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After reading many textbooks I still can't get the jargon correct. Given a spherical harmonic $u \in L^2(S^n)$ one could construct a theta function:

$$ \theta (z;u) = \sum_{ m \in \mathbb{Z}^3} u (m) e^{2\pi i \, |m|^2\,z}$$

with $z\in \mathfrak H$. The situation of interest is that $u$ is not constant. This thing is a $\Gamma_0(4)$ cusp form. Could these theta functions be automorphic forms?

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    $\begingroup$ It's definitely not clear what's being asked. My I'd like to know about the relationship between theta functions and cuspidal representations. $\endgroup$
    – john mangual
    Sep 18, 2017 at 18:46
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    $\begingroup$ As @reuns comments, perhaps partly rhetorically, one should only take homogeneous harmonic polynomials, and one must know Hecke's identity about Fourier transforms of such harmonic functions $\times$ Gaussians. This gives the usual modularity for $z\to z+1$ and $z\to -1/z$... $\endgroup$
    – paul garrett
    Sep 18, 2017 at 23:45
  • $\begingroup$ I'm confused (on several things about the question, but first): are you working with theta functions of 3 variables or $n$ variables? $\endgroup$
    – Kimball
    Sep 19, 2017 at 0:44
  • $\begingroup$ For some homogeneous polynomial $P$ of degree $n$ there is another polynomial $Q$ such that $P(x_1,x_2,x_3) e^{-\pi (x_1^2+x_2^2+x_3^2) t} = C \ t^{-n} \ Q(\partial_{x_1},\partial_{x_2},\partial_{x_3}) e^{-\pi (x_1^2+x_2^2+x_3^2) t}$ whose Fourier transform is $C \ (-2i \pi/t)^n t^{-3/2} Q(\xi_1,\xi_2,\xi_3) e^{-\pi (\xi_1^2+\xi_2^2+\xi_3^2) /t}$. If $P$ is harmonic then $P = Q$ and $\sum_m P(m) e^{2i \pi |m|^2 z}$ is a modular form of weight $n+3/2$ ? @paulgarrett $\endgroup$
    – reuns
    Sep 19, 2017 at 4:04
  • $\begingroup$ @reuns, yes, indeed. A complication in the question as it stands is that the quadratic form is in an odd number of variables, so, as "half-integral weight" modular form, Hecke theory works differently than for integral-weight. $\endgroup$
    – paul garrett
    Sep 19, 2017 at 20:03

1 Answer 1

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"Cusp form" means "cuspidal automorphic form" by definition. So yes, $\theta(z;u)$ is an automorphic form. But it is not a form lying in a cuspidal automorphic representation, because it is not a Hecke eigenform. (Actually, this also depends on the context, see the Added section for clarification.)

Cuspidal automorphic representations are irreducible subspaces of the relevant cuspidal automorphic $L^2$-space, so most cusp forms are linear combinations of forms coming from distinct (usually infinitely many) cuspidal automorphic representations. For non-cuspidal forms the picture is even more complicated: linear combination is replaced by an integral with respect to some spectral measure which includes various (Hecke-)Eisenstein series. The general theory is due to Langlands (spectral decomposition of $L^2$-automorphic forms). (Again, see the Added section for clarification. The point is that the notion of irreducibility depends on the group, and one can think of the group over the reals $\mathbb{R}$ but also over the rational adeles $\mathbb{A}_\mathbb{Q}$.)

Added. I realize now that a serious confusion can arise from considering (irreducible) cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{R})$ vs. considering those of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$. I like to consider the adelic group $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$, because it has more structure and is more relevant for number theory. For simplicity, let us restrict to automorphic forms spherical at the archimedean place (e.g. a classical Maass form on the upper half-plane). Then, a vector from a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$ corresponds to a cusp form on $\mathrm{GL}_n(\mathbb{R})$, which is an eigenfunction of the local spherical Hecke algebra at all but finitely many non-archimedean places, and vice versa. One can refine this notion to talk about newforms and oldforms, by considering non-spherical Hecke algebras (but Hecke algebras corresponding to a level). In the end, the notion of "automorphic form" and "automorphic representation" varies greatly with the context. A good introduction is Borel-Jacquet: Automorphic forms and automorphic representations (Proc. Symp. Pure Math. 53 (1979), 189-202), which discusses both $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$.

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    $\begingroup$ Sorry for this very naive question, but is it the "Hecke eigenformness" that makes the representation a form lies in irreducible ? I can ask a separate question if needed. $\endgroup$
    – Sylvain JULIEN
    Sep 18, 2017 at 21:12
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    $\begingroup$ @SylvainJULIEN: It is not a naive question but a good one. I will try to say a few words about this in an added section. $\endgroup$
    – GH from MO
    Sep 18, 2017 at 21:32
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    $\begingroup$ I have lost so much time on elementary or semantic issues like these. $\endgroup$
    – john mangual
    Sep 21, 2017 at 0:24

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