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I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the Fourier-expansions and constant terms. Another preference is that reference be written in the language of the reals instead of the Adeles, though this isn't so important.

Thanks!

UPDATE: Thanks for the references so far! I should have specified, I am interested mostly in the real analytic Eisenstein series, which are induced from the Borel.

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  • $\begingroup$ Since you said "anything would be welcome" I can mention Schwermer's "On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties". At one point I had to think a bit about Eisenstein cohomology of $A_2$ and this paper was quite useful. $\endgroup$
    – Dan Petersen
    Mar 16, 2016 at 6:59
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    $\begingroup$ This question is slightly ambiguous. Are you interested in Eisenstein series in the sense of Langlands (automorphic forms) or the more classical story of holomorphic functions in Siegel 3-space? The issue is that the dictionary between eigenforms and automorphic representations is more subtle in the Eisenstein case -- even for GL(2) [for example the classical Eisenstein series E_4 is holomorphic but E_2 is not so you see both in the automorphic theory but only one in the holomorphic theory] $\endgroup$
    – znt
    Mar 16, 2016 at 7:48
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    $\begingroup$ Siegel's 1939 paper "Einfuhrung..." discusses the rationality of the Fourier coefficients of (convergent) holomorphic Eisenstein series, certainly in classical terms. Is that the sort of thing you'd want? One of the appendices in Langlands' 544 shows how to obtain degenerate Eis by taking residues of minimal-parabolic Eis, in a very classical (and formulaic) manner... ? $\endgroup$
    – paul garrett
    Mar 16, 2016 at 20:15
  • $\begingroup$ Have you looked in Klingen's book? van der Geer's notes from the 1-2-3 of modular forms refer to a paper of Maass for computation of Fourier coefficients of Klingen Eisenstein series. $\endgroup$
    – Kimball
    Mar 17, 2016 at 21:16
  • $\begingroup$ Perhaps this paper of Haruki -- link.springer.com/article/10.1007%2FBF02678184 -- which discusses Eisenstein series associated to the Siegel parabolic in classical language is helpful, and maybe also Shimura's 1983 Duke paper referenced therein. $\endgroup$
    – Martin Dickson
    May 17, 2016 at 15:39

2 Answers 2

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For some reason I did just now remember that an appendix in R. Langlands' SLN 544 does show a classical, non-adelic method to treat (non-cuspidal data) Eisenstein series for $GL_n$ and $Sp_n$. The actual immediate point seemed/seems to be to show how to do this over number fields, paying attention to class numbers greater than $1$, etc., but I recall that when I first saw that years ago there was no easily accessible alternative source that I knew or had heard rumors of.

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I am going to assume you are asking about the (degenerate) Siegel Eisenstein series on $\mathrm{Sp}(2n)$. If that is the case, I think you should check out

Piatetski-Shapiro and Rallis, "Rankin triple products", section 4

and

Kudla and Rallis, "Poles of Eisenstein series and $L$-functions".

As far as I am aware, these are the canonical references for the harder analytic properties of the constant term and Fourier expansions of these Eisenstein series. But they are written in the adelic language, and offer perhaps a more in-depth treatment than you are looking for. So, you should be able to find an easier treatment just for the case of $\mathrm{Sp}(4)$, but I do not know a reference at the moment.

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