Borcherds products associated with certain Thompson series

Local Borcherds Products for Unitary Groups

For a discrete subgroup of an indefinite unitary group $U(1,n+1)$, $n\geq 1$, consider the attached modular variety. Using local Borcherds products, we study Heegner divisors in the local Picard group over a boundary component the compactified variety. We obtain a criterion for local Heegner divisors to be torsion elements in the local Picard group. As an application, we find that the obstructions for a local Heegner divisor to be a torsion element can be described through spaces of vector valued elliptic cusp forms spanned by certain theta-series.

Holomorphic Borcherds products of singular weight for simple lattices of arbitrary level

We classify the holomorphic Borcherds products of singular weight for all simple lattices of signature $(2,n)$ with $n \geq 3$. In addition to the automorphic products of singular weight for the simple lattices of square free level found by Dittmann, Hagemeier and the second author, we obtain several automorphic products of singular weight $1/2$ for simple lattices of signature $(2,3)$. We interpret them as Siegel modular forms of genus $2$ and explicitly describe them in terms of the ten even theta constants. In order to rule out further holomorphic Borcherds products of singular weight, we d…

The Fourier coefficients of the McKay-Thompson series and the traces of CM values

The elliptic modular j function enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. In this article, we are concerned with analogues of Kaneko's result for the McKay-Thompson series of square-free level.

正交 Eisenstein 级数与 Theta 提升

Orthogonal Eisenstein Series and Theta Lifts

本文证明了向量值非全纯 Eisenstein 级数的加性 Borcherds 提升是 $O(2, l)$ 的正交非全纯 Eisenstein 级数，据此证明了其亚纯延拓、Fourier 展开与函数方程，并给出了提升满射的充分条件。

Generalized moonshine II: Borcherds products

The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. The Lie algebras that we construct conjecturally appear in an orbifold conformal field theory with symmetries given by the monster simple group. We introduce vector-valued modular functions attached to families of modular functions of different levels, and we prove infinite product identities for a distinguished class of automorphic…

Mathieu Moonshine 的 McKay-Thompson 级数模二

The McKay-Thompson series of Mathieu Moonshine modulo two

本文研究 Mathieu moonshine 的 McKay-Thompson 级数系数的奇偶性，并据此证明了 Cheng-Duncan-Harvey 关于 Umbral moonshine 的猜想在 Mathieu 情形下的成立。

$O(2,1)$型广义Borcherds乘积的Hecke等变性

Hecke equivariance of generalized Borcherds products of type $O(2,1)$

本文证明了$O(2,1)$型广义Borcherds提升算子与对数微分算子在特定Hecke算子扩展下的等变性，并以此推导出模素数幂的奇异模扭迹关系及扭类数同余式。

Coefficients of McKay-Thompson series and distributions of the moonshine module

In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module $V^\natural = \bigoplus_n V_n^\natural$. Their results show that as $n$ tends to infinity, $V_n^\natural$ is dominated by direct sums of copies of the regular representation. That is, if we view $V_n^\natural$ as a module over the group ring $\mathbb{Z}[\mathbb{M}]$, the free-part dominates. A natural problem, posed at the end of the aforementioned paper, is to char…

秩二 Nahm 和、模性障碍与 McKay–Thompson 级数 $T_{31A}$

Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series $T_{31A}$

本文猜想秩二 Nahm 和的正四移位恒等式，通过 Rogers–Ramanujan 函数实现 Monster 级数 $T_{31A}$，证明各分量的非模性并验证组合的渐近行为，提出对偶 Fricke 定律及月光恒等式。

On recursive properties of certain p-adic Whittaker functions

We investigate recursive properties of certain p-adic Whittaker functions (of which representation densities of quadratic forms are special values). The proven relations can be used to compute them explicitly in arbitrary dimensions, provided that enough information about the orbits under the orthogonal group acting on the representations is available. These relations have implications for the first and second special derivatives of the Euler product over all p of these Whittaker functions. These Euler products appear as the main part of the Fourier coefficients of Eisenstein series associated…

到不定正交群若干上同调表示的 theta 提升

Theta lifts to certain cohomological representations of indefinite orthogonal groups

本文利用 Borcherds 方法将全纯 Mp₂(ℝ) 尖点形式 theta 提升到不定正交群的上同调表示 Π 中，并通过 Fourier 展开计算与尖点支集分析确认所得自守形式的平方可积性。

Special cycles and derivatives of Eisenstein series

This article sketches relations among algebraic cycles for the Shimura varieties defined by arithmetic quotients of symmetric domains for O(n,2), theta functions, values and derivatives of Eisenstein series and values and derivatives of certain L-functions. In the geometric case, results of joint work with John Millson imply that the generating functions for the classes in cohomology of certain algebraic cycles of codimension r are Siegel modular forms of genus r and weight n/2+1. A result of Borcherds shows that, for r=1, the same is true for the generating function for the classes of such di…

至多有限多个奇异模数是S-单位

There are at most finitely many singular moduli that are S-units

本文证明对任意有限素数集S，至多有限多个奇异模数是S-单位，并推广至相关模函数，且特化出无穷多个代数单位的模函数必为弱模单位。