Collected works of John Tate. Part I (1951–1975). Edited by Barry Mazur and Jean-Pierre Serre.

*(English)*Zbl 1407.01030
Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9092-9/hbk; 978-1-4704-3021-4/ebook). xxvii, 716 p. (2016).

Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his “Fourier analysis in number fields and Hecke’s zeta-functions”, Tate’s thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305–347 (1967)], where Tate worked out Emil Artin’s suggestion to derive the functional equation of Hecke’s zeta functions using the newly developed tool of ideles.

Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin’s reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162–203 (1967; Zbl 1179.11041)] in the Brighton proceedings.

In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201–211 (1971; Zbl 0229.12013)] and Milnor groups [with H. Bass, Lect. Notes Math. 342, 349–446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with B. Mazur, Invent. Math. 22, 41–49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179–206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate’s collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate’s collected works was published by J. S. Milne [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551–558 (2017; Zbl 1369.00040)]; see also [J. S. Milne, in: The Abel Prize 2008–2012. Heidelberg: Springer. 259–340 (2014; Zbl 1317.01011)].

The individual articles are: “Fourier analysis in number fields and Hecke’s zeta-functions”, “A note on finite ring extensions” (with E. Artin) [Zbl 0043.26701], “On the relation between extremal points of convex sets and homomorphisms of algebras” [Zbl 0043.11403], “Genus change in inseparable extensions of function fields” [Zbl 0047.03901], “On Chevalley’s proof of Lüroth’s theorem” (with S. Lang) [Zbl 0047.03802], “The higher dimensional cohomology groups of class field theory” [Zbl 0047.03703], “The cohomology groups in algebraic number fields” [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2–9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66–67 (1954)], “On the Galois cohomology of unramified extensions of function fields in one variable” (with Y. Kawada) [Zbl 0068.03402], “On the characters of finite groups” (with R. Brauer) [Zbl 0065.01401], “Homology of Noetherian rings and local rings” [Zbl 0079.05501], “WC-groups over \(p\)-adic fields” [Zbl 0091.33701], “On the inequality of Castelnuovo-Severi” (with A. Mattuck) [Zbl 0081.37604], “On the inequality of Castelnuovo-Severi, and Hodge’s theorem” [unpublished], “Principal homogeneous spaces over abelian varieties” (with S. Lang) [Zbl 0097.36203], “Principal homogeneous spaces for abelian varieties” [Zbl 0116.38201], “A different with an odd class” (with A. Fröhlich and J.-P. Serre) [Zbl 0105.02903], “Nilpotent quotient groups” [Zbl 0125.01503], “Duality theorems in Galois cohomology over number fields” [Zbl 0126.07002], “Ramification groups of local fields” (with S. Sen) [Zbl 0136.02702], “Formal complex multiplication in local fields” (with J. Lubin) [Zbl 0128.26501], “Algebraic cycles and poles of zeta functions” [Zbl 0213.22804], “Elliptic curves and formal groups” (with J. Lubin and J.-P. Serre) [unpublished], “On the conjectures of Birch and Swinnerton-Dyer and a geometric analog” [Zbl 0199.55604], “Formal moduli for one-parameter formal Lie groups” (with J. Lubin) [Zbl 0156.04105], “The cohomology groups of tori in finite Galois extensions of number fields” [Zbl 0146.06501], “Global class field theory” [Zbl 1179.11041], “Endomorphisms of Abelian varieties over finite fields” [Zbl 0147.20303], “The rank of elliptic curves” (with I. R. Shafarevich) [Zbl 0168.42201], “Residues of differentials on curves” [Zbl 0159.22702], “\(p\)-divisible groups” [Zbl 0157.27601], “The work of David Mumford” [Zbl 0333.01015], “Classes d’isogénie des variétés abéliennes sur un corps fini (d’après Z. Honda)” [Zbl 0212.25702], “Good reduction of abelian varieties” [Zbl 0172.46101], “Group schemes of prime order” (with F. Oort) [Zbl 0195.50801], “Symbols in arithmetic” [Zbl 0229.12013], “Rigid analytic spaces” [Zbl 0212.25601], “The Milnor ring of a global field” [Zbl 0299.12013], “Appendix to The Milnor ring of a global field” [unpublished], “Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology” [Zbl 0284.12004], “Points of order \(13\) on elliptic curves” (with B. Mazur) [Zbl 0268.14009], “The arithmetic of elliptic curves” [Zbl 0296.14018], “The 1974 Fields Medals. I: An algebraic geometer” [Zbl 1225.01087], “Algorithm for determining the type of a singular fiber in an elliptic pencil” [Zbl 1214.14020].

Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin’s reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162–203 (1967; Zbl 1179.11041)] in the Brighton proceedings.

In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201–211 (1971; Zbl 0229.12013)] and Milnor groups [with H. Bass, Lect. Notes Math. 342, 349–446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with B. Mazur, Invent. Math. 22, 41–49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179–206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate’s collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate’s collected works was published by J. S. Milne [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551–558 (2017; Zbl 1369.00040)]; see also [J. S. Milne, in: The Abel Prize 2008–2012. Heidelberg: Springer. 259–340 (2014; Zbl 1317.01011)].

The individual articles are: “Fourier analysis in number fields and Hecke’s zeta-functions”, “A note on finite ring extensions” (with E. Artin) [Zbl 0043.26701], “On the relation between extremal points of convex sets and homomorphisms of algebras” [Zbl 0043.11403], “Genus change in inseparable extensions of function fields” [Zbl 0047.03901], “On Chevalley’s proof of Lüroth’s theorem” (with S. Lang) [Zbl 0047.03802], “The higher dimensional cohomology groups of class field theory” [Zbl 0047.03703], “The cohomology groups in algebraic number fields” [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2–9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66–67 (1954)], “On the Galois cohomology of unramified extensions of function fields in one variable” (with Y. Kawada) [Zbl 0068.03402], “On the characters of finite groups” (with R. Brauer) [Zbl 0065.01401], “Homology of Noetherian rings and local rings” [Zbl 0079.05501], “WC-groups over \(p\)-adic fields” [Zbl 0091.33701], “On the inequality of Castelnuovo-Severi” (with A. Mattuck) [Zbl 0081.37604], “On the inequality of Castelnuovo-Severi, and Hodge’s theorem” [unpublished], “Principal homogeneous spaces over abelian varieties” (with S. Lang) [Zbl 0097.36203], “Principal homogeneous spaces for abelian varieties” [Zbl 0116.38201], “A different with an odd class” (with A. Fröhlich and J.-P. Serre) [Zbl 0105.02903], “Nilpotent quotient groups” [Zbl 0125.01503], “Duality theorems in Galois cohomology over number fields” [Zbl 0126.07002], “Ramification groups of local fields” (with S. Sen) [Zbl 0136.02702], “Formal complex multiplication in local fields” (with J. Lubin) [Zbl 0128.26501], “Algebraic cycles and poles of zeta functions” [Zbl 0213.22804], “Elliptic curves and formal groups” (with J. Lubin and J.-P. Serre) [unpublished], “On the conjectures of Birch and Swinnerton-Dyer and a geometric analog” [Zbl 0199.55604], “Formal moduli for one-parameter formal Lie groups” (with J. Lubin) [Zbl 0156.04105], “The cohomology groups of tori in finite Galois extensions of number fields” [Zbl 0146.06501], “Global class field theory” [Zbl 1179.11041], “Endomorphisms of Abelian varieties over finite fields” [Zbl 0147.20303], “The rank of elliptic curves” (with I. R. Shafarevich) [Zbl 0168.42201], “Residues of differentials on curves” [Zbl 0159.22702], “\(p\)-divisible groups” [Zbl 0157.27601], “The work of David Mumford” [Zbl 0333.01015], “Classes d’isogénie des variétés abéliennes sur un corps fini (d’après Z. Honda)” [Zbl 0212.25702], “Good reduction of abelian varieties” [Zbl 0172.46101], “Group schemes of prime order” (with F. Oort) [Zbl 0195.50801], “Symbols in arithmetic” [Zbl 0229.12013], “Rigid analytic spaces” [Zbl 0212.25601], “The Milnor ring of a global field” [Zbl 0299.12013], “Appendix to The Milnor ring of a global field” [unpublished], “Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology” [Zbl 0284.12004], “Points of order \(13\) on elliptic curves” (with B. Mazur) [Zbl 0268.14009], “The arithmetic of elliptic curves” [Zbl 0296.14018], “The 1974 Fields Medals. I: An algebraic geometer” [Zbl 1225.01087], “Algorithm for determining the type of a singular fiber in an elliptic pencil” [Zbl 1214.14020].

Reviewer: Franz Lemmermeyer (Jagstzell)

##### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

11-03 | History of number theory |

14-03 | History of algebraic geometry |

11Gxx | Arithmetic algebraic geometry (Diophantine geometry) |

14Gxx | Arithmetic problems in algebraic geometry; Diophantine geometry |

14Kxx | Abelian varieties and schemes |