Mallick and Mishra [‘Links between the solvability of matrix and scalar Diophantine equations’, Bull. Aust. Math. Soc., 10.1017/S0004972726101087] observed that solutions of the matrix Diophantine equation $X^n + Y^n = 2Z^n$, where $X, Y, Z$ are $2 \times 2$ matrices in a certain matrix class $G_2(d, l)$, correspond to solutions of the scalar equation $\alpha ^n + \beta ^n = 2\gamma ^n$ over $\mathbb {Z}[\sqrt {d}]$ and assert that there are only trivial solutions. We show that this fails when $d = -1$ with $4 \mid n$ and $d = -3$ with $3 \mid n$, and determine the values of $l$ that admit solutions. We extend the analysis to $k \times k$ matrices, and show that for every totally real field $K$ and every even $n$, all solutions are trivial. We obtain some results for odd $n$, but the problem for odd $n$ in totally real fields of degree $k \geqslant 3$ remains open.

We prove a conjectural formula for the Brumer–Stark units. Dasgupta and Kakde have shown the formula is correct up to a bounded root of unity. In this paper, we resolve the ambiguity in their result. We also remove an assumption from Dasgupta–Kakde’s result on the formula.

We consider a concrete family of $2$-towers $(\mathbb {Q}(x_n))_n$ of totally real algebraic numbers for which we prove that, for each $n$, $\mathbb {Z}[x_n]$ is the ring of integers of $\mathbb {Q}(x_n)$ if and only if the constant term of the minimal polynomial of $x_n$ is square-free. We apply our characterization to produce new examples of monogenic number fields, which can be of arbitrary large degree under the ABC-Conjecture.

Let S be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb {Z}_{S}$-points on integral models of Hilbert modular varieties, extending a result of D. Helm and F. Voloch about modular curves. Let L be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre’s conjecture for mod $\ell $ representations of the absolute Galois group of L, we prove that the same holds also for the $\mathcal {O}_{L,S}$-points.