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Background: I am a theoretical computer scientist (PhD candidate) and have done graduate level courses in Algebra.

I want to understand the following theorem from the book "Symmetric Bilinear Forms" by J. Milnor and D. Husemoller.

Theorem (9.5, pp 46) For any dimension n there exists a positive definite inner product space $X$ of type $1$ and rank $n$ with $\min_{x \in X \setminus {0}} x.x \geq n ($n \rightarrow \infty$)$.

In particular, the theorem implies the existence of a self-dual lattice with shortest vector \geq \sqrt{n}.

The proof of this theorem is a byproduct of Siegel's theorem, which can be seen as a bound on the "average" number of solutions to a quadratic equation. The equation of interest for me is x.x = k i.e., the number of vectors of a particular length $k$ from an inner-product space $X$.

The above mentioned book does not give a proof for the Siegel's theorem. Additionally, the proof of Theorem 9.5 uses some results over p-adic integers and others on genus of bilinear form spaces to show that the number of solutions of the equation x.x=k, summed for k \in {1, \dots, n} is <2 if averaged over all distinct linear product space in the genus $I_n$.

My aim is to understand this proof.

What are the books which I should start with to understand the proof ? The Milnor book is too short and seems to be aimed at experts. Note that I do not understand the p-adic integers and so it is very difficult for me to make sense of the proof.

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2 Answers

up vote 1 down vote accepted

Hi there, I think we had better give you a start here...

You have combined together a few ideas that come from very different areas of inquiry.

In one direction, kissing numbers and Minkowski-Hlawka (Milnor and Husemoller, page 31) see Table 1.3 on pages 15-17 of SPLAG, that is Sphere Packings, Lattices and Groups by Conway and Sloane. Self-dual is typically called "unimodular," see the bottom of page 53. Easier introductions that will still lie in comfortable territory are Lattices and Codes by Wolfgang Ebeling, also From Error Correcting Codes Through Sphere Packings to Simple Groups by Thomas M. Thompson. Note that your Thompson is J. G. Thompson.

See http://en.wikipedia.org/wiki/Unimodular_lattice

As to exact numbers of representations, I actually recommend a much earlier book, The Arithmetic Theory of Quadratic Forms by Burton W. Jones. I see that in Rational Quadratic Forms by Cassels, he does three squares on page 150, Lemma 6.4, then four squares on page 152, Lemma 6.5.

In a few famous cases, notably four squares and eight squares, the exact number of representations has a fairly clean expression, due to Jacobi. http://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem

See http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/unimodular.html

Finally, you might try http://cstheory.stackexchange.com/ with a more obviously computer-centric version of your question.

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See http://www.math.ubc.ca/~cass/siegel/INDEX.html -- it has a great collection of references.

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