We give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}\left(j\right(E\left)\right)$. More precisely, let $K$ be an imaginary quadratic field, and let ${\mathcal{𝒪}}_{K,f}$ be an order in $K$ of conductor $f\ge 1$. Let $E$ be an elliptic curve with CM by ${\mathcal{O}}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}\left(j\right(E\left)\right)$. Let $p\ge 2$ be a prime, let $G_{\mathbb{Q}\left(j\right(E\left)\right)}$ be the absolute Galois group of $\mathbb{Q}\left(j\right(E\left)\right)$, and let ${\rho }_{E,p^{\infty }}:G_{\mathbb{Q}\left(j\right(E\left)\right)}\to \mathrm{GL}(2,{\mathbb{Z}}_p)$ be the Galois representation associated to the Galois action on the Tate module $T_p(E)$. The goal is then to describe, explicitly, the groups of $\mathrm{GL}(2,{\mathbb{Z}}_p)$ that can occur as images of ${\rho }_{E,p^{\infty }}$, up to conjugation, for an arbitrary order ${\mathcal{O}}_{K,f}$.

We introduce a class of modules over Kac–Moody superalgebras; we call these modules snowflake modules. These modules are characterized by invariance property of their characters with respect to a certain subgroup of the Weyl group. Examples of snowflake modules appear as admissible modules in representation theory of affine vertex algebras and in the classification of bounded weight modules. Using these modules we prove Arakawa’s theorem for the Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}{\left(1\right|2\ell )}^{\left(1\right)}$.

We use the arithmetic of ideals in orders to parametrize the roots $\mu (\mathrm{mod}m)$ of the polynomial congruence $F(\mu )\equiv 0(\mathrm{mod}m)$, $F(X)\in \mathbb{Z}[X]$ monic, irreducible and degree $d$. Our parametrization generalizes Gauss’s classic parametrization of the roots of quadratic congruences using binary quadratic forms, which had previously only been extended to the cubic polynomial $F(X)=X^3-2$. We show that only a special class of ideals are needed to parametrize the roots $\mu (\mathrm{mod}m)$, and that in the cubic setting, $d=3$, general ideals correspond to pairs of roots ${\mu }_1(\mathrm{mod}{m}_1)$, ${\mu }_2(\mathrm{mod}{m}_2)$ satisfying $\gcd (m_1,m_2,{\mu }_1–{\mu }_2)=1$. At the end we illustrate our parametrization and this correspondence between roots and ideals with a few applications, including finding approximations to $\frac{\mu }{m}\in ℝ∕\mathbb{Z}$, finding an explicit Euler product for the cotype zeta function of $\mathbb{Z}[2^{1∕3}]$, and computing the composition of cubic ideals in terms of the roots ${\mu }_1(\mathrm{mod}{m}_1)$ and ${\mu }_2(\mathrm{mod}{m}_2)$.

We prove that the normal bundle of a general Brill–Noether space curve of degree $d$ and genus $g\ge 2$ is stable if and only if $(d,g)\notin \{(5,2),(6,4)\}$. When $g\le 1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this still holds in positive characteristic except when the characteristic is $2$, the genus is $0$ and the degree is even.

The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom–Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of $\mathbb{Z}$-lattices with automorphisms of finite order and with cyclic generators. The calculations are involved. They use fine properties of unit roots, cyclotomic polynomials, their resultants and discriminants.