The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type ${\mathbf{A}}_4$.

Let $N\ge 1$ be an integer. Let $X_0\left(N\right)$ be the modular curve over $\mathbf{Z}$, as constructed by Katz and Mazur. The minimal resolution of $X_0\left(N\right)$ over $\mathbf{Z}[1/6]$ is computed. Let $p\ge 5$ be a prime, such that $N=p^{2}M$, with $M$ prime to $p$. Let $n=(p^2-1)/2$. It is shown that $X_0\left(N\right)$ has stable reduction at $p$ over $\mathbf{Q}\left[\sqrt[n]{p}\right]$, and the fibre at $p$ of the stable model is computed.

Let ${𝒪}_K$ be a discrete valuation ring of mixed characteristics $(0,p)$, with residue field $k$. Using work of Sekiguchi and Suwa, we construct some finite flat ${𝒪}_K$-models of the group scheme ${\mu }_{p^n,K}$ of $p^n$-th roots of unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic properties of this construction. When $k$ is perfect and ${𝒪}_K$ is a complete totally ramified extension of the ring of Witt vectors $W\left(k\right)$, we provide a parallel study of the Breuil-Kisin modules of finite flat models...