On explicit isomorphisms of spaces of Jacobi forms of lattice-index

Abstract

Let $\underline{L}$ be a positive definite, even lattice. In Dieckmann (Abh Math Sem Univ Hamburg 8(5):197–213, 2015), the author studied the $\mathcal O(\mathcal H)$-module $\mathcal E(\underline{L})$ of holomorphic functions $\phi$ on the Jacobi half-space $\mathcal H \times (L \otimes \mathbb {C})$, which satisfy the elliptic transformation law

$$\begin{aligned} \phi (z, w+ lz + l^\prime ) = \text {e}^{-2\pi i \left( Q(l) \cdot z + B(l, w) \right) } \cdot \phi (z,w), \quad l,l^\prime \in L. \end{aligned}$$

Given another positive definite, even lattice $\underline{L_0}$ and an isometry $\iota : \underline{L_0} \longrightarrow \underline{L}$, there is a operator $[\iota ]: \mathcal E(\underline{L}) \longrightarrow \mathcal E(\underline{L_0})$ defined by a natural pullback procedure. This operator induces homomorphisms between (finite-dimensional) spaces of Jacobi forms $J_{k,\underline{L}}$ and $J_{k,\underline{L_0}}$. In distinguished cases, pullback operators induce isomorphisms of spaces of Jacobi forms. We utilise the transformation matrices to obtain explicit descriptions of the inverse maps on the basis of the attached spaces of vector-valued modular forms with respect to (sub-)representations of the corresponding Weil representations. These maps provide liftings of classical Jacobi forms of index 1 or 2 in the sense of Eichler and Zagier (The theory of Jacobi forms, Progress in mathematics, vol. 55, Birkhäuser, Boston, 1985) to Jacobi forms of higher rank index. We give explicit dimension formulas that partly were already known, cf. Krieg (Math Ann 276:675–686, 1987; Math Ann 289:663–681, 1991) and Gehre (Quaternionic Modular Forms of Degree two over ${\mathbb {Q}}(-3, -1)$. Ph.D. Thesis, Aachen, 2012). This paper summarises and extends parts of the author’s thesis.