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The Life and Work of R.A. Rankin (1915–2001)

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Number Theory and Modular Forms

Part of the book series: Developments in Mathematics ((DEVM,volume 10))

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Abstract

Robert Alexander Rankin, an eminent Scottish number theorist and, for several decades, one of the world’s foremost experts in modular forms, died on January 27, 2001 in Glasgow at the age of 85. He was one of the founding editors of The Ramanujan Journal. For this and the next two issues of the The Ramanujan Journal, many well-known mathematicians have prepared articles in Rankin’s memory. In this opening paper, we provide a short biography of Rankin and discuss some of his major contributions to mathematics. At the conclusion of this article, we provide a complete list of Rankin’s doctoral students and a complete bibliography of all of Rankin’s writings divided into five categories: (1) Research and Expository Papers; (2) Books; (3) Books Edited; (4) Obituaries; (5) Other Writings.

Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.

Research partially supported by the an Alfred P. Sloan Foundation Research Fellowship, the National Science Foundation, a David and Lucile Packard Foundation Research Fellowship, an H.I. Romnes Fellowship and a John S. Guggenheim Fellowship.

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Articles

  1. Robert A. Rankin, “The difference between consecutive prime numbers,” J. London Math. Soc. 13 (1938), 242–247.

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  2. Robert A. Rankin, “Contributions to the theory of Ramanujan’s function r(n) and similar arithmetical functions. I. The zeros of the function, r(n)ln` on the line Rs = 13/2,” Proc. Cambridge Philos. Soc. 35 (1939), 351–356.

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  3. Robert A. Rankin, “Contributions to the theory of Ramanujan’s function r(n) and similar arithmetical functions. II. The order of the Fourier coefficients of integral modular forms,” Proc. Cambridge Philos. Soc. 35 (1939). 357–373.

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  4. Robert A. Rankin, “Contributions to the theory of Ramanujan’s function r(n) and similar arithmetical functions. III. A note on the sum function of the Fourier coefficients of integral modular forms,” Proc. Cambridge Philos. Soc. 36 (1940), 150–151.

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  5. Robert A. Rankin, “The difference between consecutive prime numbers. II,” Proc. Cambridge Philos. Soc. 36 (1940), 255–266.

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  6. Robert A. Rankin, “On the representations of a number as a sum of squares and certain related identities,” Proc. Cambridge Philos. Soc. 41 (1945) 12.

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  7. Robert A. Rankin, “A note on a particular type of asymptotic series,” Philos. Mag. 36 (7) (1945), 860–861.

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  8. Robert A. Rankin, “A certain class of multiplicative functions.” Duke Math. J. 13 (1946), 281–306.

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  9. Robert A. Rankin, (with D.G. Kendall) “On the number of Abelian groups of a given order,” Quart. J. Math., Oxford Ser. 18 (1947), 197–208.

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  10. Robert A. Rankin, “On the closest packing of spheres in n dimensions.” Ann. Math. 48 (2) (1947), 1062–1081.

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  11. Robert A. Rankin, “The difference between consecutive prime numbers. III,’ J. London Math. Soc. 22 (1947), 226–230.

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  12. Robert A. Rankin, “A campanological problem in group theory.” Proc. Cambridge Philos. Soc. 44 (1948), 17–25.

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  13. Robert A. Rankin, “On sums of powers of linear forms. III,” Nederl. Akad. Wetensch., Proc. 51 (1948), 846–853.

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  14. Robert A. Rankin, (under the name Rob Alasdair Mac Fhraing) “The numbering of Fionn’s and Dubhan’s men, and the story of Josephus and the forty Jews” (Gaelic), Proc. Roy. Irish Acad. Sect. A. 52 (1948), 87–93.

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  15. Robert A. Rankin, “The mathematical theory of the motion of rotated and unrotated rockets,’ Philos. Trans. Roy. Soc. London Ser. A 241 (1949), 457–585.

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  16. Robert A. Rankin, “On sums of powers of linear forms. I,” Ann. Math. 50 (2) (1949), 691–698.

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  17. Robert A. Rankin, “On sums of powers of linear forms. II,’ Ann. Math. 50 (2) (1949), 699–704.

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  18. Robert A. Rankin, “The difference between consecutive prime numbers. IV,” Proc. Amer Math. Soc. 1 (1950), 143–150.

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  19. Robert A. Rankin, “The scalar product of modular forms,” Proc. London Math. Soc. 2 (3) (1952), 198–217.

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  20. Robert A. Rankin, “A problem concerning the product of the differences of n variables,” Norske Vid. Selsk. Forh., Trondheim 25 (1952), 50–53.

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  21. Robert A. Rankin, “The anomaly of convex bodies.” Proc. Cambridge Philos. Soc. 49 (1953), 54–58.

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  22. Robert A. Rankin, “A problem concerning three-dimensional convex bodies.” Proc. Cambridge Philos. Soc. 49 (1953), 44–53.

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  23. Robert A. Rankin, “On positive definite quadratic forms,” J. London Math. Soc. 28 (1953), 309–314.

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  24. Robert A. Rankin, (with D.G. Kendall) “On the number of points of a given lattice in a random hypersphere.” Quart. J. Math., Oxford Ser. 4 (2) (1953), 178–189.

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  25. Robert A. Rankin, “A minimum problem for the Epstein zeta-function,” Proc Glasgow Matt. Assoc. 1 (1953), 149–158.

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  26. Robert A. Rankin, (with J.M. Rushforth) “The coefficients of certain integral modular forms,” Proc. Cambridge Philos. Soc. 50 (1954), 305–308.

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  27. Robert A. Rankin, “On horocyclic groups.” Proc. London Math. Soc. 4 (3) (1954), 219–234.

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  28. Robert A. Rankin, “Chebyshev polynomials and the modulary group of level p,” Math. Scand. 2 (1954), 315–326.

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  29. Robert A. Rankin, “Van der Corput’s method and the theory of exponent pairs,” Quart. J. Math., Oxford Ser. 6 (2) (1955), 147–153.

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  30. Robert A. Rankin, “The closest packing of spherical caps in n dimensions,” Proc. Glasgow Math. Assoc. 2 (1955), 139–144.

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  31. Robert A. Rankin, “On packings of spheres in Hilbert space.” Proc. Glasgow Math. Assoc. 2 (1955), 145–146.

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  32. Robert A. Rankin, “On the minimal points of positive definite quadratic forms,” Mathematika 3 (1956), 15–24.

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  33. Robert A. Rankin, “The construction of automorphic forms from the derivatives of a given form;’ J. Indian Math. Soc. (N.S.) 20 (1956), 103–116.

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  34. Robert A. Rankin, “Diophantine approximation and horocyclic groups.” Canad J. Math. 9 (1957). 277–290.

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  35. Robert A. Rankin, “The construction of automorphic forms from the derivatives of given forms,’ Michigan Math. J. 4 (1957), 181–186.

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  36. Robert A. Rankin, “An inequality,” Math. Gaz. 42 (1958), 39–40.

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  37. Robert A. Rankin, “The construction of branched covering Riemann surfaces,” Proc. Glasgow Math. Assoc. 3 (1958), 199–207.

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  38. Robert A. Rankin, “Sir Edmund Whittaker’s work on automorphic functions,” Proc. Edinburgh Math. Soc. 11 (1958), 2530.

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  39. Robert A. Rankin, “The Schwarzian derivative and uniformization,” J. Analyse Math. 6 (1958), 149–167.

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  40. Robert A. Rankin, (with J.A.C. Burlak and A.P. Robertson) “The packing of spheres in the space l p,” Proc. Glasgow Math. Assoc. 4 (1958), 22–25.

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  41. Robert A. Rankin, “A cyclic inequality,” Proc. Edinburgh Math. Soc. 12 (2) (1961), 139–147.

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  42. Robert A. Rankin, “The differential equations associated with the uniformization of certain algebraic curves,” Proc. Roy. Soc. Edinburgh Sect. A 65 (1962), 35–62.

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  43. Robert A. Rankin, “Representations of a number as the sum of a large number of squares,” Proc. Roy. Soc. Edinburgh Sect. A 65 (1962), 318–331.

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  44. Robert A. Rankin, “Sets of integers containing not more than a given number of terms in arithmetical progression,” Proc. Roy. Soc. Edinburgh Sect. 65 (1962), 332–344.

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  45. Robert A. Rankin, “The divisibility of divisor functions,” Proc. Glasgow Math. Assoc. 5 (1961), 35–40.

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  46. Robert A. Rankin, “A crystal dislocation problem,” Proc. Cambridge Philos. Soc. 57 (1961), 898–899.

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  47. Robert A. Rankin, “On sequences of integers containing no arithmetical progressions,” Bull. Malayan Math. Soc. 8 (1961), 43–52.

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  48. Robert A. Rankin, “On the representation of a number as the sum of any number of squares, and in particular of twenty,” Acta Arith. 7 (1962), 399–407.

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  49. Robert A. Rankin “Multiplicative functions and operators of Hecke type,” Acta Math. Acad. Sci. Hangar. 13 (1962), 81–89.

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  50. Robert A. Rankin, “Change of variable in an indefinite integral,” Math. Gaz. 43 (1962), 14–17.

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  51. Robert A. Rankin, “The difference between consecutive prime numbers. V,” Proc. Edinburgh Math. Soc. 13 (2) (1963), 331–332.

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  52. Robert A. Rankin, “On the minimal points of perfect quadratic forms,” Math. Z. 84 (1964), 228–232.

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  53. Robert A. Rankin, “Difference sets,” Acta Arith. 9 (1964), 161–168.

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  54. Robert A. Rankin, “Sums of squares and cusp forms,” Amer. J. Math. 87 (1965), 857–860.

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  55. Robert A. Rankin, “Functions whose powers have non-negative Taylor coefficients,” Proc. London Math. Soc. 14 (3) (1965), 239–248.

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  56. Robert A. Rankin, A campanological problem in group theory. II, “ Proc. Cambridge Philos. Soc. 62 (1966), 11–18.

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  57. Robert A. Rankin, “Isomorphic congruence groups and Hecke operators,” Proc. Glasgow Math. Assoc. 7 (1966), 168.

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  58. Robert A. Rankin, “Functions whose powers have non-negative Taylor coefficients. II. Corrigenda and further results,” Proc. London Math. Soc. 16 (3) (1966), 766–768.

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  59. Robert A. Rankin, “Common transversals,” Proc. Edinburgh Math. Soc. 15 (2) (1967), 147–154.

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  60. Robert A. Rankin, “Hecke operators on congruence subgroups of the modular group,” Math. Ann. 168 (1967), 40–58.

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  61. Robert A. Rankin, “Lattice subgroups of free congruence groups,” Invent. Math. 2 (1967), 215–221.

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  62. Robert A. Rankin, “The zeros of Eisenstein series,” Publ. Ramanujan Inst. No. 1 (1968/1969), 137–144.

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  63. Robert A. Rankin, “Ramanujan’s function r(n),” in 1970 Symposia on Theoretical Physics and Mathematics, Inst. Math. Sci., Madras, 1969, Vol. 10, pp. 37–45.

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  64. Robert A. Rankin, Subgroups of the modular group generated by parabolic elements of constant amplitude,“ Acta Arith. 18 (1971), 145–151.

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  65. Robert A. Rankin, “An 12-result for the coefficients of cusp forms,” Math. Ann. 203 (1973), 239–250.

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  66. Robert A. Rankin, “Subgroups of the modular group defined by a single linear congruence,” Acta Arith. 24 (1973), 313–323.

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  67. Robert A. Rankin, “Elementary proofs of relations between Eisenstein series,” Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 107–117.

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  68. Robert A. Rankin, “Ramanujan’s unpublished work on congruences,” in Modular Functions of One Variable, V (Proc. Second Internat. Conf, Univ. Bonn, Bonn, 1976) (J.-P. Serre and D.B. Zagier, eds.), Lecture Notes in Math., Vol. 601, Springer-Verlag, Berlin, 1977, pp. 3–15.

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  69. Robert A. Rankin, Hecke operators, oldforms and newforms,“ in Discrete Groups and Automorphic Functions (Proc. Conf, Cambridge, 1975) (W.J. Harvey, ed.), Academic Press, London, 1977, pp. 363–375.

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  70. Robert A. Rankin, “Subgroups of the unimodular group defined by a congruence,” Math. Proc. Cambridge Philos. Soc. 86 (1979), 451–459.

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  71. Robert A. Rankin, The vanishing of Poincaré series,’ Proc. Edinburgh Math. Soc. 23 (2) (1980), 15I-161.

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  72. Robert A. Rankin, “The Fourier coefficients of certain Eisenstein series,” Analysis 1 (1981), 229–239.

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  73. Robert A. Rankin, “Ramanujan’s manuscripts and notebooks,” Bull. London Math. Soc. 14 (1982), 81–97.

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  74. Robert A. Rankin, “The zeros of certain Poincaré series,” Cotnpositio Math. 46 (1982), 255–272.

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  75. Robert A. Rankin, “Sums of powers of cusp form coefficients,” Math. Ann. 263 (1983), 227–236.

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  76. Robert A. Rankin, “The first hundred years (1883–1983),” Proc. Edinburgh Math. Soc. 26 (2) (1983). 135–150.

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  77. Robert A. Rankin, “Ramanujan as a patient.” Proc. Indian Acad. Sci. (Math. Sci.) 93 (1984). 79–100.

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  78. Robert A. Rankin, “The construction of automorphic forms from the derivatives of a given form. Il,” Canad. Math. Bull. 28 (1985), 306–316.

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  79. Robert A. Rankin, “A family of newforms,” Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 461–467.

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  80. Robert A. Rankin, “Sums of powers of cusp form coefficients. II,” Math. Ann. 272 (1985), 593–600.

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  81. Robert A. Rankin, “Fourier coefficients of cusp forms,” Math. Proc. Cambridge Philos. Soc. 100 (1986), 5–29.

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  82. Robert A. Rankin, “Srinivasa Ramanujan (1887–1920),” Bull. Inst. Math. Appl. 23 (1987), 145–152.

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  83. Robert A. Rankin, “Generalized Jacobsthal sums and sums of squares,” Acta Arith. 49 (1987), 5–14.

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  84. Robert A. Rankin, “Cusp forms of given level and real weight,” J. Indian Math. Soc. (N.S.) 51 (1987), 37–48.

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  85. Robert A. Rankin, “Ramanujan’s tau-function and its generalizations,” in Ratanujan Revisited ( G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan, and R.A. Rankin, eds.), Academic Press. Boston. 1988, pp. 245–268.

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  86. Robert A. Rankin, “The adjoint Hecke operator, I,” J. Madras Univ. Sect. B 51 (1988). 22–42.

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  87. Robert A. Rankin, “The adjoint Hecke operator, II,” in Number Theory and Related Topics (Bombay, 1988 ), Oxford University Press, Bombay, 1989, pp. 161–175.

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  88. Robert A. Rankin, “Ramanujan’s manuscripts and notebooks. II.’ Bull. London Math. Soc. 21 (1989). 351–365.

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  89. Robert A. Rankin, “Diagonalizing Eisenstein series. I.” in Analytic Number Theory (Allerton Park, IL, 1989) ( B.C. Berndt, H.G. Diamond, H. Halberstam, and A. Hildebrand, eds.), Birkhäuser, Boston. 1990, pp. 429–450.

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  90. Robert A. Rankin, “The adjoint Hecke operator.” in Automorphic Functions and Their Applications (Khabarovsk, 1988) ( N. Kuznetsov and V. Bykovsky, eds.), Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990, pp. 163–166.

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  91. Robert A. Rankin, “Sums of cusp form coefficients,’ in Conference on Autotnorphic Forms and Analytic Number Theory (Montreal, PQ, 1989) ( R. Murty, ed.), Univ. Montréal, Montreal, 1990, pp. 115–121.

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  92. Robert A. Rankin, “Diagonalizing Eisenstein series. III,” in Discrete Groups and Geometry (Birmingham, 1991) (W.J. Harvey, ed.), London Mathematical Society Lecture Note Ser. 173, Cambridge University Press, Cambridge. 1992, pp. 196–208.

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  93. Robert A. Rankin, “Diagonalizing Eisenstein series, II,” in A Tribute to Emil Grosswald: Number Theory and Related Analysis (M. Knopp and M. Sheingom, eds.), Contemp. Math. No. 143, American Mathematical Society, Providence, RI, 1993, pp. 525–537.

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  94. Robert A. Rankin, “Diagonalizing Eisenstein series. IV,” in The Rademacher Legacy to Mathematics (University Park, PA, 1992) (G.E. Andrews, D.M. Bressoud. and L.A. Parson, eds.), Contemp. Math. No. 166, American Mathematical Society, Providence, RI, 1994, pp. 107–118.

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  95. Robert A. Rankin, “On certain meromorphic modular forms,” in Analytic Number Theory (Allerton Park, IL. 1995) (B.C. Berndt, H.G. Diamond, and A.J. Hildebrand, eds. ), Birkhäuser, Hildebrand, eds. ), 1996, Vol. 2, pp. 713–721.

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  96. Robert A. Rankin, `Burnside’s uniformization,“ Acta Arith. 79 (1997), 53–57.

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  97. Robert A. Rankin, “G.H. Hardy as I knew him,” Austral. Math. Soc. Gaz. 25 (1998) 73–81; revised version in Number Theory for the Millennium (M.A. Bennett, B.C. Berndt, N. Boston, H.G. Diamond, A.J. Hildebrand, and W. Philipp, eds.), A K Peters, Natick, MA, 2002, Vol. 3, pp. 191–203.

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  98. Robert A. Rankin, “Modular forms and Hecke operators,” in Number Theory and its Applications (Ankara, 1996) (C.Y. Yildirim and S.A. Stepanov, eds.), Lecture Notes in Pure and Appl. Math., 204, Dekker, New York, 1999, pp. 151–169.

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  99. Robert A. Rankin, “Newforms for the modular group on spaces of dimension 2,” in Number Theory in Progress (ZakopaneKoscielisko, 1997) (K. Györy, H. Iwaniec, and J. Urbanowicz, eds.), de Gruyter, Berlin, 1999, Vol. 2, pp. 1065–1070.

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  100. Robert A. Rankin, “Sums of squares: An elementary method,” in Number Theory ( R.P. Bambah, V.C. Dumir, and R. Hans-Gill, eds.), Hindustan Book Co., Delhi, 1999, pp. 371–399.

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  101. Robert A. Rankin, (with B.C. Berndt), “The books studied by Ramanujan in India,” Amer. Math. Monthly 107 (2000), 595–601.

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Books

  1. Robert A. Rankin, An Introduction to Mathematical Analysis, Macmillan, New York, 1963.

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  2. Robert A. Rankin, The Modular Group and its Subgroups, The Ramanujan Institute, Madras, 1969.

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  3. Robert A. Rankin, Modular Forms and Functions, Cambridge University Press, Cambridge, 1977.

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  4. Robert A. Rankin, (with B.C. Berndt) Ramanujan: Letters and Commentary, American Mathematical Society, Providence, RI, 1995; London Mathematical Society, London, 1995.

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  5. Robert A. Rankin, (with B.C. Berndt) Ramanujan: Essays and Surveys, American Mathematical Society, Providence, RI, 2001; London Mathematical Society, London, 2001.

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Books edited

  1. Robert A. Rankin, Collected Papers of G.H. Hardy (including joint papers with J.E. Littlewood and others), Vols. V, VI, VII (L.S. Bosanquet, I.W. Busbridge, M.L. Cartwright, E.F. Collingwood, H. Davenport, T.M. Flett, H. Heilbronn, A.E. Ingham, R. Rado, R.A. Rankin, W.W. Rogosinski, F. Smithies, E.C. Titchmarsh, and E.M. Wright, eds.), Clarendon Press, Oxford, 1972, 1974, 1979.

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  2. Robert A. Rankin, Modular Forms (Papers from the symposium held at the University of Durham, June 30-July 10, 1983 ) (R.A. Rankin, ed.), Wiley, New York, 1984.

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  3. Robert A. Rankin, Ramanujan Revisited (Proceedings of the Ramanujan Centenary Conference held at the University of Illinois, Urbana-Champaign, Illinois, June 1–5, 1987 ) (G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan, and R.A. Rankin, eds.), Academic Press, Boston, 1988.

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  4. Robert A. Rankin, Contributions to biographical commentaries in The Collected Papers of Hans Arnold Heilbronn (E.J. Kani and R.A. Smith, eds.), Wiley, New York, 1988.

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Obituary and biographical notices

  1. Robert A. Rankin, “Thomas Murray MacRobert,” J London Math. Soc. 39 (1964), 176–182;

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  2. Robert A. Rankin, Nature 196 (1962), 1267.

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  3. Robert A. Rankin, “George Neville Watson,” J. London Math. Soc. 41 (1966), 551–565;

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  4. Robert A. Rankin, Year Book Roy. Soc. Edinburgh (1966), 37–39.

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  5. Robert A. Rankin, “William Barry Pennington,” Bull. London Math. Soc. 1 (1969), 382–385;

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  6. Robert A. Rankin, Nature 219 (1968), 207–208.

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  7. Robert A. Rankin, “G. N. Watson,” Dictionary of Scientific Biography (Scribners) 14 (1976), 188–189.

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  8. Robert A. Rankin, Thomas S. Graham, “ Proc. Edinburgh Math. Soc. 21 (2) (1979), 187–188.

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  9. Robert A. Rankin, Robert Pollock Gillespie, “ Year Book Roy. Soc. Edinburgh (1978), 31–32.

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  10. Robert A. Rankin, Richard Alexander Robb, “ Year Book Roy. Soc. Edinburgh (1978), 52–53.

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  11. Robert A. Rankin, Edward Thomas Copson, “ Bull. London Math. Soc. 13 (1981), 564–567;

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  12. Robert A. Rankin, Year Book Roy. Soc. Edinburgh(1981), 8–10.

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  13. Robert A. Rankin, “Gertrude Katherine Stanley,” Bull. London Math. Soc. 14 (1982), 554–555.

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  14. Robert A. Rankin, (John) Charles Burkill,“ The Independent 24 April 1993.

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Other writings

  1. Robert A. Rankin, “Oran na Comhachaig,” Trans. Gaelic Soc. Glasgow 5 (1958), 122–171.

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  2. Robert A. Rankin, “A missing manuscript of Robert Boyd,” College Courant 25 (1973), 10–17.

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  3. Robert A. Rankin, “George Campbell Hay as I knew him,” Chapman 40 (1985), 1–12.

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  4. Robert A. Rankin, Mathematics,“ in A Faculty for Science: A Unified Diversity: A Century of Science in the University of Glasgow (R.Y. Thomson, ed. ), University of Glasgow, 1993, pp. 19–31.

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  5. Robert A. Rankin, March Stones in the Kilpatrick Hills,Clydebank District Libraries and Museums Department, Glasgow, 1993.

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  6. Robert A. Rankin, “George A. Gibson and the Gibson lectureship in the history of mathematics.” Brit. Soc. Hist. Math. Newsletter 29 (1995), 7–8.

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  7. Robert A. Rankin, The University of Glasgow Library“ (British Libraries #12), Brit. Soc. Hist. Math. Newsletter 31 (1996), 44–46.

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  8. Robert A. Rankin, “More on Maclaurin,” Math. Intel. 18 (2) (1996), 5.

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  9. Robert A. Rankin, My Cambridge years, “ Brit. Soc. Hist. Math. Newsletter 38 (1998), 30–34.

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  10. Robert A. Rankin, “Place-names in the Comhachag and other similar poems,” Scottish Gaelic Studies 18 (1998), 111–130.

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  11. Robert A. Rankin, “Addendum to Place-Names in the Comhachag, Scottish Gaelic Studies, 19 (1999), 257.

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  12. Robert A. Rankin, Hugh Blackburn: A little-known mathematical friend of Lord Kelvin, “ Brit. Soc. Hist. Math. 43 (2001), 7–14.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois, 61801, USA

    Bruce C. Berndt

  2. Mathematisches Institut, Universität Heidelberg, INF 288, D-69120, Heidelberg, Germany

    Winfried Kohnen

  3. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin, 53726, USA

    Ken Ono

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  1. Bruce C. Berndt
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  2. Winfried Kohnen
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  3. Ken Ono
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Editors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, Illinois, USA

    Bruce Berndt

  2. Department of Mathematics, University of Wisconsin, Madison, Wisconsin, USA

    Ken Ono

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Berndt, B.C., Kohnen, W., Ono, K. (2003). The Life and Work of R.A. Rankin (1915–2001). In: Berndt, B., Ono, K. (eds) Number Theory and Modular Forms. Developments in Mathematics, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6044-6_1

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