Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I\_n$ be the $n\times n$ identity matrix, and let $A\_n$ be the $n\times n$ diagonal matrix having coefficient $1,q,...,q$ on the diagonal.


Note that $I\_n$ and $A\_n$ are always congruent over the $p$-adic integers for all $p$ not dividing $q$. Further assume that $I\_n$ and $A\_n$ are congruent over $\mathbf{Q}\_p$ for $p$ dividing $q$ (which is the case when $q=10,12$ or $15$). Does it follow that $I\_n$ is congruent to $A\_n$ over $\mathbf{Z}[1/q]$ ?


$\textbf{Motivation}$: A friend explained the paper "On the non-existence of certain finite projective planes", and a naïve hope is to push a bit further their strategy by answering the above question in the negative. This has for sure already been tried, and I would like to know some references about that.


$\textbf{Edit}$:In view of the question [What's in the genus of the cubic lattice?](https://mathoverflow.net/questions/230069/whats-in-the-genus-of-the-cubic-lattice/230141#230141), let me restate my question: the two quadratic forms over $\mathbf{Z}$ given by $\sum \limits\_{i=1}^{111} x\_i^2$ and $x\_1^2+ \sum \limits\_{i=2}^{111} 10x\_i^2$ are in the same genus. But are they actually isomorphic over $\mathbf{Z}[1/10]$? (As Noam notes in the comments, they cannot be isomorphic over $\mathbf{Z}$ as determinant do not match). Can I hope for an answer from Magma, or would it be overwhelmed by the number of variables ?



Are those $2$ quadratic forms congruent over $\mathbb{Z}[1/q]$
The answer to the main question is yes ! (The question has been answered by the commentators).


As a consequence, it is not possible to push the Bruck-Ryser theorem by looking at congruences over rings of the form $\mathbf{Z}[1/q]$.


