Related papers: Reflective modular forms: A Jacobi forms approach

We analyze holomorphic Jacobi forms of weight one with level. One such form plays an important role in umbral moonshine, leading to simplifications of the statements of the umbral moonshine conjectures. We prove that non-zero holomorphic…

We consider local non-Gorenstein rings of the form $(S_i,\mathfrak{n}_i)=k[X, Y_1, \ldots ,Y_i]/\left(X^2, (Y_1, \ldots, Y_i)^2\right), $ where $i\geq 2.$ We show that every totally reflexive $S_i$-module has a presentation matrix of the…

We associate a Jacobi form over a rank s lattice to N=2, D=4 heterotic string compactifications which have s Wilson lines at a generic point in the vector multiplet moduli space. Jacobi forms of index m=1 and m=2 have appeared earlier in…

We classify all the symmetric integer bilinear forms of signature (2,1) whose isometry groups are generated up to finite index by reflections. There are 8595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability…

Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\frac12$. Since the only modular forms of weight $\frac12$ on congruence subgroups of…

We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower…

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This…

We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices $1,2$ with classical Jacobi forms (JF) of weight $k$ and indices $1,2,4$. Using the embedding into JF, upper bounds for the order of vanishing of HJF at the…

The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity relating the number of $\mathrm{GL}_2 \mathbb{Z}$-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants $D$, $-27D$; it…

Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional…

We propose a classification of the solutions of the graded reflection equations to the $U_{q}[spo(2n|2m)]$ vertex model. We find twelve distinct classes of reflection matrices such that four of them are diagonal. In the non-diagonal…

Let $L$ be a positive definite even lattice. We introduce theta type Jacobi forms and construct three towers of Jacobi forms with a particular easy pullback-structure. We use theta type Jacobi forms to explain the existence of a cusp form…

In this paper, we establish two types of upper bounds on the vanishing order of Jacobi forms at infinity. The first type is for classical Jacobi forms, which is optimal in a certain sense. The second type is for Jacobi forms of lattice…

If a Jacobi matrix $J$ is reflectionless on $(-2,2)$ and has a single $a_{n_0}$ equal to 1, then $J$ is the free Jacobi matrix $a_n\equiv 1$, $b_n\equiv 0$. I'll discuss this result and its generalization to arbitrary sets and present…

We construct a model of moonshine phenomenon based on the use of N=2 superconformal algebra. We consider an extremal Jacobi form of weight 0 and index 2, and expand it in terms of N=2 massless and massive representations. We find 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…

Let $l\geq 3$ and $F$ be a modular form of weight $l/2-1$ on $\mathrm{O}(l,2)$ which vanishes only on rational quadratic divisors. We prove that $F$ has only simple zeros and that $F$ is anti-invariant under every reflection fixing a…

Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]], we further study reflection principles in connection with the L\"owenheim-Skolem Theorems of stationary logics. In this paper, we mainly analyze the…

In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that…