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In this paper, we introduce the notion of an Evertse–Ferretti Nevanlinna constant and compare it with the birational Nevanlinna constant introduced by the authors in a recent joint paper. We then use it to recover several previously known results. This includes a 1999 example of Faltings from his Baker’s Garden article. We also extend the theory of these Nevanlinna constants to what we call “multidivisor Nevanlinna constants,” which allow the proximity function to involve the maximum of Weil functions for finitely many divisors.

The purpose of this paper is to modify the notion of the Nevanlinna constant $\operatorname{Nev}(D)$, recently introduced by the first author, for an effective Cartier divisor on a projective variety $X$. The modified notion is called the birational Nevanlinna constant and is denoted by $\operatorname{Nev}_{\text{bir}}(D)$. By computing $\operatorname{Nev}_{\text{bir}}(D)$ using the filtration constructed by Autissier in 2011, we establish a general result (see the General Theorem in the Introduction), in both the arithmetic and complex cases, which extends to general divisors the 2008 results of Evertse and Ferretti and the 2009 results of the first author. The notion $\operatorname{Nev}_{\text{bir}}(D)$ is originally defined in terms of Weil functions for use in applications, and it is proved later in this paper that it can be defined in terms of local effectivity of Cartier divisors after taking a proper birational lifting. In the last two sections, we use the notion $\operatorname{Nev}_{\text{bir}}(D)$ to recover the proof of an example of Faltings from his 2002 Baker's Garden article., Comment: 56 pages; latex

In 1986 and 1991, Serge Lang defined holomorphic, diophantine, and geometric exceptional sets of a complete variety over C, over a number field, or over a field of characteristic zero, respectively, and conjectured that they should coincide when defined.

This paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties. Combined with the main theorem of the first part of this paper, this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety, minus a divisor. In addition, the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors., Comment: 31 pages, amstex with rsfs fonts

We give a mild generalization of Cartan's theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes. This generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author: Given hyperplanes in projective space in general position, there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union (even linearly degenerate curves) satisfy the inequality of Cartan's theorem, except for the ramification term. In addition, it is shown how these methods can lead to a shorter proof of Nochka's theorem on Cartan's conjecture and (in the number field case) how Nochka's theorem gives a short proof of Wirsing's theorem on approximation of algebraic numbers by algebraic numbers of bounded degree.

In recent years, due to work of Vojta, Lang, Osgood, etc., people have started to realize that there is a close relationship between Nevanlinna theory and Diophantine approximation. P. Vojta has compiled a dictionary (see Vojta [V1]) providing correspondences for the concepts fundamental to the two theories. The second main theorem for meromorphic functions is the most important result in Nevanlinna theory, and its counterpart in number theory, via Vojta’s dictionary, is Roth’s theorem. In 1929, R. Nevanlinna conjectured that the second main theorem remains correct if distinct points in P1(C) are replaced by distinct holomorphic maps ai : C → P1(C) with Tai (r) = o(Tf (r)), where Tf (r) is the Nevanlinna characteristic function of the holomorphic map f : C → P1(C) (see [RS 1] for the definition). After many attempts by various people, Osgood proved the conjecture in 1985, which is now known as the Second Main Theorem with Moving Targets. In 1986, Steinmetz gave another simple and elegant proof. Recently, Vojta ([V 2] and [V 3]) formulated and proved its counterpart in number theory: Theorem A (Roth’s theorem with moving targets). Let k be a number field and let S be a finite set of places of k . Let q ∈ Z>0, and let > 0. Then there does not exist an infinite sequence of distinct tuples

This paper states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a multiplier ideal sheaf. This new conjecture trivially implies earlier conjectures in Nevanlinna theory or diophantine approximation, and in fact is equivalent to these conjectures. Although it does not provide anything new, it may be a more convenient formulation for some applications.

Beginning with the work of Osgood [65], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth’s theorem on diophantine approximation.

In 1929 Siegel proved a celebrated theorem on finiteness for integral solutions of certain diophantine equations. This theorem applies to systems of polynomial equations which either (a) describe an irreducible curve whose projective closure has positive genus, or (b) describe an irreducible curve of genus zero with at least three points at infinity. For such systems, Siegel's theorem says that there are only finitely many solutions in the ring of integers of any given number field. Siegel's proof used the method of diophantine approximations, as pioneered by Thue in 1909 [T]. To give an example, if (x, y) is a large integral solution of the equation x3 2y3 = 1, then

In 1982-83, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in $\Bbb P^n$ relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subseqently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides a mild simplification of the proof of the construction of the weights, by construing the ``Nochka diagram'' of Ru and Wong as a convex hull., Comment: 6 pages, 1 figure. Reorganized to reflect the submitted version; added a reference to a paper of Nochka

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split function field case of characteristic zero, and in the case of holomorphic curves (Nevanlinna theory). The proof of the latter involves a geometric generalization of the classical lemma on the logarithmic derivative, due to McQuillan. This lemma may be of independent interest., 28 pages, 1 figure Some minor errors fixed; updated references to previous work in the field

1 History; integral and rational points 2 Siegel’s lemma 3 The index 4 Sketch of the proof of Roth’s theorem 5 Notation 6 Derivatives 7 Proof of Mordell, with some simplifications by Bombieri 8 Proof using Gillet-Soule Riemann-Roch 9 The Faltings complex 10 Overall plan 11 Lower bound on the space of sections 12 More geometry of numbers 13 Arithmetic of the Faltings complex 14 Construction of a global section 15 Some analysis 16 More derivatives 17 Lower bound for the index 18 The product theorem

This addendum corrects the attribution for some of the major results of P. Vojta, Amer. J. Math. 122 (2000), 843-872.

Let C be a curve defined over a number field k. Let h(P) denote the height of an algebraic point P ∈ C \( P \in C(\overline k ) \) relative to some fixed divisor of degree 1. For a number field F, let $$ d(F) = \frac{{\log \left| {{D_{{F/Q}}}} \right|}}{{\left[ {F:Q} \right]}} $$ , and for an algebraic point P on C, let $$ d(P) = d(k(P)) $$ .

This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterl\'e and the author's diophantine conjecture for algebraic points of bounded degree. It also shows that the new conjecture is implied by the earlier conjecture. As with most of the author's conjectures, this new conjecture stems from analogies with Nevanlinna theory; in this case it corresponds to a Second Main Theorem in Nevanlinna theory with truncated counting functions. The original abc conjecture of Masser and Oesterl\'e corresponds to the Second Main Theorem with truncated counting functions on P^1 for the divisor [0]+[1]+[\infty].

It is shown that the groups A 6 , A 7 , and A 7 are Q -admissible. This is done by showing the existence of certain Laguerre polynomials whose splitting fields over Q have the desired properties. These polynomials are found by using a computer to show that certain elliptic equations have solutions which satisfy a variety of arithmetical conditions.

Faltings’ long awaited proof of the Mordell conjecture completes, roughly speaking, the question of whether a given curve has only finitely many integral or rational points. Indeed, if a complete curve has genus g ≥ 2, then it has finitely many rational points; any affine curve whose projective closure is a curve of genus at least two will, a fortiori, have only finitely many integral points. A curve of genus 1 is an elliptic curve; it will have infinitely many rational points over a sufficiently large ground field, but no affine subvariety has an infinite number of integral points. Finally, a curve of genus zero is, after a base change, the projective line, which has an infinite number of rational points; affine sub-varieties omitting at most two points will have infinitely many integral points over a sufficiently large ring; but affine sub-varieties omitting at least three points will have only finitely many integral points. Thus the answer to the finiteness question is given entirely by the structure of the curve over the complex numbers.