^{n-2+\varepsilon}\) uniformly in \(F\).
Theorem 9. For any absolutely irreducible form \(F(x_1,\dots, x_4)\) of degree \(d\geq 2\), we have \(N(F; B)\ll_{\varepsilon, d} B^{2+\varepsilon}\) uniformly in \(F\).
In higher dimensions, for \(d\geq 3\), an analogous estimate has not been proved. The author proposes the following conjecture.
Conjecture. Suppose that \(d\geq 3\) and \(n\geq 5\). For any absolutely irreducible form \(F(\vec x)\) we have \(N(F; B)\ll_{\varepsilon, n,d} B^{n-2+\varepsilon}\) uniformly in \(F\).
Theorem 10. For any nonsingular form \(F(x_1,\dots, x_4)\) of degree \(d\), we have \(N_1(F; B)\ll_{\varepsilon, d} B^{4/3+ 16/9d+ \varepsilon}\) uniformly in \(F\), where \(N_1(F; B)\) is defined to count the same rational points as does \(N(F; B)\), but excluding any that lie on lines in the surface \(F(\vec x)= 0\).
The paper is rather well written and can be read by any mathematician interested in Diophantine geometry.

Reviewer: B. Z. Moroz (Bonn)

MSC:

11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05	Rational points
11H06	Lattices and convex bodies (number-theoretic aspects)

Keywords:

rational points; hypersurfaces; lattices