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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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

Introduction to finite fields and their applications: Preface

These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995 The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs (X, V ) whereX is a complex manifold and V a holomorphic subbundle of TX  If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody’s criterion) We describe a construction of projectivized kjet bundles PkV , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature More precisely, πk : PkV → X is a tower of projective bundles over X and carries a canonical line bundle OPkV (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on OPkV (−1) for k large enough The direct images (πk)⋆OPkV (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when OPkV (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of “partial projective projection” and the associated Wronskian operators, substantial improvements on Nadel’s degree estimate will be achieved here 2 J-P Demailly, Kobayashi hyperbolic projective varieties and jet differentials

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential −∆ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.

/pdf/an-overview-of-periodic-elliptic-operators-uqeh1n79ah.pdf

An overview of periodic elliptic operators

A photonic crystal, or photonic band gap (PBG) optical material, is an artificially created periodic low-loss dielectric medium in which electromagnetic waves of certain frequencies cannot propagate The range of the prohibited frequencies is called the complete band gap A simple example of such a medium is a dielectric background material with a periodic array of air bubbles The reason why the band gap arises (if it does) is the coherent multiple scattering of waves and destructive interference To put it simply, if a wave of a prohibited frequency somehow managed to propagate in the medium, it would reflect and self-interfere in such a way that it would cancel itself completely It is expected that industrial manufacturing of photonic crystals will bring about a new technological revolution in optics, computing, information transmission, and other areas The idea of photonic crystals was coined in [107, 192], though simpler versions of such materials like layered media and optical gratings have been known for a long time We will not dwell much on the physics aspects of this field of research, since the reader can refer to the surveys and proceedings [33, 101, 108, 134, 154, 156, 167, 168, 182, 189] devoted to this topic, and especially to the lovely book [106] The bibliography [153] and the collection of photonic crystal links [190] are also very useful The area of photonic crystal research presents a bonanza of beautiful, important, and hard problems for a mathematician, most of which are still unexplored or explored only tangentially The number of mathematics publications dealing with PBG materials is growing (see [3, 7, 8, 21, 41, 42, 44, 45, 50, 51], [61]–[81], [86, 87, 100, 126, 127, 155, 188]) but is probably still not sufficient We hope that

The mathematics of photonic crystals

Periodic binary harmonic functions on lattices

We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z\times A^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective $G_a$-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].

New examples of cylindrical Fano fourfolds

Schottky-Landau growth estimates for s-normal families of holomorphic mappings.

A Danilov-Gizatullin surface is a normal affine surface V, which is a complement to an ample section S in a Hirzebruch surface of index d. By a surprising result due to Danilov and Gizatullin, V depends only on the self-intersection number of S and neither on d nor on S. In this note we provide a new and simple proof of this Isomorphism Theorem.

On the Danilov-Gizatullin Isomorphism Theorem

We show that the affine cones over any Fano--Mukai fourfold of genus 10 are flexible;
in particular, the automorphism group of such a cone acts highly transitively outside the vertex. 
Furthermore, any Fano--Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to the affine four-space.

Affine cones over Fano-Mukai fourfolds of genus 10 are flexible

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