What is the relationship between modular forms and Maass forms? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:11:29Z http://mathoverflow.net/feeds/question/52744 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms What is the relationship between modular forms and Maass forms? 7-adic 2011-01-21T08:08:46Z 2011-01-22T09:51:42Z <p>Modular forms are defined here: <a href="http://en.wikipedia.org/wiki/Modular_form#General_definitions" rel="nofollow">http://en.wikipedia.org/wiki/Modular_form#General_definitions</a></p> <p>Maass forms are defined here: <a href="http://en.wikipedia.org/wiki/Maass_wave_form" rel="nofollow">http://en.wikipedia.org/wiki/Maass_wave_form</a></p> <p>I wonder if modular forms can be transfered into Maass forms. Or they two are different categories of automorphic forms.</p> http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms/52753#52753 Answer by Francesco Polizzi for What is the relationship between modular forms and Maass forms? Francesco Polizzi 2011-01-21T09:55:33Z 2011-01-22T09:51:42Z <p>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 <a href="http://www.ams.org/notices/201011/index.html" rel="nofollow">Notices of the AMS, December 2010</a>, which seems related to your question.</p> <p>Let $M \colon \mathbb{H} \to \mathbb{C}$ be a <em>smooth</em> function that transforms like a weight $k$ modular form and such that $\Delta_k(M)=0$. Then we say that $M$ is a <em>weight $k$ harmonic Maass form</em>.</p> <p>Any harmonic Maass form can be uniquely written as</p> <p>$M=M^{+} + M^{-}$,</p> <p>where $M^+$ is the <em>holomorphic part</em> and $M^-$ is the <em>non-holomorphic part</em>. Then the modular forms are exactly those harmonic Maass forms such that $M^-=0$. </p> <p>In the general case, the holomorphic part of a harmonic Maass form is <em>not</em> a modular form, but it is still a very interesting object. For instance, when $k=1/2$ it is a so-called <em>mock theta function</em>.</p> <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms/52766#52766 Answer by GH for What is the relationship between modular forms and Maass forms? GH 2011-01-21T13:47:18Z 2011-01-21T13:47:18Z <p>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. </p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms/52791#52791 Answer by Richard Borcherds for What is the relationship between modular forms and Maass forms? Richard Borcherds 2011-01-21T20:43:42Z 2011-01-21T20:43:42Z <p>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 $G$. </p>