The group structure of the normalizer of $Γ_0(N)$

Normalizers of Congruence Groups in $SL_{2}(\mathbb{R})$ and Automorphisms of Lattices

We determine the normalizer in $SL_{2}(\mathbb{R})$ of several families of congruence subgroups of $SL_{2}(\mathbb{Z})$. In addition, we show how these tools can be used to evaluate the groups of automorphisms and the discriminant kernels of a large family of isotropic lattices of signature (2,1).

On the multiple holomorph of a finite almost simple group

Let $G$ be a group. Let $\mathrm{Perm}(G)$ denote its symmetric group and write $\mathrm{Hol}(G)$ for the normalizer of the subgroup of left translations in $\mathrm{Perm}(G)$. The multiple holomorph $\mathrm{NHol}(G)$ of $G$ is in turn defined to be the normalizer of $\mathrm{Hol}(G)$ in $\mathrm{Perm}(G)$. In this paper, we shall show that the quotient group $\mathrm{NHol}(G)/\mathrm{Hol}(G)$ has order two when $G$ is finite and almost simple. As an application of our techniques, we shall also develop a method to count the number of Hopf-Galois structures of isomorphic type on a finite almos…

超几何层与偶特征下的超特殊群

Hypergeometric sheaves and extraspecial groups in even characteristic

本文精确刻画了特征2下具有超特殊正规化子的不可约超几何层，解决了有限几何单值群情形下局部单值性判定的最后一例。

Bielliptic and Hyperelliptic modular curves X(N) and the group Aut(X(N))

We determine all modular curves X(N) (with $N\geq 7$) which are hyperelliptic or bielliptic. We make available a proof that the automorphism group of X(N) coincides with the normalizer of $Γ(N)$ in $\operatorname{PSL}_2(\mathbb{R})$.