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Learning about Quadratic Forms

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For Graduate Students

The Arithmetic theory of Quadratic Forms can be thought of in several different ways, each of which offers its own insights.  Here are a list of accessible references for learning about these topics:

  1. Quadratic Forms over Fields — This describes the theory of quadratic forms over various fields, from very simple (e.g. complete local fields: $\Bbb C$, $\Bbb R$, $\Bbb Q_p$; number fields) to very complicated (e.g. higher-dimensional geometric fields like $\Bbb Q(x,y,z,w)/???$).  In this setting, one can try to understand geometric properties (e.g. when do we represent zero non-trivially?, do we represent all numbers?, etc. ) and define invariants for testing when two quadratic forms are  equivalent.
  2. Quadratic Forms over Rings — This involves the study of quadratic forms over familiar rings (usually $\Bbb Z$ rings of integers of number fields $K/\Bbb Q$ and function fields $\Bbb F_q[t]$), usually by a “local-global” principle to study quadratic forms over all local completions (e.g. over $\Bbb Z_p$, $\R$, power series rings).
  3. Theta Series of representations of Quadratic Forms — These are generating functions for the number of ways that one can represent a number (or even a quadratic form) by another quadratic form over the integers $\Bbb Z$ (or even the ring of integers of a number field).  These “theta series” are an important example of modular forms, and one can study these representation numbers by using the theory of modular forms.

Other Topics: Fall 2011 UBC Seminar on Siegel’s formula and Tamagawa Numbers (Webpage)

For Undergraduates

  • Conway “The Sensual Quadratic Form” (Google Books)
  • Cohn “Advanced number theory” (Google Books, Amazon)
  • My PROMYS 2010 Course notes on Quadratic Forms and Modular Forms

For General Audiences

  • Exact formulas for the sum of 2 and 4 squares
  • Arithmetic of Quadratic Number fields and norm forms


Written by jonhanke

December 10th, 2011 at 4:57 pm

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