Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1401.7189

Negative index Jacobi forms and quantum modular forms

Kathrin Bringmann·Thomas Creutzig·Larry Rolen
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we consider the Fourier coefficients of a special class of meromorphic Jaocbi forms of negative index. Much recent work has been done on such coefficients in the case of Jacobi forms of positive index, but almost nothing is known for Jacobi forms of negative index. In this paper we show, from two different perspectives, that their Fourier coefficients have a simple decomposition in terms of partial theta functions. The first perspective uses the language of Lie super algebras, and the second applies the theory of elliptic functions. In particular, we find a new infinite family of rank-crank type PDEs generalizing the famous example of Atkin and Garvan. We then describe the modularity properties of these coefficients, showing that they are "mixed partial theta functions", along the way determining a new class of quantum modular partial theta functions which is of independent interest.

Explore related subjects

math.NThep-th

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kathrin Bringmann, Thomas Creutzig, Larry Rolen. 2015-08-18. Negative index Jacobi forms and quantum modular forms. https://arxiv.org/abs/1401.7189

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove a two-term asymptotic formula for the eigenvalues of L, show their algebraic simplicity, sign alternation pattern, and decrease in absolute value. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.

math.NT↗
arXiv2026

Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Zábrádi's functor

Let $ρ$ be a smooth $n$-dimensional representation of $\mathcal{G}_{\mathbb{Q}_p}$ over $\overline{\mathbb{F}_p}$. When $ρ$ is generic and a good conjugate, the article "Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$", by Breuil-Herzig-Hu-Morra-Schraen, introduces the notion of an admissible representation $Π$ of $\mathrm{GL}_n(\mathbb{Q}_p)$ compatible with $ρ$. In loc. cit., the five authors also question whether there exists some $Π$ compatible with $ρ$ from which Zábrádi's functor $\mathbf{V}_Δ$ recovers a specific representation $\overline{L}^{\boxtimes}(ρ)$ of $\mathcal{G}_{\mathbb{Q}_p}^{n-1}$, constructed from $ρ$. We give a range of results about how badly $\mathbf{V}_Δ(Π)$ behaves for an arbitrary $Π$ satisfying some weaker compatibilities with $ρ$. In particular, when $ρ$ is reducible and $n\geq 3$, no representation $Π$ compatible with $\widetilde{P}_ρ$ can satisfy $\mathbf{V}_Δ(Π)\simeq \overline{L}^{\boxtimes}(ρ)$.

math.NT↗
arXiv2026

Curves of genus two with maps of every degree to a fixed elliptic curve

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve.

math.NT↗