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The Sensual Quadratic Form

The Sensual Quadratic Form

John Horton Conway
Assisted by Francis Y. C. Fung
Volume: 26
Copyright Date: 1997
Edition: 1
Pages: 167
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  • Book Info
    The Sensual Quadratic Form
    Book Description:

    John Horton Conway’s unique approach to quadratic forms was the subject of the Hedrick Lectures given by him in August of 1991 at the Joint Meetings of the Mathematical Association of America and the American Mathematical Society in Orono, Maine. This book presents the substance of those lectures. The book should not be thought of as a serious textbook on the theory of quadratic forms—it consists rather of a number of essays on particular aspects of quadratic forms that have interested the author. The lectures are self-contained and will be accessible to the generally informed reader who has no particular background in quadratic form theory. The minor exceptions should not interrupt the flow of ideas. The Afterthoughts to the Lectures contain discussion of related matters that occasionally presuppose greater knowledge.

    eISBN: 978-1-61444-025-3
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-x)
    J.H. Conway
  3. Table of Contents
    (pp. xi-xii)
  4. Note to the Reader
    (pp. xiii-xiv)
  5. THE FIRST LECTURE Can You See the Values of 3x²+6xy−5y²?
    (pp. 1-26)

    This question will lead us into the theory of quadratic forms. This is an old subject, and indeed A.-M. Legendre published most of the theory of binary quadratic forms in hisEssai sur la théorie des nombres(1798) [Leg], while Carl Friedrich Gauss, in his monumentalDisquisitiones Arithmeticae[Gau1] of 1801, essentially completed that theory. In this lecture, we shall present a very visual new method to display the values of any binary quadratic form. This will lead to a simple and elegant method of classifying all integral binary quadratic forms, and answering some basic questions about them.

    Before we...

  6. AFTERTHOUGHTS PSL₂(Z) and Farey Fractions
    (pp. 27-34)

    The afterthoughts following our lectures will add more detail, introduce some related topics, or just put our ideas into some other context. We shall occasionally presume some knowledge of more standard treatments. The underlying subject of this lecture is the group PSL₂(Z), which can be regarded as the set of all maps

    $z \mapsto \frac{{az + b}}{{cz + d}}$,$a,b,c,d \in Z,ad - bc = 1$.

    from the upper half-plane to itself. It is interesting to see how our topograph is drawn in the upper half plane$H = \{ x + iy|y > 0\} $.

    The picture showsHdivided into fundamental regions for the group PSL₂(Z) = Γ. The solid edges form a tree with three edges per...

  7. THE SECOND LECTURE Can You Hear the Shape of a Lattice?
    (pp. 35-52)

    Our title is intended to recall Mark Kac’s famous question “Can one hear the shape of a drum?” Kac’s article [Kac] drew wide attention to an important old problem which was first raised about 100 years ago. In physical language, we may state this as “do the frequencies of the normal modes of vibration determine the shape of the drum?” Of course this is a purely mathematical problem:—do the eigenvalues of the Laplacian for the Dirichlet problem determined by a planar domain determine the shape of that domain?

    When the titles for the lectures on which this book is...

  8. AFTERTHOUGHTS Kneser’s Gluing Method: Unimodular Lattices
    (pp. 53-60)

    The main topic of this book is classification of quadratic forms| this Lecture has been a digression whose relevance will only become apparent later. The First Lecture classified 2-dimensional forms, the Third will classify definite 3-dimensional forms, and the Fourth will classify indefinite forms in all dimensions greater than 2.

    There is no hope of classifying positive definite quadratic forms in high dimensions. However, Kneser obtains many integral lattices of small determinant by “gluing” root lattices to each other (or themselves). Milnor’s toroidal “drums” used the 16-dimensional even unimodular lattices$E_8^2$and$D_{16}^ + $. A more spectacular application was Niemeier’s enumeration...

  9. THE THIRD LECTURE ... and Can You Feel Its Form?
    (pp. 61-84)

    When we discussed binary quadratic forms in the First Lecture, there was a marked difference between the definite and indefinite cases. This persists in higher dimensions. The values of a positive definite form are best regarded as squared lengths of vectors in a lattice, and we classify such forms by discussing the shape of this lattice geometrically.

    In the indefinite case, when the dimension is at least 3, there is a complete classification, due to Eichler, in terms of an arithmetical invariant called the spinor genus, which is defined in terms of a simpler and more important invariant, the genus....

  10. AFTER THOUGHTS Feeling the Form of a Four-Dimensional Lattice
    (pp. 85-90)

    In the body of the lecture, we showed that in 2 or 3 dimensions, the shape of the Voronoi cell was determined by the positions of the conorms of value 0. In 2 dimensions, the cell is rectangular or hexagonal, according as there is or is not a 0 conorm. In 3 dimensions, when the number of 0 conorms is

    1 the cell is a truncated octahedron

    2 the cell is a hexarhombic dodecahedron

    3 (in line) the cell is a rhombic dodecahedron

    3 (not) the cell is a hexagonal prism

    4 the cell is a rectangular parallelipiped.

    However, conorms...

  11. THE FOURTH LECTURE The Primary Fragrances
    (pp. 91-116)

    In the Third Lecture, we classified definite forms in up to three dimensions by a process which is equivalent to examining their Voronoi cells geometrically. But the essential essences of a quadratic form are arithmetical! By considering congruences modulo powers of primes, it is possible to write down arithmetical invariants that tell us a lot about the form. More precisely, they completely solve the problem of rational equivalence for all forms.

    There is a way to enlarge the field Q of rationals to certain larger fields Qpof “p-adic rationals”, one for each primep. Although thep-adics are the...

  12. AFTERTHOUGHTS More About the Invariants: The p-Adic Numbers
    (pp. 117-126)

    We have not yet proved that thep-signatures (or equivalently thep-excesses) are invariants of rational equivalence. We can do this by showing that for any type II integral formFthat is rationally equivalent tof, thep-adic Gauss mean ofFis${\zeta ^{{\sigma _2}\left( f \right)}}$or${\zeta ^{ - {e_p}\left( f \right)}}$times a real positive number, according aspis or is not 2.

    If F₁ and F₂ are two type II integral forms equivalent tof, then we can get from one to the other by a chain of steps, each of which replaces a lattice by a sublattice of prime indexP...

  13. POSTSCRIPT A Taste of Number Theory
    (pp. 127-142)

    In this Postscript we shall prove three famous theorems. These are the notorious quadratic reciprocity law, the fact that the signature of an even unimodular quadratic form is a multiple of 8, and Legendre’s celebrated three squares theorem. We shall derive some consequences of Legendre’s theorem, including the universality of certain forms in four variables, and finish by explaining why no rational positive definite ternary form is universal.

    For an odd numbernwhose prime factorization ispqr. . . , Jacobi defined his symbol to be

    >$\left( {\frac{a}{n}} \right) = \left( {\frac{a}{p}} \right)\left( {\frac{a}{q}} \right)\left( {\frac{a}{r}} \right)...,$

    a product of Legendre symbols. Although it was clear from its...

  14. References
    (pp. 143-146)
  15. Index
    (pp. 147-152)