Soo-bong Lim

In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples $f(q)$, Ramanujan claimed that there is a collection $\{ G_j\}$ of modular forms such that for each root of unity $\zeta$, there is a $j$ such that \[ \lim _{q \to \zeta }(f(q) - G_j(q)) = O(1).\] Moreover, Ramanujan claimed that this collection must have size larger than $1$. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms.

We define odd-balanced unimodal sequences and show that their generating function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V left-parenthesis x comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo str

Motivated by the recent works of Bringmann, Guerzhoy, Kent, and Ono [4] and Bringmann, Fricke, and Kent [3], we introduce $L$-series for vector-valued weakly holomorphic cusp forms, and mock modular period polynomials for vector-valued harmonic weak Maass forms. In particular, we will discuss an integral representation of this new $L$-series and the limiting behavior of special values. Moreover, we also give relations between mock modular periods and $L$-series for vector-valued harmonic weak Ma

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and

In this paper, we prove that if the Fourier coefficients of a vector-valued modular form satisfy the Hecke bound, then it is cuspidal. Furthermore, we obtain an analogous result with regard to Jacobi forms by applying an isomorphism between vector-valued modular forms and Jacobi forms. As an application, we prove a result on the growth of the number of representations of $m$ by a positive definite quadratic form $Q$.

Zagier proved that the generating functions of traces of singular values of Jm(z) are weight 3 2 weakly holomorphic modular forms. In this paper we prove that there is the sign-periodicity of traces of singular values of Jm(z).

Kohnen proved a non-vanishing result for $L$-functions associated to Hecke eigenforms of integral weights on the full group. In this paper, we show a non-vanishing result for the averages of $L$-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms of weight $k \in \frac{1}{2} \mathbb{Z}$ on the full group. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we