Related papers: Borcherds Products on Unitary Group $U(2,1)$

We prove divisibility results for the Fourier coefficients of canonical basis elements for the spaces of weakly holomorphic modular forms of weight $0$ and levels $6, 10, 12, 18$ with poles only at the cusp at infinity. In addition, we show…

We show that a modular unit on two copies of the upper half-plane is a Borcherds product if and only if its boundary divisor is a special boundary divisor. Therefore, we define a subspace of the space of invariant vectors for the Weil…

We show that every Fricke invariant meromorphic modular form for $\Gamma_0(N)$ whose divisor on $X_0(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a…

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…

Let $\mathbf{U}$ be a quantum group of symmetric type. We introduce the {\it thickening realization} to realize (a suitable approximation of) the tensor product ${^{\omega}\Lambda_{\lambda_1}}\otimes \Lambda_{\lambda_2}$ of a simple…

We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $\textbf{U}^{\imath}$-module and a based $\textbf{U}$-module is a based $\textbf{U}^{\imath}$-module. This is then used to formulate a…

Let $p$ be a prime for which the congruence group $\Gamma_0(p)^*$ is of genus zero, and $j_p^*$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on $\Gamma_0(4p)$ which satisfies some congruence…

In this paper, we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of Fourier coefficients of the modular forms. We use the results to construct automorphic correction of the…

We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated to root lattices. We give a uniform construction of $147$ hyperplane arrangements on type IV symmetric domains…

We show that the space of invariants for the Weil representation for discriminant groups which contain self-dual isotropic subgroups is spanned by the characteristic functions of the self-dual isotropic subgroups. As an application, we…

Given a quantum group, we prove that the canonical bases of the tensor products of its integrable highest weight modules can be obtained from the canonical bases of the integrable highest weight modules of a bigger quantum group. As a…

We use the description of the Picard modular surface for discriminant $-3$ as a moduli space of curves of genus $3$ to generate all vector-valued Picard modular forms from bi-covariants for the action of ${GL}_2$ on the space of pairs of…

For all positive powers of primes $p\geq 5$, we prove the existence of infinitely many linear congruences between the exponents of twisted Borcherds products arising from a suitable scalar-valued weight $1/2$ weakly holomorphic modular form…

Based on the notion of strong modular form developed in Part I, we propose to structure the family of cuspidal modular form spaces $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and to determine bases for each of these spaces, once known bases…

We construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2,1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

For $\imath$quantum covering groups $(\mathbf{U}, \mathbf{U}^\imath)$ of super Kac-Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as…

We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that constructs holomorphic Borcherds Products from certain Jacobi forms that are theta blocks without theta denominator. The proof uses generalized valuations from formal…

For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the…

We prove that there are only finitely many isometry classes of even lattices $L$ of signature $(2,n)$ for which the space of cusp forms of weight $1+n/2$ for the Weil representation of the discriminant group of $L$ is trivial. We compute…

The twisted elliptic genera of a $K3$ surface associated with the conjugacy classes of the Mathieu group $M_{24}$ are known to be weak Jacobi forms of weight $0$. In 2010, Cheng constructed formal infinite products from the twisted elliptic…