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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.

Rdrr
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  • Of course, I meant I'm particularly interested in $k=2$ and $k=3$ – Rdrr Jan 04 '21 at 18:17
  • 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". – François Brunault Jan 04 '21 at 19:24
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    Relevant: https://mathoverflow.net/questions/311879 – François Brunault Jan 04 '21 at 19:37
  • 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? – Rdrr Jan 04 '21 at 20:13
  • 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. – markvs Jan 04 '21 at 20:31
  • Ideally as well known as possible. Particularly as an element of the Hilbert class field or ring class field. – Rdrr Jan 04 '21 at 21:59
  • @Rdrr: If you can formulate the question coherently, send it to Don Zagier. – markvs Jan 05 '21 at 00:17

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