Diophantine Approximation and Nevanlinna Theory Paul Vojta Abstract As was originally observed by C. F. Osgood and further developed by the author, there is a formal analogy between Nevanlinna theory in complex analysis and certain results in diophantine approximation. These notes describe this analogy, after briefly introducing the theory of heights and Weil function in number theory and the main concepts of Nevanlinna theory. Parallel conjectures are then presented, in Nevanlinna theory (“Griffiths’ conjecture”) and the author’s conjecture on ratio- nal points. Following this, recent work is described, highlighting work of Corvaja, Zannier, Evertse, and Ferretti on combining Schmidt’s Subspace Theorem with ge- ometrical constructions to obtain partial results on this conjecture. Counterparts of these results in Nevanlinna theory are also given (due to Ru). These notes also de- scribe parallel extensions of the conjectures in Nevanlinna theory and diophantine approximation, to involve finite ramified coverings and algebraic points, respec- tively. Variants of these conjectures involving truncated counting functions are also introduced, and the relations of these various conjectures with the abc conjecture of Masser and Oesterle´ are also described. 0 Introduction Beginning with the work of C. F. Osgood [1981], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution the- ory) has many similarities with Roth’s theorem on diophantine approximation. This was extended by the author [Vojta, 1987] to include an explicit dictionary and to in- clude geometric results as well, such as Picard’s theorem and Mordell’s conjecture (Faltings’ theorem). The latter analogy ties in with Lang’s conjecture that a projec- Paul Vojta Department of Mathematics, University of California, 970 Evans Hall #3840, Berkeley, CA 94720- 3840 e-mail: [email protected] Partially supported by NSF grant DMS-0500512 113 114 Paul Vojta tive variety should have only finitely many rational points over any given number field (i.e., is Mordellic) if and only if it is Kobayashi hyperbolic. This circle of ideas has developed further in the last 20 years: Lang’s conjecture on sharpening the error term in Roth’s theorem was carried over to a conjecture in Nevanlinna theory which was proved in many cases. In the other direction, Bloch’s conjectures on holomorphic curves in abelian varieties (later proved; see Section 14 for details) led to proofs of the corresponding results in number theory (again, see Section 14). More recently, work in number theory using Schmidt’s Subspace Theorem has led to corresponding results in Nevanlinna theory. This relation with Nevanlinna theory is in some sense similar to the (much older) relation with function fields, in that one often looks to function fields or Nevanlinna theory for ideas that might translate over to the number field case, and that work over function fields or in Nevanlinna theory is often easier than work in the number field case. On the other hand, both function fields and Nevanlinna theory have con- cepts that (so far) have no counterpart in the number field case. This is especially true of derivatives, which exist in both the function field case and in Nevanlinna theory. In the number field case, however, one would want the “derivative with re- spect to p,” which remains as a major stumbling block, although (separate) work of Buium and of Minhyong Kim may ultimately provide some answers. The search for such a derivative is also addressed in these notes, using a potential approach using successive minima. It is important to note, however, that the relation with Nevanlinna theory does not “go through” the function field case. Although it is possible to look at the function field case over C and apply Nevanlinna theory to the functions representing the rational points, this is not the analogy being described here. Instead, in the analogy presented here, one holomorphic function corresponds to infinitely many rational or algebraic points (whether over a number field or over a function field). Thus, the analogy with Nevanlinna theory is less concrete, and may be regarded as a more distant analogy than function fields. These notes describe some of the work in this area, including much of the nec- essary background in diophantine geometry. Specifically, Sections 1–3 recall basic definitions of number theory and the theory of heights of elements of number fields, culminating in the statement of Roth’s theorem and some equivalent formulations of that theorem. This part assumes that the reader knows the basics of algebraic number theory and algebraic geometry at the level of Lang [1970] and Hartshorne [1977], respectively. Some proofs are omitted, however; for those the interested reader may refer to Lang [1983]. Sections 4–6 briefly introduce Nevanlinna theory and the analogy between Roth’s theorem and the classical work of Nevanlinna. Again, many proofs are omitted; references include Shabat [1985], Nevanlinna [1970], and Goldberg and Ostrovskii [2008] for pure Nevanlinna theory, and Vojta [1987] and Ru [2001] for the analogy. Sections 7–15 generalize the content of the earlier sections, in the more geometric context of varieties over the appropriate fields (number fields, function fields, or C). Again, proofs are often omitted; most may be found in the references given above. Diophantine Approximation and Nevanlinna Theory 115 Section 14 in particular introduces the main conjectures being discussed here: Conjecture 14.2 in Nevanlinna theory (“Griffiths’ conjecture”) and its counterpart in number theory, the author’s Conjecture 14.6 on rational points on varieties. Sections 16 and 17 round out the first part of these notes, by discussing the func- tion field case and the subject of the exceptional sets that come up in the study of higher dimensional varieties. In both Nevanlinna theory and number theory, these conjectures have been proved only in very special cases, mostly involving subvarieties of semiabelian va- rieties. This includes the case of projective space minus a collection of hyperplanes in general position (Cartan’s theorem and Schmidt’s Subspace Theorem). Recent work of Corvaja, Zannier, Evertse, Ferretti, and Ru has shown, however, that using geometric constructions one can use Schmidt’s Subspace Theorem and Cartan’s the- orem to derive other weak special cases of the conjectures mentioned above. This is the subject of Sections 18–22. Sections 23–27 present generalizations of Conjectures 14.2 and 14.6. Conjecture 14.2, in Nevanlinna theory, can be generalized to involve truncated counting func- tions (as was done by Nevanlinna in the 1-dimensional case), and can also be posed in the context of finite ramified coverings. In number theory, Conjecture 14.6 can also be generalized to involve truncated counting functions. The simplest nontrivial case of this conjecture, involving the divisor [0] + [1] + [¥] on P1, is the celebrated “abc conjecture” of Masser and Oesterle.´ Thus, Conjecture 22.5 can be regarded as a generalization of the abc conjecture as well as of Conjecture 14.6. One can also generalize Conjecture 14.6 to treat algebraic points; this corresponds to finite ramified coverings in Nevanlinna theory. This is Conjecture 24.1, which can also be posed using truncated counting functions (Conjecture 24.3). Sections 28 and 29 briefly discuss the question of derivatives in Nevanlinna the- ory, and Nevanlinna’s “Lemma on the Logarithmic Derivative.” A geometric form of this lemma, due to R. Kobayashi, M. McQuillan, and P.-M. Wong, is given, and it is shown how this form leads to an inequality in Nevanlinna theory, due to McQuil- lan, called the “tautological inequality.” This inequality then leads to a conjecture in number theory (Conjecture 29.1), which of course should then be called the “tauto- logical conjecture.” This conjecture is discussed briefly; it is of interest since it may shed some light on how one might take “derivatives” in number theory. The abc conjecture infuses much of this theory, not only because a Nevanlinna- like conjecture with truncated counting functions contains the abc conjecture as a special case, but also because other conjectures also imply the abc conjecture, and therefore are “at least as hard” as abc. Specifically, Conjecture 24.1, on algebraic points, implies the abc conjecture, even if known only in dimension 1, and Con- jecture 14.6, on rational points, also implies abc if known in high dimensions. This latter implication is the subject of Section 30. Finally, implications in the other di- rection are explored in Section 31. 116 Paul Vojta 1 Notation and Basic Results: Number Theory We assume that the reader has an understanding of the fundamental basic facts of number theory (and algebraic geometry), up through the definitions of (Weil) heights. References for these topics include [Lang, 1983] and [Vojta, 1987]. We do, however, recall some of the basic conventions here since they often differ from author to author. Throughout these notes, k will usually denote a number field; if so, then Ok will denote its ring of integers and Mk its set of places. This latter set is in one-to-one correspondence with the disjoint union of the set of nonzero prime ideals of Ok, the set of real embeddings s : k ,! R, and the set of unordered complex conjugate pairs (s;s) of complex embeddings s : k ,! C with image not contained in R. Such elements of Mk are called non-archimedean or finite places, real places, and complex places, respectively. The real and complex places are collectively referred to as archimedean or infi- nite places. The set of these places is denoted S¥. It is a finite set. To each place v 2 Mk we associate a norm k · kv, defined for x 2 k by kxkv = 0 if x = 0 and (1.1) 8 (O : p)ordp(x) if v corresponds to p ⊆ O ; <> k k kxkv = js(x)j if v corresponds to s : k ,! R; and > :js(x)j2 if v is a complex place, corresponding to s : k ,! C if x 6= 0.