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I am reading Shimura's paper "The Special Values of the Zeta Functions Associated With Hilbert Modular Forms" and I do not exactly understand his definition of the Eisenstein series in section 3.

Can someone refer me to another paper that explains this construction of Eisenstein series of a Hilbert modular form of totally real field extensions more clearly? I am mostly interested in the case of a quadratic field extension.

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    $\begingroup$ May be section 2 of Samit Dasgupta, Henri Darmon, and Robert Pollack Hilbert modular forms and the Gross-Stark conjecture Annals of Math. (2) 174 (2011), no. 1, 439-484. In the homepage of Henri Darmon there is a pdf... $\endgroup$
    – Xarles
    Commented Jul 25, 2018 at 21:34
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    $\begingroup$ Tim Atwill and I described this construction in our paper "Newform theory for Hilbert Eisenstein series" (available at benjaminlinowitz.com/papers/hilberteisenstein.pdf). See for instance sections 2 and 3. I also agree with Xarles that the paper of Dasgupta, Darmon and Pollack contains a very nice (and concrete) description of the construction of Hilbert Eisenstein series. $\endgroup$
    – user1073
    Commented Jul 25, 2018 at 23:21
  • $\begingroup$ @Xarles Thank you! Dasgupta, Darmon, and Pollack explicitly compute the constant term of the Fourier expansion of the Eisenstein series. Do you know if/where I can find a similar computation of the non constant terms? $\endgroup$
    – R.T.
    Commented Jul 31, 2018 at 15:01

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