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At the beginning of Eichler and Zagier's book on Jacobi Forms, there is the following diagram, summarizing part of the special role played by Jacobi forms of index 1. One of the things confusing me …

Given a holomorphic Jacobi form $\varphi_{k,1} \in \mathbb{J}_{k,1}$ of weight $k$ and index 1, we know we can use the Hecke operators $V_{m}$ to lift $\varphi_{k,1}$ to a Siegel modular form $$\math …

Given an extremal K3 surface $S$ over $\mathbb{Q}$ (i.e. a K3 surface with maximal Picard rank) there is a 2-dimensional Galois representation on the transcendental lattice $T(S)$, and an associated c …

Let $J_{k,m}(N)$ be the space of Jacobi forms of weight $k$, index $m$, and congruence subgroup $\Gamma_{0}(N) \rtimes \mathbb{Z}^{2}$. I do not believe it is relevant here to specify what type of Ja …

In something I've been thinking about recently, the following object appears: $$\mathcal{F}_{g} = \sum_{n=0}^{\infty} Q^{n} T_{n}\big( \phi_{2g-2}(\tau, z) \big)$$ where $T_{n}$ is the $n$-th Hecke …

On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function $$\Theta_{x_{0}}(\tau, z) = \sum_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, …

A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying $$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d …

So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think …

The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it resemb …

Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form $$\begin{bmatrix} * & *N & * & *\\ * & * …

I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4} …

This is something which I feel must be out in the literature somewhere, but I have been unable to find anything. If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A …

Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector …