I'm currently reading a paper of Georges Gras on the Reflection Principle. The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting, but I want to get some practical motivation (from mathematicians working on the theory, or number theorists who use the theory, etc) on the Order Theory.
Thank you.
Below are two [Edit: four] examples where orders in central simple algebras are useful. I might add more when I have the time.
Before giving them, let me first motivate in general why one would consider orders, and maximal orders in particular: They generalize rings of integers in field extensions.
In more detail, if one is given a Dedekind domain $R$ with fraction field $K$ and $L/K$ is a finite separable extension, then the integral closure of $R$ in $L$ forms a finite ring extension $S$ of $R$, which allows one to exhibit some arithmetic properties of $L$. However, if $L$ is replaced with a central simple algebra $A$ over $K$, then the $R$-integral elements do not usually form a subring, and one common remedy is to use orders in order to reveal arithmetic properties of $A$. From this point of view, maximal orders are the closest thing one has to a ring of integers, and remarkably, they share many properties with the Dedekind domain $R$, e.g., their one-sided ideals are projective and their two-sided ideals admits unique factorization as products of primes. (Item 4 below is a nice application of this approach.)
A more geometric reason why one might want to consider orders is that a central simple algebra $A$ is an objects defined over $\mathrm{Spec}\,K$ --- a sheaf of algebras, say ---, and an order $\Lambda$ in $A$ can be regarded as an extension $A$ to a sheaf of algebras over $\mathrm{Spec}\, R$ whose stalk at the generic point is $A$. Here, at least a priori, there is no bias toward particular types of orders, but some choices are more convenient than others, depending on the application. (For example, the group $R$-scheme $\mathbf{Aut}_R(\Lambda)$ can be smooth for some orders, and non-smooth for others.) In many cases, one would like to have an Azumaya order, i.e., an order that, similarly to $A$, is locally a matrix algebra relative to the etale topology on $\mathrm{Spec}\,R$. Such orders do not exist in general, but often orders which are "maximal" in some sense provide a sufficient approximation.
[Edited.]
1. One classical use of orders is the classification of central division algebras over non-Archimedean local fields (section 14 in Reiner's book), which is done by studying their unique maximal order. These algebras are classified by an invariant taking values in $\mathbb{Q}/\mathbb{Z}$.
This allows a description of the Brauer group of global fields (Reiner's book section 32): If $K$ is a global field and $V$ is its set of places, then there is an exact sequence $$ 0\to \mathrm{Br}\,K \to \bigoplus_{v\in V}\mathrm{Br}\,K_v\xrightarrow{\sum_v\mathrm{inv}_v} \mathbb{Q}/\mathbb{Z}\to 0 $$ in which the first map is given by base change to all completions and $\mathrm{inv}_v$ is the aforementioned invariant when $v$ is non-Archimedean. This is known as the Albert-Brauer-Hasse-Noether theorem.
2. Another, very different, reason why orders are interesting is because they are lattices of matrices. For example, if $D$ is a division $\mathbb{Q}$-algebra such that $D\otimes\mathbb{R}\cong \mathrm{M}_n(\mathbb{R})$ (or $\mathrm{M}_n(\mathbb{C})$) and $\Lambda$ is a $\mathbb{Z}$-order in $D$, then $\Lambda$ is a lattice in $\mathrm{M}_n(\mathbb{R})$ consisting entirely of invertible matrices (excepting $0$). Such lattices have practical applications to communications, which I know very little about, but I do know that engineers and mathematicians tried to look for such maximal orders with desired properties.
3. Orders also occur naturally in Bruhat--Tits theory, when describing the buildings of classical groups. For example, if $R$ is complete discrete valuation ring, $K$ is its fraction field and $A$ is a central simple algebra, then one possible description of the building of the algebraic group $\mathbf{G}:=\mathbf{PGL}_1(A)$ over $K$ is as the set of hereditary $R$-orders in $A$, ordered under reversed inclusion --- this is a contractible simplicial complex one which $\mathbf{G}(K)=A^\times /K^\times$ acts faithfully via conjugation.
(One reference is Bruhat and Tits' paper from 1984, but it is difficult to read. Another is Abramson and Nebe's "Lattice chain models for affine buildings of classical type".)
4. There is a proof of Jacobi's formula for the number of integral solutions of $n=x^2_1+x_2^2+x_3^2+x_2^4$ which uses quaternations. This proof essentially analyzes the prime factorization in the ring of integral quaternions, which is a non-maximal $\mathbb{Z}$-order in the rational quaternions. (However, it becomes maximal, and even Azumaya, after inverting $2$. The ramification at the prime $2$ gives a conceptual explanation to why the formula is more complicated when $n$ is even.)
See, for instance, "Elementary Number Theory, Group Theory and Ramanjuan Graphs" by Davidoff, Sarnakand and Valette.
