# Jonathan Hanke's Website

Research and Software in Number Theory

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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)

• 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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