The geometry of numbers is q branch of number theory created by Minkowski.

The idea of this theory is to consider the intersection of some set with a lattice in the Euclidean space to get the existence or non-existence of some particular number.

Start with the basics. Suppose is a lattice in (i.e. a discret subgroup of the additive group of without point of accumulation, which viewed as a set of vectors in generate the whole space). Using the fact that has a finite set as basis, we deduce easily that is finitely generated. Using the fact that is a principal domain, we see that has, in fact, a finite set as -basis(what is more, the cardinality of this set is independent of the choice of the basis, it is an invariant under isomorphism, also a property of the abelian group). That is we can assume that looks like . What is more, we can for any other basis of , the matrix of these expressed with respect to these lies in the group . After choosing the canonical basis of , we can calculate the determinant of the matrix . The absolute value of this determinant is, coincidently, the volume of the fundamental domain of the lattice , or in other words, the volume of equipped with the quotient measure(we denote it by , the co-volume of ). This value is, in some sense, the generator of the image of the homomorphism of groups .

The fundamental result in the geometry of numbers concerns the intersection of a convex set with the lattice. We say a convex subset is symmetric if for all , we have . So if , we have that $latex\frac{x-y}{2}$. This is really a simple fact, but it is also a really important observation. Now we are ready to state the **theorem of Minkowski**:

Suppose that a convex symmetric subset of , and is a lattice of the latter. What is more, . Then there is a non-zero element which lies in also. That is to say, .

The point of the proof is to express the above inequality as a comparison between two volumes. Noting that if we dilate the lattice by , then we have that . So the inequality turns into . Then we can say that there are two distinct elements such that the half of their difference . If it is not the case, then after choosing a measurable representation of in , we can express . And so has the same volume as . But is now a measurable subset of (because is a measurable subset, see here), so we have , which is a contradiction to the assumption. So we get a non-zero element , thus proved the theorem.

Now come to the applications of this fundamental result. As a first example, we consider one theorem of Fermat’s, that is a prime number if and only if can be written as a sum of two squares of integers, . Note that is the same as that is quadratic residue mod . So we can find some integer such that . We will use this to construct some lattice in . That is, ze define . Clearly this is a lattice(a routine way of showing this is, ). What is more, the indice of in is (just consider the group homomorphism , which is surjective, with kernel . So we have ).And thus . Now consider the disk . This disk is clearly convex and symmetric, with volume . Now that the assumptions are satisfied just by noting that , so there is . Note that since , we have , in other words, for some integer .But recall the definition of , we have that , so the only possibility is that .

So by carefully choosing a lattice, we proved the theorem due to Fermat. This is a small success of the geometry of numbers. In fact, the above reasoning can be generalized to any polynomials like where is a positive integer.

The following is one such result:

Suppose that integer is a quadratic residue mod , then at least one of the numbers ( is the largest integer such that ) can be expressed in the form .

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