Integral quadratic forms and volume formulas

UBC graduate seminar, winter 2011

This seminar will feature talks by students. The principal subject will be the representation of positive integers by positive definite integral quadratic forms, and the main result will be Siegel's formula for the number of ways a given positive integer can be expressed in terms of a given form and other forms in its class. This turns out to be a special if most attractive case of a very general result about the relation between values of $L$-functions and volumes of arithmetic quotients. The course will be broken down into specific topics:

Part I. Deciphering Siegel's Theorem

  • Statement of Siegel's formula, and some examples (Siegel/Morgan, Cassels: Appendix)
  • The sizes of spheres over finite fields (Artin, notes)
  • Siegel's local factors for p-adic fields (Cassels, notes)
  • Euclidean spheres and Gamma functions (notes)
  • Values of L-functions (Borevitch-Shafarevitch, notes)
  • Classes and genera (Cassels: Chapters 7, 8, 9)
  • Quadratic extensions of Q and binary quadratic forms (Borevitch-Shafarevitch)
  • Dirichlet's class number formula (Borevitch-Shafarevitch)
  • Sums of three squares (Grosswald: Chapter 3, Cassels: Chapter 9.6, Gauss: DA sections 266-292)

Part II. Proving Siegel's Theorem

  • Introduction to theta functions (Gunning)
  • Brief introduction to modular forms (Gunning)
  • Ramanujan's conjecture (Gunning)

Part III. Adèles and volumes

  • Siegel's formula and adèles (Tamagawa)
  • Elementary divisors and the volume of SL2(Z)\SL2(R)(Voskresenskii)
  • Volumes of adèle groups (Tamagawa, Voskresenski, Weil)

    Reading material

    My own notes:

    The relevant literature is scattered. Only small independent parts of each of the following will be relevant. I have paper and electronic copies of many of them. The one text that comes closest to being adequate is Cassels, of which Chapters 7, 8, 9, 12, 14, and Appendix B are of interest.

    I put a star * to mark items I know to be available on the Internet, either through UBC or in public access.


    • Emil Artin, Geometric algebra, Wiley, 1957.
    • Emil Grosswald, Representations of numbers as sums of squares, Springer, 1985.

      Chapter 4 is a good introduction to integral quadratic forms in general, and sums of three squares in particular.

    • J. W. S. Cassels, Rational quadratic forms, Dover edition 2008.

      Appendix B is a good summary of Siegel's formula and related methods.

    • Z. I. Borevitch and I. R. Shafarevitch, Number theory, Academic Press, 1967. Good discussion of binary forms and quadratic extensions of Q, also of values of L-functions.
    • * Carl Ludwig Siegel, Lectures on the analytical theory of quadratic forms (notes in English by Morgan Ward), Peppmüller, 1963.

      This is the main reference in English for Siegel's original work.

    • Robert Gunning, Lectures on modular forms Princeton University Press, 1962.
    • Martin Eichler, Introduction to the theory of algebraic numbers and functions, Academic Press, 1966.

      Appendix 1 to Chapter1 is the clearest introduction to the relationship between theta functions and quadratic forms.

    • André Weil, `Sur la théorie des forme quadratiques', 9-22 in Colloq. Théorie des groupes algébriques, Bruxelles, 1962

      The original account relating Siegel's formula to adele groups.

    • * Tsuneo Tamagawa, `Adèles', in Algebraic groups and discontinuous subgroups, the proceedings of the Boulder conference, Symposia in Pure Mathematics 9, 1966. Edited by A. Borel and D. Mostow.

      Maybe the only place where the relationship between Siegel's formula and the volume of adele quotients is explained in relatively elementary terms. The entire proceedings are available for free download from the AMS web page Algebraic groups and discontinuous subgroups.

    • * André Weil, Adeles and algebraic groups, Birkhaüser, 1982.

      Originally a set of mimeographed notes from the Institute in Princeton from around 1965. Computes adelic volume formulas for several algebraic groups.

    • * Alex Eskin, Zeev Rudnik, and Peter Sarnak, `A proof of Siegel's weight formula', International Mathematics Research Notices (1991) no 5, 65-69.
    • * King Fai Lai, `Tamagawa numbers of reductive algebraic groups', Compositio Mathematicae 41 (1980), 153-188.
    • * Paul Garrett's notes on the volumes of classical arithmetic quotients.
    • * R. P. Langlands, `The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups', in Algebraic groups and discontinuous subgroups, the proceedings of the Boulder conference, Symposia in Pure Mathematics 9, 1966. Edited by A. Borel and D. Mostow.
    • * R. E. Kottwitz, `Tamagawa numbers', Annals of Mathematics 127 (1988), 629-626.
    • * Voskresenskii, `Adele groups and Siegel-Tamagawa formulas', Journal of Mathematical Sciences 73 (1995), 47-67.

      This is essentially the unique general survey of Tamagawa numbers, with one example being representation by sums of squares. There is even a long, more or less self-contained, account of Langlands' computation of Tamagawa numbers in terms of Eisenstein series, a very ambitious effort. Overall, it's pretty good, although the English is shaky and there are a few fine points about integral quadratic forms he doesn't get quite right. The bibliography is thorough, although much recent work of Kottwitz has been missed.


    Much of the charm of this subject is that it lies at the origin of modern number theory in the early nineteenth century. The story starts with Gauss, and then passes through Jacobi, Dirichlet, Eisenstein, H. J. Smith, Minkowski, Hurwitz, Hecke, Siegel, Maass, Weil, Langlands, and Kottwitz. The beginning is elementary enough that you can read the classic documents without too much trouble.
    • Carl Friedrich Gauss, Disquisitiones arithmeticae, Yale University, 1966. Republished later by Springer.

      This is the only English translation of Gauss' original Latin. Sections 266-292 deal with ternary quadratic integral forms, and give Gauss' formula for r3(n). Much of this is explained more clearly in Grosswald's book, but everyone should try to read Gauss sometime.

    • * H. J. Smith, Collected mathematical papers, Oxford Press, 1894.

      Available online from a link on the Wikipedia page about Smith. His long `Report on the theory of numbers' is easy reading.

    • Hermann Minkowski, `Grundlagen für eine Theorie quadratischen Formen mit ganzahligen Koefficienten', in Gesammelte Abhandlungen, 3-145.

      This is Minkowski's prize essay, written when he was 18 years old.

    • * Hermann Minkowski, `Diskontinuätsbereich für arithmetische Äquivalenz', Crelles Journal für die reine und angewandte Mathematik 129 (1905), 220-274.

      This was Minkowski's last paper in number theory. It describes in some detail the action of SLn(Z) on the space of positive definite quadratic forms.

    • * Carl Ludwig Siegel, `Über die analytischen Theorie der quadratischer Formen' I, II, III, in the Annals of Mathematics 1935-1937. Also in Siegel's Gesammelte Abhandlungen.

      My own translation of one of Siegel's prefaces.

    • * Carl Ludwig Siegel, `The volume of the fundamental domain for some infinite groups', Transactions of the American Mathematical Society 39 (1936), 209-218.

      He finds a simple proof of the volume calculation of SL(n)\Xn, the subject of the last paper of Minkowski, and generalizes it.

    • Paul Garrett, Siegel's integral.

      A simple calculation of the key integral appearing in Siegel's paper on SL(n,Z)\SL(n,R).

    • * Rudolf Scharlau's short history of quadratic forms (slides)

      He points out that Martin Kneser seems to have been the first to point out the connection between Siegel's formula and adele groups (in 1956). Here is Kneser's footnote:

      which unfortunately doesn't say much.

    • * Scharlau's article on Martin Kneser

      This is the final published text of his talk listed above. It is from the A. M. S. book Quadratic Forms--Algebra, Arithmetic, and Geometry edited by Ricardo Baeza et al.