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The Bogomolov-Miyaoka-Yau inequality for the arithmetical surfaces and its applications. (English) Zbl 0706.14015

Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 299-312 (1988).
[For the entire collection see Zbl 0653.00005.]
The central point of the paper is the inequality \[ (9)\omega_{V/B}\cdot \omega_{V/B}\leq 3\cdot \sum_{v\in B\cup B_{\infty}}\delta '_ v\cdot \epsilon_ v +(2g-2)\log | D_{K/{\mathbb{Q}}}|. \] This is the first explicit announcement of an arithmetic analogon of the well-known Bogomolov-Miyaoka-Yau inequality for algebraic surfaces of general type. Here V is a regular scheme of dimension 2, \(B=Spec({\mathfrak O}_ K)\), \({\mathfrak O}_ K\) the ring of integers of a number field K, V/B denotes a proper morphism \(V\to B\) with geometrically irreducible non-singular general fibre of genus \( g>1,\) \(B\cup B_{\infty}\) the compactification of B \((B_{\infty}\) consists of all archimedean places of K), \(\omega_{V/B}\) the relative canonical class of the corresponding arithmetic surface introduced by S. Yu. Arakelov in Math. USSR, Izv. 8(1974), 1167-1180 (1976), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179-1192 (1974; Zbl 0355.14002), \(\omega_{V/B}\cdot \omega_{V/B}\) the selfintersection in the sense of Arakelov’s intersection theory, \(D_{K/{\mathbb{Q}}}\) the discriminant of K, \(\epsilon_ v=\log (\#({\mathfrak O}_ v/P_ v)) (=1, 2)\) if v is a non- (real, complex) archimedean place of K, and \(\delta '_ v\) is an invariant of the fibres \(X_ v\), which has been introduced by G. Faltings [Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005)] for archimedean v.
It seems to be reasonable to expect that a refined weaker version \((9')\), say, of (9), published in Math. USSR, Sb. 66, No.1, 249-264 (1990); translation from Mat. Sb. 180, No.2, 244-259 (1989; Zbl 0702.14017), is true for all arithmetic surfaces of fibre genus \(g>1.\) Some important consequences are discussed together with an outline of proofs:
1. The intersection numbers of sections with the relative canonical class \(\omega_{V/B}\) can be bounded by an effectively computable constant c(V/B).
2.i) \(\Delta_{\min}(E)\leq N(E)^{c_ 1}\) (Szpiro conjecture),
2.ii) \(H(A)\leq c_ 3(D_{K/{\mathbb{Q}}}N(A))^{c_ 2}\). - Here E/\({\mathbb{Q}}\) and A/K are elliptic curves, \(\Delta_{\min}\) the minimal discriminant, N(E) the conductor, H(A) is Faltings’ canonical (exponential) height, and \(c_{1,2,3}\) are effectively computable absolute constants.
The proofs of the implications (9), (9’) \(\Rightarrow\) 1.,2.i);2.ii) need the Kodaira-Parshin construction (Parshin trick) of special branched coverings \(X_ P\) of \(V_ K\), \(P\in V(K)\). For the proof of 2. the inequality (9) (or (9’)) is applied to the arithmetic surface of a moduli scheme of elliptic curves with level structure. For more detailed proofs we refer to Parshin’s article [loc. cit.]. As the author remarks, the Szpiro conjecture, hence the proof of (9’), has strong consequences: the asymptotic Fermat theorem [G. Frey, Ann. Univ. Sarav., Ser. Math. 1, No.1 (1986; Zbl 0586.10010)], and the boundedness of the torsion of elliptic curves (Oesterlé).
Reviewer: R.-P.Holzapfel

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14H52 Elliptic curves
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
11D25 Cubic and quartic Diophantine equations