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Modular forms are defined here: http://en.wikipedia.org/wiki/Modular_form#General_definitions

Maass forms are defined here: http://en.wikipedia.org/wiki/Maass_wave_form

I wonder if modular forms can be transfered into Maass forms. Or they two are different categories of automorphic forms.

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3 Answers

up vote 24 down vote accepted

In the more common terminology modular forms on the upper half-plane fall into two categories: holomorphic forms and Maass forms. In fact there is a notion of Maass forms with weight and nebentypus, which includes holomorphic forms as follows: if $f(x+iy)$ is a weight $k$ holomorphic form, then $y^{k/2}f(x+iy)$ is a weight $k$ Maass form.

There are so-called Maass lowering and raising operators that turn a weight $k$ Maass form into a weight $k-2$ or weight $k+2$ Maass form. Using these, the weight $k$ holomorphic forms can be understood as those that are "new" for weight $k$: for $k\geq 2$ the raising operator isometrically embeds the space of weight $k-2$ Maass forms into the space of weight $k$ Maass forms, and the orthogonal complement is the subspace coming from weight $k$ holomorphic forms as described in the previous paragraph; also, the lowering operator acts as an inverse on the image of the raising operator and annihilates the mentioned orthogonal component.

All these connections can be better understood in the language of representation theory. I learned this material from Bump: Automorphic Forms and Representations, see especially Theorem 2.7.1 on page 241. Another good reference (from the classical perspective) is Duke-Friedlander-Iwaniec (Invent Math. 149 (2002), 489-577), see Section 4 there.

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Automorphic forms correspond to representations that occur in $L^2(G/\Gamma)$. In the case when $G$ is $SL_2$, holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of $G$, while Maass wave forms correspond to (spherical vectors of) continuous series representations of $G$.

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I'm not a specialist in the field, but recently it happened to me to read the beautiful paper by Ono "The last words of a genius" on the Notices of the AMS, December 2010, which seems related to your question.

Let $M \colon \mathbb{H} \to \mathbb{C}$ be a smooth function that transforms like a weight $k$ modular form and such that $\Delta_k(M)=0$. Then we say that $M$ is a weight $k$ harmonic Maass form.

Any harmonic Maass form can be uniquely written as

$M=M^{+} + M^{-}$,

where $M^+$ is the holomorphic part and $M^-$ is the non-holomorphic part. Then the modular forms are exactly those harmonic Maass forms such that $M^-=0$.

In the general case, the holomorphic part of a harmonic Maass form is not a modular form, but it is still a very interesting object. For instance, when $k=1/2$ it is a so-called mock theta function.

Mock theta functions were first described by Ramanujan in a famous letter to Hardy, written on his deathbed, but only very recently their deep connections with real-analytic modular forms were discovered by S. Zwegers, in his Ph.D. thesis written under D. Zagier.

For further detail you can look at Ono's paper or at the article "What is... a mock modular form?" by Amanda Folsom in the same issue of the Notices of the AMS.

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