Definition Of A Shadow

Question

The Wikipedia page for 'Mock Modular Forms' says: "The mock modular form h is holomorphic but not quite modular, while h + g* is modular but not quite holomorphic. The space of mock modular forms of weight k contains the space of nearly modular forms ("modular forms that may be meromorphic at cusps") of weight k as a subspace. The quotient is (antilinearly) isomorphic to the space of holomorphic modular forms of weight $(2 − k)$. The weight $(2 − k)$ modular form g corresponding to a mock modular form h is called its shadow". Further more it says: "Any modular form of weight k is a mock modular form of weight k with shadow $0$" Can someone please explain what this means? Specifically the definition of a shadow. If the shadow is '$0$' does that mean the shadow is just some real number?

Answer 1

That Wikipedia page could certainly be improved.

A proper (and blind – I am just translating what is on Wikipedia) wording would be: let $\mathcal{M}_k$ be the space of mock modular forms of weight $k$ (let’s omit level structure for the sake of simplicity).

Then there is an injection $j: \mathcal{N}_k \rightarrow \mathcal{M}_k$, where $\mathcal{N}_k$ is the set of holomorphic functions $\mathbb{H} \rightarrow \mathbb{C}$ that transform as modular forms of weight $k$ under the action of $SL_2(\mathbb{Z})$, and are meromorphic at cusps.

What Wikipedia states is that the quotient $\mathcal{M}_k/j(\mathcal{N}_k)$ is (anti-linearly) isomorphic to $M_{2-k}$, where $M_n$ is the space of actual modular forms of weight $n$. The “shadow” of some $F \in \mathcal{M}_k$ is the image under this isomorphism of the class of $F$ in $\mathcal{M}_k/j(\mathcal{N}_k)$.

Saying that a mock modular form $F$ has shadow $0$ means that under this isomorphism, the image of $F$ is the modular form zero (mapping any $\tau \in \mathbb{H}$ to zero). In other words, $F$ is in $j(\mathcal{N}_k)$, ie is a nearly holomorphic modular form.