Lifting Subgroups of Symplectic Groups over $\mathbb{Z} / \ell \mathbb{Z}$

Symplectic representations of inertia groups

Suppose $\ell$ is a prime number, $\ell >3$, $K$ is a field that is an unramified finite extension of the field $\Q_\ell$ of $\ell$-adic numbers, and $G$ is a finite group that is a semi-direct product of a normal $\ell'$-subgroup $H$ and a cyclic $\ell$-group $L$. Suppose that the group algebra $K[H]$ is decomposable. If there exists an embedding of $G$ in the symplectic group $\Sp_{2d}(K)$ for some positive integer $d$, then there exists an embedding of $G$ in $\Sp_{2d}({\mathcal O}_K)$, where ${\mathcal O}_K$ is the ring of integers of $K$.

来自 Ehrhart 多项式的辛 Hecke 特征基

Symplectic Hecke eigenbases from Ehrhart polynomials

本文研究提取格多面体Ehrhart多项式系数的函数，证明其在偶数维度下为辛Hecke特征函数，并应用球函数及zeta函数理论推导出相关算术函数的解析、渐近与组合结果。

Isotropy in Group Cohomology

The analogue of Lagrangians for symplectic forms over finite groups is studied, motivated by the fact that symplectic G-forms with a normal Lagrangian N<G are in one-to-one correspondence, up to inflation, with bijective 1-cocycle data on the quotients G/N. This yields a method to construct groups of central type from such quotients, known as Involutive Yang-Baxter groups. Another motivation for the search of normal Lagrangians comes from a non-commutative generalization of Heisenberg liftings which require normality. Although it is true that symplectic forms over finite nilpotent groups alway…

Classifying Toric and Semitoric Fans by Lifting Equations from ${\rm SL}_2({\mathbb Z})$

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric …

K3 surfaces with a symplectic automorphism of order 11

We classify possible finite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground field must be equal to 11. The complete list of such groups consists of five groups: the cyclic group of order 11, $11\rtimes 5$, $L_2(11)$ and the Mathieu groups $M_{11}$, $M_{22}$. We also show that a surface $X$ admitting an automorphism $g$ of order 11 admits a $g$-invariant elliptic fibration with the Jacobian fibration isomorphic to one of explicitly given elliptic K3 surfaces.