All Questions

For which $n$ is the "principal congruence subgroup" $\Gamma(n)\le \mathrm{SL}_2(\mathbb{Z})$, the subgroup consisting of matrices congruent to the identity modulo $n$, characteristic? I.e., for ...

Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\...

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...

Definition: Let $h$ be a polynomial in $n$ variables, then : $\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$ Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...

$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer: $$ C(K)=\{ B \in \GL(n,\...