This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.

Analytic Functions of Several Complex Variables

https://projecteuclid.org/download/pdf_1/euclid.bams/1183553166

Hyperbolic and Diophantine analysis

introduction to complex analysis in several variables is available in our book collection an online access to it is set as public so you can download it instantly. Our books collection spans in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the introduction to complex analysis in several variables is universally compatible with any devices to read.

Introduction To Complex Analysis In Several Variables

Le present article decrit a l'aide de fibrations fonctorielles intrinseques, la structure geometrique (et conjecturalement la pseudometrique de Kobayashi ainsi que l'arithmetique dans le cas projectif) des varietes Kahleriennes compactes Les varietes speciales sont tout d'abord definies comme etant les varietes Kahleriennes compactes ne possedant pas d'application meromorphe surjective sur une orbifolde de type general, la structure d'orbifolde de la base provenant du diviseur des fibres multiples On montre que les varietes Kahleriennes compactes qui sont, soit rationnellement connexes, soit a dimension de Kodaira nulle sont speciales Nous construisons ensuite fonctoriellement, pour toute telle variete X, l' unique fibration c X : X → C(X) (le coeur de X) dont les fibres sont speciales et dont la base orbifolde est soit de type general, soit un point (ce dernier cas se produisant si et seulement si X est speciale) Le coeur de X est ensuite canoniquement decompose comme une tour de fibrations a fibres soit κ-rationnellement engendrees (une version faible de la connexite rationnelle), soit a dimension de Kodaira nulle En particulier, les varietes speciales sont donc de telles tours de fibrations L'ingredient technique essentiel des demonstrations est une version orbifolde de la conjecture C n,m d'Iitaka, etablie lorsque la base orbifolde est de type general Le coeur de X permet de donner une description qualitative conjecturale tres simple de la pseudometrique de Kobayashi de X et de la distribution de ses points K-rationnels (si X est projective, definie sur le corps K de type fini sur Q), description se reduisant a celle de Lang lorsque X est de type general

/pdf/orbifolds-special-varieties-and-classification-theory-2eojelm24v.pdf

Orbifolds, special varieties and classification theory

Geometric Function Theory in Several Complex Variables

Nevanlinna's lemma on logarithmic derivatives played an essential role in the proof of the second main theorem for meromorphic functions on the complex plane C (c£, e.g., [17]). In [19, Lemma 2.3] it was generalized for entire holomorphic curves /: C->M in a compact complex manifold M (Lemma 2.3 in [19] is still valid for non-Kahler M). Here we call, in general, a holomorphic mapping from a domain of C or a Riemann surface into M a holomorphic curve in M, and sometimes use it in the sense of its image if no confusion occurs. Applying the above generalized lemma on logarithmic derivatives to holomorphic curves /: C -> V in a complex projective algebraic smooth variety V and making use of Ochiai [22, Theorem A], we had an inequality of the second main theorem type for / and divisors on V (see [19, Main Theorem] and [20]). Other generalizations of Nevanlinna's lemma on logarithmic derivatives were obtained by Nevanlinna [16], Griffiths-King [10, § 9] and Vitter [23]. In this paper we first deal with holomorphic curves /: J* -* M from the punctured disc J* = {\\z\\ >̂ 1} with center at the infinity oo of the Riemann sphere into a compact Kahler manifold M. Our first aim is to prove the following lemma on logarithmic derivatives which is a generalization of Nevanlinna [16, IΠ, p. 370] and will play a crucial role in §§ 3 and 4 (see § 1 as to the notation):

/pdf/lemma-on-logarithmic-derivatives-and-holomorphic-curves-in-3d5uaa3sum.pdf

Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties

We establish the second main theorem with the best truncation level one Tf (r;L(D)) N1(r; f∗D) + Tf (r)|| for an entire holomorphic curve f : C → A into a semi-abelian variety A and an arbitrary effective reduced divisor D on A; the low truncation level is important for applications. We will actually prove this for the jet lifts of f . Finally we give some applications, including the solution of a problem posed by Mark Green.

Junjiro Noguchi

Papers

Geometric Function Theory in Several Complex Variables

Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties

The second main theorem for holomorphic curves into semi-Abelian varieties

A higher dimensional analogue of Mordell's conjecture over function fields

Holomorphic curves in algebraic varieties