Narrow Results

The transformation behavior of the vector-valued theta function of a positive-definite even lattice under the metaplectic group ${Mp}_2(\mathbb{Z})$ is described by the Weil representation. We show that the invariants of this representation are induced from five fundamental invariants. As an application we give simple generating sets for Jacobi forms of singular weight.

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field ${𝒦}=\mathbb{Q}({\varsigma }_8)$ over $\mathbb{Q}$. We employ binary codes to generate a lattice over $\mathbb{Q}({\varsigma }_8)$, which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels.

In this paper, we show that the graded ring of Siegel paramodular forms of degree $2$ with level $N=2,3,4$ has a very simple unified structure, taking with character. All are generated by six modular forms. The first five are obtained by a kind of Maass lift. The last one is obtained by a kind of Rankin–Cohen–Ibukiyama differential operator from the first five. This result is similar to the case of the graded ring of Siegel modular forms of degree $2$ with respect to ${\mathrm{\Gamma }}_0(N)$.

In this paper, we give an upper bound of the orders of Jacobi forms. For each fixed weight $k$, the growth rate of our upper bound divided by the index $m$ as $m$ goes to infinity is zero. Applying this result to Siegel paramodular forms, we know that the space of all formal series of Jacobi forms is finite dimensional.

We use recent results of Rolen, Zwegers, and the first author to study the characters of irreducible (highest weight) modules for the vertex operator algebra $L_{{𝔰𝔩}_{\ell }}(-{\mathrm{\Lambda }}_0)$. We establish asymptotic behaviors of characters for the (ordinary) irreducible $L_{{𝔰𝔩}_{\ell }}(-{\mathrm{\Lambda }}_0)$-modules. As a consequence, we prove that their quantum dimensions are one, as predicted by the representation theory. We also establish a full asymptotic expansion of irreducible characters for ${𝔰𝔩}_3$. Finally, we determine a decomposition formula for the full characters in terms of unary theta and false theta functions which allows us to study their modular properties.

In this paper we study a family of quaternary forms which represent almost all binary forms of a certain type. The result follows from the representation number by the genus of ternary forms and a correspondence among theta series.

We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.

In previous work, we introduced harmonic Maass–Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass–Jacobi–Poincaré series, as well as Zwegers' real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass–Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass–Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.

We consider a generalization of Jacobi theta series and show that every such function is a quasi-Jacobi form. Under certain conditions we establish transformation laws for these functions with respect to the Jacobi group and prove such functions are Jacobi forms. In establishing these results, we construct other functions which are also Jacobi forms. These results are motivated by applications in the theory of vertex operator algebras.

We prove that a nonzero Jacobi form of level $N$ (an odd integer) and square-free index $m_1{m}_2$ with $m_1|N$ and $(N,m_2)=1$ has a nonzero theta component $h_{\mu }$ with either $(\mu ,2m_1{m}_2)=1$ or $(\mu ,2m_1{m}_2)$$\nmid$$2m_2$. As an application, we prove that a nonzero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin–Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.

We take a Galois ring $GR(p^m,g)$ and discuss about the self-dual codes and its properties over the ring. We will also describe the relationship between Clifford-Weil group and Jacobi forms by constructing the invariant polynomial ring with the complete weight enumerator.

We study Jacobi forms associated to even positive definite unimodular lattices, in particular E₈ lattice and the Leech lattice. We require the Jacobi forms to be invariant under the orthogonal group of the lattice, in particular E₈ Weyl group and Conway group Co₀. These objects are of interest in both number theory and string theory, for example elliptic genera of E-strings, Monster CFT and so on. We establish the explicit structure theorem and determine the generators for the E₈ case with index t ≤ 13 and the Leech case with index t ≤ 3. This is a short summary of two joint papers with Haowu Wang.