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

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
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