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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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 objective of the present work was to study the blood pressure with respect to some anthropometric characteristics in adol The study comprised 1225 subjects (634 males and 591 females) aged 10-18 years. The range of average blood pressure (systolic/diastolic) was 116.33-134.52/ 78.23-92.62 mmHg in males and 117.71-128.75/ 77.97-88.85 mm Hg in females of 10 to 18 years of age respectively. It was observed that 11 and 12 years of age groups were more susceptible for pre-hypertension (4.49% and 4.24% respectively). Whereas, 10 years and 14 years age groups were more susceptible for hypertension (6.12% and 4.40% respectively). According to multivariate regression analysis, systolic and diastolic blood pressure was significantly associated with weight, BMI and waist circumference. The results of the present analysis support the hypothesis that BMI is a significant predictor of blood pressure in adolescent age groups.

List of Publications of

Inverse cyclotomic polynomials

Despite their classical nature, continued fractions are a neverending research area, with a body of results accessible enough to suit a wide audience, from researchers to students and even amateur enthusiasts. Neverending Fractions brings these results together, offering fresh perspectives on a mature subject. Beginning with a standard introduction to continued fractions, the book covers a diverse range of topics, from elementary and metric properties, to quadratic irrationals, to more exotic topics such as folded continued fractions and Somos sequences. Along the way, the authors reveal some amazing applications of the theory to seemingly unrelated problems in number theory. Previously scattered throughout the literature, these applications are brought together in this volume for the first time. A wide variety of exercises guide readers through the material, which will be especially helpful to readers using the book for self-study, and the authors also provide many pointers to the literature.

Neverending Fractions: An Introduction to Continued Fractions

Coefficients of ternary cyclotomic polynomials

Let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\Phi_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p 0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1 (2/3-\epsilon)p_j$ for $j\ge 1$.

Ternary cyclotomic polynomials having a large coefficient

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A cyclotomic polynomial8n.x/ is said to be ternary if nD pqr , with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behavior of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family8 pqr.x/ with p< q < r , p and q fixed and r a free prime.

/pdf/the-family-of-ternary-cyclotomic-polynomials-with-one-free-1rxkf83069.pdf

The family of ternary cyclotomic polynomials with one free prime

A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers. 
As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic polynomials with an optimally large set of coefficients.

Yves Gallot

Papers

Ternary cyclotomic polynomials having a large coefficient

Ternary cyclotomic polynomials having a large coefficient

The family of ternary cyclotomic polynomials with one free prime

Neighboring ternary cyclotomic coefficients differ by at most one

Neighboring ternary cyclotomic coefficients differ by at most one