The Alternating Group of degree 6 in Geometry of the Leech Lattice and K3 Surfaces

Siegel modular forms of weight 13 and the Leech lattice

For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the ${\rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${\rm Leech}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${\rm Sp}_{2g}(\mathbb{Z})$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard ${\rm L}$-functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular fo…

Extensions of the alternating group of degree 6 in the geometry of K3 surfaces

We shall determine the uniquely existing extension of the alternating group of degree 6 (being normal in the group) by a cyclic group of order 4, which can act on a complex K3 surface.

Noncommutative Riemannian Geometry of the Alternating Group A_4

We study the noncommutative Riemannian geometry of the alternating group $A_4=(Z_2 \times Z_2)\cross Z_3$ using a recent formulation for finite groups. We find a unique `Levi-Civita' connection for the invariant metric, and find that it has Ricci-flat but nonzero Riemann curvature. We show that it is the unique Ricci-flat connection on $A_4$ with the standard framing (we solve the vacuum Einstein's equation). We also propose a natural Dirac operator for the associated spin connection and solve the Dirac equation. Some of our results hold for any finite group equipped with a cyclic conjugacy cl…

交错多重线性映射的 Turán 与 Ramsey 问题

Turán and Ramsey problems for alternating multilinear maps

本文研究交错多重线性映射的 Turán 与 Ramsey 问题，证明在维数为 s·t⁴ 的空间上，交错双线性映射必含全零 s 维子空间或满秩 t 维子空间，并给出群论与几何推论。

The geometry of the Coble cubic and orbital degeneracy loci

The Coble cubics were discovered more than a century ago in connection with genus two Riemann surfaces and theta functions. They have attracted renewed interest ever since. Recently, they were reinterpreted in terms of alternating trivectors in nine variables. Exploring this relation further, we show how the Hilbert scheme of pairs of points on an abelian surface, and also its Kummer fourfold, a very remarkable hyper-Kähler manifold, can very naturally be constructed in this context. Moreover, we explain how this perspective allows us to describe the group law of an abelian surface, in a strik…

Geometry of the Wiman-Edge monodromy

The Wiman-Edge pencil is a pencil of genus $6$ curves for which the generic member has automorphism group the alternating group $\mathfrak{A}_5$. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group $\mathfrak{S}_5$. Farb and Looijenga proved that the monodromy of the Wiman-Edge pencil is commensurable with the Hilbert modular group $\mathrm{SL}_2(\mathbb{Z}[\sqrt{5}])$. In this note, we give a complete description of the monodromy by congruence conditions modulo $4$ and $5$. The congruence condition modulo $4$ is new, and this answers a question of Fa…