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The real Eisenstein series

$G_s^* = \frac{\Gamma(s)}{\pi^s} \sum'_{m,n}\frac{Im(\tau)}{|m+n \tau|^{2s}}$

admits the following integral representation (their Mellin transform):

$G_s^* = \frac{1}{2} \int_0^\infty (\Theta_\tau(t)-1) t^{s-1} dt$

where $\Theta_\tau(t)$ is the associated theta series $\Theta_\tau(t) = \sum_{m,n}e^{-\pi Q_\tau(m,n)t}$ . In general, we can construct real modular invariant quantities by considering:

$\hat M(\tau) = \frac{1}{2} \int_0^\infty (\Theta_\tau(t)-1) M(t) dt$

Does anyone know if it is possible to invert this relation? I.e. given a real modular form $\hat M(\tau)$, is it possible to find $M(t)$? Are there any references I could look at? Thank you very much!

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Yes, this is essentially the $L^2$ spectral decomposition of the orthogonal complement to cuspforms (waveforms). It is spanned (for example) by pseudo-Eisenstein series $E_\phi$, formed by winding up functions of the form $\Phi(x+iy)=\phi(y)$ with $\phi\in C^o_c(0,\infty)$. The more mundane spectral decomposition of $\phi(y)$ on $(0,+\infty)\sim \mathbb R$ by ordinary Fourier transform, and a few further manipulations, expresses pseudo-Eisenstein series as integrals nearly of the sort given in the question. (It may be better to remove the factor $\zeta(2s)$ from the expression as posed.)

The previous scenario produces functions $M({1\over 2}+it)$ in the Paley-Wiener space as functions of $t\in \mathbb R$; then one proves that the map is an $L^2$ isometry on this part of the spectrum, and extends to full Plancherel by extending by continuity.

Given an automorphic form $f$ in the orthogonal complement to the cuspforms, the $s$-th spectral component is $\langle f, E_s\rangle$, where $E_s$ is the Eisenstein series with the $\zeta(2s)$ and Gamma factor removed, so that its Fourier expansion begins $y^s+c_sy^{1-s}+\ldots$ where $c_s=\xi(2s-1)/\xi(2s)$, with $\xi(s)$ the completed zeta.

One version of that discussion is at http://www.math.umn.edu/~garrett/m/mfms/notes_c/cont_afc_spec.pdf I first saw the argument in Godement's articles in the Boulder Conference, AMS Proc Symp Pure Math IX.

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  • $\begingroup$ Thank you very much Paul! the link you provided me with seems to give an error, could you please check it? $\endgroup$
    – fernando
    Nov 23, 2013 at 21:11
  • $\begingroup$ Ah, sorry, typo in link, fixed now. $\endgroup$ Nov 23, 2013 at 21:42
  • $\begingroup$ Dear Paul, sorry for taking your time, but being a non-mathematician I am finding these notes a little hard to follow. Do you know if anyone has worked out examples of the above inversion procedure? thanks so much! $\endgroup$
    – fernando
    Nov 24, 2013 at 19:52
  • $\begingroup$ @fernando, my intention in those notes was to treat the simplest possible case, with least possible prerequisites, (in which I may have failed, of course!), so I think it's hard to make it simpler. The best-known examples of computing the integral of a waveform against $E_s$ are $\langle fg,E_s\rangle$ with $f,g$ two cuspforms, and then the outcome is the Rankin-Selberg $L$-function attached to the two. Not an elementary thing! For number theorists, this non-elementariness is very interesting, but much less elementary than one might have hoped. Perhaps I misunderstand what you want/need? $\endgroup$ Nov 24, 2013 at 20:37
  • $\begingroup$ Dear Paul, I think your notes are great! it is just that my background is really poor. In my specific problem I only have $\hat M(\tau)$ numerically, but I know it is modular invariant, real, and I have it in the whole fundamental region. So I was interested in computing (numerically as well) the corresponding $M(t)$. $\endgroup$
    – fernando
    Nov 25, 2013 at 20:00

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