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Counting rational points on cubic surfaces. (English) Zbl 0844.14008

Ciliberto, Ciro (ed.) et al., Classification of algebraic varieties. Algebraic geometry conference on classification of algebraic varieties, May 22-30, 1992, University of L’Aquila, L’Aquila, Italy. Providence, RI: American Mathematical Society. Contemp. Math. 162, 371-379 (1994).
Let \(V\) be a Fano variety defined over a number field \(K\) and \(h\) be a height function on \(V(K)\) associated to the anti-canonical class. In joint papers Yu. I. Manin studied [cf. V. V. Batyrev and Yu. I. Manin, Math. Ann. 286, No. 1-3, 27-43 (1990; Zbl 0679.14008) and J. Franke, Yu. I. Manin and Yu. Tschinkel, Invent. Math. 95, No. 2, 421-435 (1989; Zbl 0674.14012)] the behaviour of the number \(N(X)\) of \(P \in V(K)\) such that \(h(P) < X\) as \(X \to \infty\). It may happen that most of the points of \(V(K)\) are concentrated on finitely many subvarieties, so instead of \(V\) he considered an open subset \(W\) obtained by excluding finitely many subvarieties and conjectured that \(N(X,W) \sim CX (\log X)^{r - 1}\) as \(s \to 1\), where \(C\) is some constant and \(r = \text{rank} (\text{Pic} (V))\). He also introduced an alternative counting function \(H(s) = \sum_{P \in W(K)} h(P)^{- s}\). Under reasonable hypothesis on \(N(X)\) and \(H(s)\) he showed that his original conjecture is equivalent to \(\lim_{s \to 1} (s - 1)^r H(s) = (r - 1)! C\).
D. Heath-Brown has given [Mat. Comput. 59, No. 200, 613-623 (1992; Zbl 0778.11017)] a heuristic formula for \(C\) as well as significant supporting evidence. In this paper the author recasts this formula for \(C\) in order to generalize it. Actually he studies the particular case where \(V\) is the cubic surface defined by an equation \(f(x,y,z,t) = 0\) over \(\mathbb{Q}\). The height function \(h(P)\) on \(V (\mathbb{Q})\) is obtained by first giving a finite set of linear forms \(L_i\) which do not vanish at any point of \(V \times_\mathbb{Q} \overline \mathbb{Q}\) so that for each prime \(p\) and \((x,y,z,t) \in \mathbb{P}^3 (\mathbb{Q})\) we have \(h(P) = \prod_p h_p \max_i (|L_i (x,y,z,t) |_p)\). Supposing that \(r = 1\) the formula for \(C\) is given by \(C = {1 \over 2} \tau (V,h) \lim_{s \to 2} (L(V,s)/(s - 2))\), where \(L(V,s)\) is the \(L\)-series derived from the 2-cohomology of \(V\) and \(\tau (V,H)\) is a kind of Tamagawa number. In this set-up this formula for \(C\) can be rewritten in terms of the counting function \(H(s,V)\) as \(H(s,V) \sim {1 \over 2} \tau (V,h) L(V,s + 1)\) as \(s \to 1\). Moreover, the author gives evidence that if \(r = 2\) the fraction \(1/2\) in the latter formula has to be replaced by \(1/4\). Finally in the last section the author computes the function \(H(s,V)\) for some Fano surfaces \(V\).
For the entire collection see [Zbl 0791.00020].

MSC:

14G05 Rational points
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14J45 Fano varieties
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