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The Eisenstein series \begin{equation*} G_{2k}(\tau) = \sum_{(m,n) \in \mathbb{Z} \setminus (0,0)} \frac{1}{(m + n \tau)^{2k}} \end{equation*} is absolutely convergent to a holomorphic function of $\tau$ in the upper half-plane (see https://en.wikipedia.org/wiki/Eisenstein_series).

I am wondering why this function is restricted to the upper half-plane. Is that just a convention, because the information on the lower half-plane is redundant? Or is the analytic continuation not well-defined?

My questions:

  1. Is there a unique analytic continuation on the whole complex plane (up to poles)?
  2. If yes, does $G_{2k}(\bar\tau) = \overline{G_{2k}(\tau)}$ hold for the analytic continuation?
  3. Is each singularity of $G_{2k}(\tau)$ a pole on the real line? (in particular: no essential singularity, no branch point)
  4. Where are the poles and zeros, of which multiplicity?

I understand that some questions may not have a short answer, but I would like to have an overview of what is well-known to experts in this area.

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    $\begingroup$ Remember that this $\tau$ here is the ratio of two complex numbers that generate a lattice. Therefore we could just assume $\tau\in\mathbb{H}$ since $\mathbb{Z}\tau_1=\mathbb{Z}(-\tau_1)$. I think you may find this helpful: math.stackexchange.com/questions/1730352/… $\endgroup$
    – justadzr
    Jul 30 '20 at 10:21
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    $\begingroup$ It converges on the lower half-plane too. However, the real line is a natural boundary, you cannot analytically continue the function over it anywhere. $\endgroup$ Jul 30 '20 at 10:44
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    $\begingroup$ @AnginaSeng Thank you. That basically kills the question, but rightfully so. Is more to be known about the behavior of $G_{2k}(\tau)$ as $\tau$ goes close to the real line? Does it diverges like for an essential singularity? I am just curious, I have never dealt with such functions before, I have no intuition yet. $\endgroup$
    – Loic
    Jul 30 '20 at 11:54
  • $\begingroup$ @Loic one can see that it behaves like $(-cz+a)^(-2k) 2 \zeta(2k)$ near the rational point $a/c$ (in lowest terms), using the invariance under the weighted action of $SL(2,\mathbb{Z})$ by considering its Fourier expansion. $\endgroup$
    – assaferan
    Oct 4 '20 at 12:06

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