# Basic Properties

• Martin Eichler
• Don Zagier
Part of the Progress in Mathematics book series (PM, volume 55)

## Abstract

The definition of Jacobi forms for the full modular group Γ1 = SL2(ℤ) was already given in the Introduction. In order to treat subgroups Γ ⊂ Γ1 with more than one cusp, we have to rewrite the definition in terms of an action of the groups SL (ℤ) and ℤ2 on functions . This action, analogous to the action
$$({\left. f \right|_k}M)(\tau ): = {(c\tau + d)^{ - k}}f\left( {\frac{{a\tau + b}}{{c\tau + d}}} \right)\quad \left( {M = \left( {\begin{array}{*{20}{c}} {a\quad b} \\ {c\quad d} \end{array}} \right) \in {\Gamma _1}} \right)$$
(1)
in the usual theory of modular forms, will be important for several later constructions (Eisenstein series, Hecke operators). We fix integers k and m and define
$$(\phi {{|}_{{k,m}}}[\begin{array}{*{20}{c}} {a\,b} \\ {c\,d} \\ \end{array} ])(\tau ,z): = {{(c\tau + d)}^{{ - k}}}{{e}^{m}}(\frac{{ - c{{z}^{2}}}}{{c\tau + d}})\phi (\frac{{a\tau + b}}{{c\tau + d}},\frac{z}{{c\tau + d}})((\begin{array}{*{20}{c}} {a\,b} \\ {c\,d} \\ \end{array} ) \in {{\Gamma }_{1}})$$
(2)
and
$$\left( {\phi \left| {_m\left[ {\lambda \;\mu } \right]} \right.} \right)(\tau ,z)\quad : = {e^m}({\lambda ^2}\tau + 2\lambda z)\phi (\tau ,z + \lambda \tau + \mu )\left( {\left( {\lambda \;\mu } \right) \in {\mathbb{Z}^2}} \right),$$
(3)
where em (x) = e2πimx (see “Notations”). Thus the two basic transformation laws of Jacobi forms can be written
$$\phi \left| {_{k,m}M = } \right.\phi \quad \left( {M \in {\Gamma _1}} \right),\quad \phi \left| {_mX = \phi \quad \left( {X \in {\mathbb{Z}^2}} \right)} \right.,$$
where we have dropped the square brackets around M or X to lighten the notation. One easily checks the relations
$$\begin{array}{*{20}{c}} {\left( {\phi \left| {_{k,m}M} \right.} \right)\left| {_{k,m}M' = \phi \left| {_{k,m}\left( {MM'} \right),\quad \left( {\phi \left| {_mX} \right.} \right){{\left| {_mX' = \phi } \right|}_m}\left( {X + X'} \right)} \right.} \right.,} \\ {\left( {\phi \left| {_{k,m}M} \right.} \right)\left| {_mXM = } \right.\left( {\phi \left| {_mX} \right.} \right)\left| {_{k,m}M,} \right.\quad \left( {M,M' \in {\Gamma _1},\;X,X' \in {\mathbb{Z}^2}} \right)} \end{array}$$
(4)

## Keywords

Modular Form Fourier Coefficient Eisenstein Series Cusp Form Jacobi Form