Jonathan Hanke's Website

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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.
• Lam “Introduction to Quadratic Forms over Fields“
• Garibaldi, Merkurjev, Serre “Cohomological Invariants in Galois Cohomology“
• Elman, Karpenko, Merkurjev “The  Algebraic and Geometric Theory of Quadratic Forms“
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).
• Cassels “Rational Quadratic Forms“
• Gerstein “Basic Quadratic Forms“
• O’Meara “Introduction to Quadratic Forms” (Amazon)
• Knus “Quadratic and Hermitian Forms over Rings” (Amazon)
• Scharlau “Quadratic and Hermitian Forms” (Amazon)
• Milnor “Symmetric Bilinear Forms” (Amazon)
• Kitaoka “The Arithmetic of Quadratic Forms” (Google Books)
• Shimura “The Arithmetic of Quadratic Forms” (Google Books)
• Knebusch “Specialization of Quadratic and Symmetric Bilinear Forms” (Amazon, Google Books)
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.
• Iwaniec “Topics in Classical Automorphic Forms” (Google Books)
• Miyake “Modular Forms” (Google Books)

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