The Theory of Jacobi Forms

The functions studied in this monograph are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jacobi form on SL₂(ℤ) to be a holomorphic function and having a Fourier expansion of the form Here k and m are natural numbers, called the weight and index of Φ, respectively. Note that the function Φ(τ,0) is an ordinary modular form of weight k, while for fixed τ the function z → Φ(τ,z) is a function of the type normally used to embed the elliptic curve ℂ/ℤτ + ℤ into a projective space.

We use ℕ to denote the set of natural numbers, ℕ₀ for ℕ∪{0}. We use Knuth’s notation ⌊x⌋ (rather than the usual [x]) for the greatest-integer function max{n∈ℤ|n ≦x} and similarly ⌈x⌉ = min{n∈ℤ|n≧x} = −⌊−x⌋. The symbol ☐ denotes any square number. By d‖n we mean d|n and \( \left( {d,\frac{n}{d}} \right)\; = \;1. \). In sums of the form \( \mathop \sum \limits_{d\left| n \right.} \) or \( \mathop \sum \limits_{ad\; = \;\ell } \) it is understood that the summation is over positive divisors only. The function \( \mathop \sum \limits_{d\left| n \right.} {d^\upsilon } \) (de IN) is denoted σ_υ (n).

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 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 and where e^m (x) = e^2πimx (see “Notations”). Thus the two basic transformation laws of Jacobi forms can be written where we have dropped the square brackets around M or X to lighten the notation. One easily checks the relations

In §2 we showed that the coefficients c(n,r) of a Jacobi form of index m depend only on the “discriminant” r²-4nm and on the value of r(mod 2m), i.e.

The object of this and the following section is to obtain as much information as possible about the algebraic structure of the set of Jacobi forms, in particular about We will study only the case of forms on the full Jacobi group \( \Gamma _1^J \) (and usually only the case of forms of even weight), but many of the considerations could be extended to arbitrary Γ.