Quadratic Forms and Automorphic Forms

Chapter
Part of the Developments in Mathematics book series (DEVM, volume 31)

Abstract

These notes give a friendly four-part introduction to various aspects of the arithmetic and analytic theories of quadratic forms, aimed at a graduate-level audience. The main themes discussed are: geometry and local-global methods, theta functions and Siegel’s theorem, Clifford algebras and spin groups, and adelic theta liftings via the Weil representation.

Key words

Quadratic forms Automorphic forms Local densities Mass formulas Theta series Neighbor method Clifford algebra Spin group Theta lifting 

MSC Classification(s):

11E08 11E12 11E41 11E45 11E88 11F12 11F27 

References

  1. [:1959]
    Correspondence [signed “R. Lipschitz”]. Ann. of Math. (2), 69:247–251, 1959. Attributed to A. Weil.Google Scholar
  2. [AZ95]
    Anatolii Nikolaevich Andrianov and Vladimir Georgievich Zhuravlëv. Modular forms and Hecke operators, volume 145 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1995. Translated from the 1990 Russian original by Neal Koblitz.Google Scholar
  3. [Bak81]
    Anthony Bak. K-theory of forms, volume 98 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1981.Google Scholar
  4. [BH83]
    James William Benham and John Sollion Hsia Spinor equivalence of quadratic forms. J. Number Theory, 17(3):337–342, 1983.Google Scholar
  5. [Bha]
    Manjul Bhargava. 2009 Arizona Winter School lecture notes on “The parametrization of rings of small rank”.Google Scholar
  6. [Bum97]
    Daniel Bump. Automorphic forms and representations, volume 55 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997.Google Scholar
  7. [Cas78]
    John William Scott Cassels. Rational quadratic forms, volume 13 of London Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1978.Google Scholar
  8. [DS05]
    Fred Diamond and Jerry Shurman. A first course in modular forms, volume 228 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2005.Google Scholar
  9. [DSP90]
    William Duke and Rainer Schulze-Pillot. Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math., 99(1):49–57, 1990.Google Scholar
  10. [Duk97]
    William Duke. Some old problems and new results about quadratic forms. Notices Amer. Math. Soc., 44(2):190–196, 1997.Google Scholar
  11. [Eic52]
    Martin Eichler. Die Ähnlichkeitsklassen indefiniter Gitter. Math. Z., 55:216–252, 1952.Google Scholar
  12. [Eic66]
    Martin Eichler. Introduction to the theory of algebraic numbers and functions. Translated from the German by George Striker. Pure and Applied Mathematics, Vol. 23. Academic Press, New York, 1966.Google Scholar
  13. [EKM08]
    Richard Elman, Nikita Karpenko, and Alexander Merkurjev. The algebraic and geometric theory of quadratic forms, volume 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.Google Scholar
  14. [Eve76]
    Howard Eves. An introduction to the history of mathematics. Holt, Rinehart and Winston, fourth edition, 1976.Google Scholar
  15. [Gel75a]
    Stephen Gelbart. Automorphic forms and representations of adele groups. Department of Mathematics, University of Chicago, Chicago, Ill., 1975. Lecture Notes in Representation Theory.Google Scholar
  16. [Gel75b]
    Stephen S. Gelbart. Automorphic forms on adèle groups. Princeton University Press, Princeton, N.J., 1975. Annals of Mathematics Studies, No. 83.Google Scholar
  17. [Gel76]
    Stephen S. Gelbart. Weil’s representation and the spectrum of the metaplectic group. Lecture Notes in Mathematics, Vol. 530. Springer-Verlag, Berlin, 1976.Google Scholar
  18. [Gel79]
    Stephen Gelbart. Examples of dual reductive pairs. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pages 287–296. Amer. Math. Soc., Providence, R.I., 1979.Google Scholar
  19. [Ger08]
    Larry J. Gerstein. Basic quadratic forms, volume 90 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008.Google Scholar
  20. [GS06]
    Philippe Gille and Tamás Szamuely. Central simple algebras and Galois cohomology, volume 101 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006.Google Scholar
  21. [Han04]
    Jonathan Hanke. Some recent results about (ternary) quadratic forms. In Number theory, volume 36 of CRM Proc. Lecture Notes, pages 147–164. Amer. Math. Soc., Providence, RI, 2004.Google Scholar
  22. [Hid00]
    Haruzo Hida. Modular forms and Galois cohomology, volume 69 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2000.Google Scholar
  23. [IK04]
    Henryk Iwaniec and Emmanuel Kowalski. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004.Google Scholar
  24. [Iwa87]
    Henryk Iwaniec. Spectral theory of automorphic functions and recent developments in analytic number theory. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 444–456, Providence, RI, 1987. Amer. Math. Soc.Google Scholar
  25. [Iwa97]
    Henryk Iwaniec. Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997.Google Scholar
  26. [Jac]
    Carl Gustav Jacob Jacobi. Fundamenta nova theoriae functionum ellipticarum. Königsberg, 1829. In Latin. Reprinted with corrections in: Carl Gustav Jacob Jacobi. Gesammelte Werke. 8 volumes. Berlin, 1881–1891. 1. 49–239. reprinted new york (chelsea, 1969) and available from the american mathematical society. edition.Google Scholar
  27. [Jac89]
    Nathan Jacobson. Basic algebra. II. W. H. Freeman and Company, New York, second edition, 1989.Google Scholar
  28. [Kap03]
    Irving Kaplansky. Linear algebra and geometry. Dover Publications Inc., Mineola, NY, revised edition, 2003. A second course.Google Scholar
  29. [Kne66]
    Martin Kneser. Strong approximation. In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pages 187–196. Amer. Math. Soc., Providence, R.I., 1966.Google Scholar
  30. [Kno70]
    Marvin I. Knopp. Modular functions in analytic number theory. Markham Publishing Co., Chicago, Ill., 1970.Google Scholar
  31. [Knu91]
    Max-Albert Knus. Quadratic and Hermitian forms over rings, volume 294 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni.Google Scholar
  32. [Kob93]
    Neal Koblitz. Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.Google Scholar
  33. [KR88a]
    Stephen S. Kudla and Stephen Rallis. On the Weil-Siegel formula. J. Reine Angew. Math., 387:1–68, 1988.Google Scholar
  34. [KR88b]
    Stephen S. Kudla and Stephen Rallis. On the Weil-Siegel formula. II. The isotropic convergent case. J. Reine Angew. Math., 391:65–84, 1988.Google Scholar
  35. [Kud08]
    Stephen S. Kudla. Some extensions of the Siegel-Weil formula. In Eisenstein series and applications, volume 258 of Progr. Math., pages 205–237. Birkhäuser Boston, Boston, MA, 2008.Google Scholar
  36. [Lam05]
    Tsit Yuen Lam. Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2005.Google Scholar
  37. [Lan94]
    Serge Lang. Algebraic number theory, volume 110 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994.Google Scholar
  38. [Lan95]
    Serge Lang. Algebra. Addison-Wesley Publishing Company, Inc., Reading, MA, third edition, 1995.Google Scholar
  39. [LV80]
    Gérard Lion and Michèle Vergne. The Weil representation, Maslov index and theta series, volume 6 of Progress in Mathematics. Birkhäuser Boston, Mass., 1980.Google Scholar
  40. [Miy06]
    Toshitsune Miyake. Modular forms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2006. Translated from the 1976 Japanese original by Yoshitaka Maeda.Google Scholar
  41. [MW06]
    Carlos J. Moreno and Samuel S. Wagstaff, Jr. Sums of squares of integers. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2006.Google Scholar
  42. [Neu99]
    Jürgen Neukirch. Algebraic number theory, volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.Google Scholar
  43. [NSW08]
    Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg. Cohomology of number fields, volume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2008.Google Scholar
  44. [O’M71]
    Onorato Timothy O’Meara. Introduction to quadratic forms. Springer-Verlag, New York, 1971. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 117.Google Scholar
  45. [Ono66]
    Takashi Ono. On Tamagawa numbers. In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pages 122–132. Amer. Math. Soc., Providence, R.I., 1966.Google Scholar
  46. [Par]
    Raman Parimala. 2009 Arizona Winter School lecture notes on “Some aspects of the algebraic theory of quadratic forms”.Google Scholar
  47. [Pfe71]
    Horst Pfeuffer. Einklassige Geschlechter totalpositiver quadratischer Formen in totalreellen algebraischen Zahlkörpern. J. Number Theory, 3:371–411, 1971.Google Scholar
  48. [Pfe78]
    Horst Pfeuffer. Darstellungsmasse binärer quadratischer Formen über totalreellen algebraischen Zahlkörpern. Acta Arith., 34(2):103–111, 1977/78.Google Scholar
  49. [Pie82]
    Richard S. Pierce. Associative algebras, volume 88 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982. Studies in the History of Modern Science, 9.Google Scholar
  50. [PR94]
    Vladimir Platonov and Andrei Rapinchuk. Algebraic groups and number theory, volume 139 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen.Google Scholar
  51. [Pra93]
    Dipendra Prasad. Weil representation, Howe duality, and the theta correspondence. In Theta functions: from the classical to the modern, volume 1 of CRM Proc. Lecture Notes, pages 105–127. Amer. Math. Soc., Providence, RI, 1993.Google Scholar
  52. [Pra98]
    Dipendra Prasad. A brief survey on the theta correspondence. In Number theory (Tiruchirapalli, 1996), volume 210 of Contemp. Math., pages 171–193. Amer. Math. Soc., Providence, RI, 1998.Google Scholar
  53. [PS79]
    Ilya I. Piatetski-Shapiro. Classical and adelic automorphic forms. An introduction. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pages 185–188. Amer. Math. Soc., Providence, R.I., 1979.Google Scholar
  54. [Sah60]
    Chih-han Sah. Quadratic forms over fields of characteristic 2. Amer. J. Math., 82:812–830, 1960.Google Scholar
  55. [Sal99]
    David J. Saltman. Lectures on division algebras, volume 94 of CBMS Regional Conference Series in Mathematics. Published by American Mathematical Society, Providence, RI, 1999.Google Scholar
  56. [Ser77]
    Jean-Pierre Serre. Modular forms of weight one and Galois representations. In Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pages 193–268. Academic Press, London, 1977.Google Scholar
  57. [SH98]
    Rudolf Scharlau and Boris Hemkemeier. Classification of integral lattices with large class number. Math. Comp., 67(222):737–749, 1998.Google Scholar
  58. [Shi73]
    Goro Shimura. On modular forms of half integral weight. Ann. of Math. (2), 97:440–481, 1973.Google Scholar
  59. [Shi94]
    Goro Shimura. Introduction to the arithmetic theory of automorphic functions, volume 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original, Kano Memorial Lectures, 1.Google Scholar
  60. [Shi97]
    Goro Shimura. Euler products and Eisenstein series, volume 93 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.Google Scholar
  61. [Shi04]
    Goro Shimura. Arithmetic and analytic theories of quadratic forms and Clifford groups, volume 109 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004.Google Scholar
  62. [Shi06a]
    Goro Shimura. Integer-valued quadratic forms and quadratic Diophantine equations. Doc. Math., 11:333–367 (electronic), 2006.Google Scholar
  63. [Shi06b]
    Goro Shimura. Quadratic Diophantine equations, the class number, and the mass formula. Bull. Amer. Math. Soc. (N.S.), 43(3):285–304 (electronic), 2006.Google Scholar
  64. [Shi10]
    Goro Shimura. Arithmetic of quadratic forms. Springer Monographs in Mathematics. Springer, New York, 2010.MATHCrossRefGoogle Scholar
  65. [Sie35]
    Carl Ludwig Siegel. Über die analytische Theorie der quadratischen Formen. Ann. of Math. (2), 36(3):527–606, 1935.Google Scholar
  66. [Sie36]
    Carl Ludwig Siegel. Über die analytische Theorie der quadratischen Formen. II. Ann. of Math. (2), 37(1):230–263, 1936.Google Scholar
  67. [Sie37]
    Carl Ludwig Siegel. Über die analytische Theorie der quadratischen Formen. III. Ann. of Math. (2), 38(1):212–291, 1937.Google Scholar
  68. [Sie63]
    Carl Ludwig Siegel. Lectures on the analytical theory of quadratic forms. Notes by Morgan Ward. Third revised edition. Buchhandlung Robert Peppmüller, Göttingen, 1963.Google Scholar
  69. [SP84]
    Rainer Schulze-Pillot. Darstellungsmaße von Spinorgeschlechtern ternärer quadratischer Formen. J. Reine Angew. Math., 352:114–132, 1984.Google Scholar
  70. [SP00]
    Rainer Schulze-Pillot. Exceptional integers for genera of integral ternary positive definite quadratic forms. Duke Math. J., 102(2):351–357, 2000.Google Scholar
  71. [SP04]
    Rainer Schulze-Pillot. Representation by integral quadratic forms—a survey. In Algebraic and arithmetic theory of quadratic forms, volume 344 of Contemp. Math., pages 303–321. Amer. Math. Soc., Providence, RI, 2004.Google Scholar
  72. [Str09]
    Gilbert Strang. Introduction to Linear Algebra. Wellesley Cambridge Press, New York, fourth edition, 2009.Google Scholar
  73. [Tar29]
    W. Tartakowsky. Die gesamtheit der zahlen, die durch eine positive quadratische form \(f(x_{1},x_{2},\ldots,x_{s})\) (s ≥ 4) darstellbar sind. i, ii. Bull. Ac. Sc. Leningrad, 2(7):111–122; 165–196, 1929.Google Scholar
  74. [Tor05]
    Gonzalo Tornaria. The Brandt module of ternary quadratic lattices. PhD thesis, University of Texas, Austin, 2005.Google Scholar
  75. [Voi]
    John Voight. Computing with quaternion algebras: Identifying the matrix ring.Google Scholar
  76. [Vos98]
    Valentin Evgenévich Voskresenskiĭ. Algebraic groups and their birational invariants, volume 179 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ].Google Scholar
  77. [Wat63]
    G. L. Watson. One-class genera of positive quadratic forms. J. London Math. Soc., 38:387–392, 1963.MathSciNetMATHCrossRefGoogle Scholar
  78. [Wat84]
    G. L. Watson. One-class genera of positive quadratic forms in seven variables. Proc. London Math. Soc. (3), 48(1):175–192, 1984.Google Scholar
  79. [Wei67]
    André Weil. Basic number theory. Die Grundlehren der mathematischen Wissenschaften, Band 144. Springer-Verlag New York, Inc., New York, 1967.Google Scholar
  80. [Wei82]
    André Weil. Adeles and algebraic groups, volume 23 of Progress in Mathematics. Birkhäuser Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono.Google Scholar

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Authors and Affiliations

  1. 1.One PalmerSquarePrincetonUSA

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