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Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum $$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$ where $k$ is a positive integer greater than 1.

What is known in the literature about the values of these sums? I know they should be algebraic numbers that lie in the Hilbert/ring class field corresponding to the lattice.

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  • $\begingroup$ Of course, I meant I'm particularly interested in $k=2$ and $k=3$ $\endgroup$
    – Rdrr
    Commented Jan 4, 2021 at 18:17
  • $\begingroup$ For $k=1$ there are two ways to make sense of the series: Eisenstein summation and Kronecker/Hecke summation. This is explained in Weil's book "Elliptic functions according to Eisenstein and Kronecker". $\endgroup$
    – François Brunault
    Commented Jan 4, 2021 at 19:24
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    $\begingroup$ Relevant: mathoverflow.net/questions/311879 $\endgroup$
    – François Brunault
    Commented Jan 4, 2021 at 19:37
  • $\begingroup$ Apologies, I meant $k=2$ and $k=3$. As pointed out in Francois' link, $E_4$ and $E_6$ are known in terms of $j(\tau)$ and $\eta(\tau)$, but are the special values of these functions for CM values of $\tau$, well known? $\endgroup$
    – Rdrr
    Commented Jan 4, 2021 at 20:13
  • $\begingroup$ I do not know how well you need this to be known. This is not known to me but it may be known to Don Zagier. $\endgroup$
    – markvs
    Commented Jan 4, 2021 at 20:31

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