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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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

General Topology-I

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Brian Thompson is a Senior Nutrition Officer at the Food and Agriculture Organization of the United Nations (FAO). FAO is an intergovernmental organization, it has 191 Member Nations, two associate members and one member organization, the European Union. As a knowledge organization, FAO creates and shares critical information about food, agriculture and natural resources in the form of global public goods. FAO plays a connector role, through identifying and working with different partners with established expertise, and facilitating a dialogue between those who have the knowledge and those who need it. By turning knowledge into action, FAO links the field to national, regional and global initiatives in a mutually reinforcing cycle. Its mandate is to raise levels of nutrition and standards of living. What is the size of the malnutrition problem and what regions are most affected? There are persistently high levels of undernutrition. Nearly 870 million people in the world go to bed hungry (1 in 8 people) according to the recently released report of The State of Food Insecurity in the World 2012 (SOFI) and 2 billion have some form of micronutrient malnutrition – this constitutes a crime against humanity and is a responsibility for all of us. Prevalence of undernourishment in developing countries has declined over the past two decades, from 23 to 15 percent. In terms of total numbers in 1990– 92, around 980 million individuals were estimated to be undernourished. The number dropped to 901 million in 1999– 2001, to 885 million in 2006–06 and to 852 million in 2007–09. The financial crisis, economic downturn, persistent food price volatility, drought and other repercussions of climate change since 2006–08 may have prevented any further significant improvements in the number of people who are undernourished in developing countries since then. Africa has by far the highest prevalence, at around 23 percent in 2010–12 but though it is down from what it was in 1990–92 (27 percent) the numbers have risen from 175 million to 239 million with nearly 20 million added in the past four years. In subSaharan Africa, the modest progress achieved in recent years up to 2007 was reversed, with hunger rising 2 percent per year since then. In Asia, both prevalence and numbers dropped over the same period from 24 percent (739 million) to 14 percent (563 million). Latin America and the Caribbean (LAC) boasts the lowest rate of undernourishment (8 percent) among developing country regions but the rate of progress has slowed recently. Countries considered as leastdeveloped countries and lowincome economies have the highest prevalence rates of all around 30 percent but down from the 40 percent levels of twenty years ago.

Interview ラン・ラン

A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric spaces Eigenspace representations Solutions to exercises Bibliography Symbols frequently used Index.

Geometric Analysis on Symmetric Spaces

* Preface * Introduction Part I: Compactifications of Riemannian Symmetric Spaces * Review of Classical Compactifications of Symmetric Spaces * Uniform Construction of Compactifications of Symmetric Spaces * Properties of Compactifications of Symmetric Spaces Part II: Smooth Compactifications of Semisimple Symmetric Spaces * Smooth Compactifications of Riemannian Symmetric Spaces G / K * Semisimple Symmetric Spaces G / H * The Real Points of Complex Symmetric Spaces Defined Over R * The DeConcini-Procesi Compactification of a Complex Symmetric Space and its Real Points * The Oshima-Sekiguchi Compactification of G / K and Comparison with G/Hw (R) Part III: Compactifications of Locally Symmetric Spaces * Classical Compactifications of Locally Symmetric Spaces * Uniform Construction of Compactifications of Locally Symmetric Spaces * Properties of Compactifications of Locally Symmetric Spaces * Subgroup Compactifications of o \ G * Metric Properties of Compactifications of Locally Symmetric Spaces o \ X * References * Index

http://dept.math.lsa.umich.edu/~lji/head.pdf

Compactifications of Symmetric and Locally Symmetric Spaces

On a noncompact symmetric space $G/K$, we obtain optimal upper and lower bounds for the heat kernel $h_t(x,y)$ (as well as asymptotics and estimates of its derivatives), under the assumption that $d(x,y)=O(1+t)$. As a consequence, we get optimal global bounds (same upper and lower bound, up to positive constants), as well as full asymptotics, for other kernels, such as the Green function. This information plays a key role in the description of the Martin compactification of $G/K$, which has been worked out recently by the second author, in collaboration with Y. Guivarc'h and J.C. Taylor.

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Heat Kernel and Green Function Estimates on Noncompact Symmetric Spaces

Compactifications of symmetric spaces have been constructed by different methods for various applications. One application is to provide the so-called rational boundary components which can be used to compactify locally symmetric spaces. In this paper, we construct many compactifications of symmetric spaces using a uniform method, which is motivated by the Borel-Serre compactification of locally symmetric spaces. Besides unifying compactifications of both symmetric and locally symmetric spaces, this uniform construction allows one to compare and relate easily different compactifications, to extend the group action continuously to boundaries of compactifications, and to clarify the structure of the boundaries.

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Compactifications of Symmetric Spaces

Pour un espace localement symetrique M nous definissons une compactification M U M(∞) que nous appelons compactification geodesique. Elle est construite en ajoutant des points limites dans M(∞) a certaines geodesiques dans M. La compactification geodesique apparait dans d'autres cas. Les constructions generales de Gromov permettent, dans le cas des espaces symetriques, d'identifier le bord de la compactification de Gromov avec M(∞). De plus M(oo) se construit naturellement avec la theorie des groupes en utilisant l'immeuble de Tits. La compactification geodesique joue deux roles fondamentaux dans l'analyse harmonique de l'espace localement symetrique: 1) c'est la compactification de Martin minimale pour les valeurs negatives du laplacien et 2) elle peut etre utilisee pour parametrer les valeurs propres du laplacien dans le spectre continu sur L 2 .

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Geometry of compactifications of locally symmetric spaces

Let G be the real locus of a connected semisimple linear algebraic group G defined over Q, and Γ ⊂ G(Q) an arithmetic subgroup. Then the quotient Γ\G is a natural homogeneous space, whose quotient on the right by a maximal compact subgroup K of G gives a locally symmetric space Γ\G/K. This paper constructs several new compactifications of Γ\G. The first two are related to the Borel-Serre compactification and the reductive Borel-Serre compactification of the locally symmetric space Γ\G/K; in fact, they give rise to alternative constructions of these known compactifications. More importantly, the compactifications of Γ\G imply extension to the compactifications of homogeneous bundles on Γ\G/K, and quotients of these compactifications under non-maximal compact subgroups H provide compactifications of period domains Γ\G/H in the theory of variation of Hodge structures. Another compactification of Γ\G is obtained via embedding into the space of closed subgroups of G and is closely related to the constant term of automorhpic forms, in particular Eisenstein series.

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Lizhen Ji

Papers

Compactifications of Symmetric and Locally Symmetric Spaces

Heat Kernel and Green Function Estimates on Noncompact Symmetric Spaces

Compactifications of Symmetric Spaces

Geometry of compactifications of locally symmetric spaces

Compactifications of Locally Symmetric Spaces