Concerning the classical normalized Eisenstein series

Solution 1:

I am unable to give a proper answer, but the problem can be somewhat reduced and the example for $\tau=\sqrt{-7}$ is easy enough to demonstrate. I give those details in the hope that others can concentrate on the more intricate matters, such as higher class numbers.

Let us write $$\begin{align} \operatorname{E}_4 &= \gamma_2\eta^8 & \gamma_2 &= \mathfrak{f}^8\mathfrak{f}_1^8 + \mathfrak{f}^8\mathfrak{f}_2^8 - \mathfrak{f}_1^8\mathfrak{f}_2^8 \\ \operatorname{E}_6 &= \gamma_3\eta^{12} & \gamma_3 &= \frac{1}{2}\left(\mathfrak{f}^8 + \mathfrak{f}_1^8\right) \left(\mathfrak{f}^8 + \mathfrak{f}_2^8\right) \left(\mathfrak{f}_1^8 - \mathfrak{f}_2^8\right) \end{align}$$ where $\eta$ is the Dedekind eta function and $\mathfrak{f},\mathfrak{f}_1,\mathfrak{f}_2,\gamma_2,\gamma_3$ are modular Weber functions. Then we can split the task of evaluating the classical Eisenstein series at complex quadratic irrationals into the following subtasks:

1. Evaluate some modular (Weber) function, which gives an algebraic value. Due to algebraic interrelations, the algebraic nature carries over to the values of the other modular functions.
2. Evaluate the weight-giving eta factor, which introduces a product of Gamma function values.

Subtask 1 has been routinely done a century ago using modular equations and transformations, as well as more advanced methods that do not require knowledge of the modular equation. Useful exercises and references can be found in [BB87]. As a very easy example, combine e. g. the basic identities for Weber functions $$\begin{align} \mathfrak{f}(\tau)\,&\mathfrak{f}_1(\tau)\,\mathfrak{f}_2(\tau) = \sqrt{2} & \mathfrak{f}(-\tau^{-1}) &= \mathfrak{f}(\tau) \\ \mathfrak{f}^8(\tau) &= \mathfrak{f}_1^8(\tau) + \mathfrak{f}_2^8(\tau) & \mathfrak{f}_1(-\tau^{-1}) &= \mathfrak{f}_2(\tau) \end{align}$$ with the modular equation $$ \mathfrak{f}(\tau)\,\mathfrak{f}(7\tau) = \mathfrak{f}_1(\tau)\,\mathfrak{f}_1(7\tau) + \mathfrak{f}_2(\tau)\,\mathfrak{f}_2(7\tau)$$ and set $-\tau^{-1} = 7\tau$, then you can deduce $\mathfrak{f}^3(\tau) = 2\sqrt{2}$. As an eta quotient, $\mathfrak{f}(\tau)$ takes positive real values for purely imaginary $\tau$, therefore $\mathfrak{f}(\tau) = \sqrt{2}$. Then $\mathfrak{f}(\sqrt{-7}) = \mathfrak{f}(-\tau^{-1}) = \mathfrak{f}(\tau) = \sqrt{2}$.

Ramanujan tabled values of his class invariants $G_n$ and $g_n$ which are closely related to Weber's $\mathfrak{f}$ resp. $\mathfrak{f}_1$ at $\tau=\sqrt{-n}$. Likewise, in [Web08], which also provides the theory, the appendix contains a table with Weber's $\mathfrak{f}(\sqrt{-n})$ or $\mathfrak{f}_1(\sqrt{-n})$ for quite many positive integer values of $n$. From each such value, the corresponding values of the other Weber functions can be determined algebraically, e. g. for $\tau=\sqrt{-7}$ we arrive at $$\begin{align} \mathfrak{f}(\sqrt{-7}) &= \sqrt{2} & \gamma_2(\sqrt{-7}) &= 255 \\ \mathfrak{f}_{1,2}(\sqrt{-7}) &= \sqrt[8]{8 \pm 3 \sqrt{7}} & \gamma_3(\sqrt{-7}) &= 1539 \sqrt{7} \end{align}$$

Subtask 2 may be attempted with a ${}_2F_1$-based representation such as $$ \eta^2 = \frac{1}{\mathfrak{f}^4}{}_2F_1\left( \frac{1}{4},\frac{1}{4};1;\frac{64}{\mathfrak{f}^{24}}\right)$$ as presented elsewhere on this site. Background for that representation is given in section 5.4 around proposition 21 in [Zag08]. However, reducing that ${}_2F_1$ expression for a given algebraic value of $\mathfrak{f}^{24}$ to a product of Gamma function values seems a long-winded and barren, if not outright infeasible, route to me. Roughly speaking, each value would require some specific sequence of even more specific ${}_2F_1$ transformations, which is the direct opposite to what we actually want: A general method that does not depend much on the value of $\tau$.

I am entering unfamiliar terrain now, so let's hope I get the facts right.

Subtask 2 seems to have been boosted with a formula by Lerch (1897) that expresses a certain product of eta function values in terms of a product of Gamma function values. More than half a century later, such a thing became known as Chowla-Selberg formula[CS67]. The eta product therein contains $h(-n)$ eta factors with arguments in $\mathbb{Q}(\sqrt{-n})$ where $h(-n)$ is the class number of the ring of integers of $\mathbb{Q}(\sqrt{-n})$.

For $h(-n)=1$, the Chowla-Selberg formula can be used to solve for the value of a single eta function. In particular, for an odd prime $p$ with $h(-p)=1$, we get $$\begin{align} \eta^4(\sqrt{-p}) &= \frac{1}{2\pi p\,\mathfrak{f}^4(\sqrt{-p})} \left(\prod_{m=1}^{p-1} \Gamma\left(\frac{m}{p}\right)^{\chi(m)}\right)^{w/2} \\ &= \frac{1}{(2\pi)^{1+w\frac{p-1}{4}}p^{1-\frac{w}{4}} \mathfrak{f}^4\left(\sqrt{-p}\right)} \left(\prod_{\chi(m)=1} \Gamma\left(\frac{m}{p}\right)\right)^w \end{align}$$ where $\chi(m) = \left(\frac{m}{p}\right)_2$ is the Legendre symbol, and $w$ is the number of units in the ring of integers of $\mathbb{Q}(\sqrt{-p})$.

• Note that the above formula for $\eta^4(\sqrt{-p})$ contains $\mathfrak{f}^4(\sqrt{-p})$, so you still need subtask 1. Alternatively, note that $\mathfrak{f}(\tau)\,\eta(\tau) = (-1)^{-1/24}\eta\left(\frac{\tau+1}{2}\right)$, therefore the Chowla-Selberg formula gives you $\eta^4\left(\frac{\sqrt{-7}+1}{2}\right)$ directly.

For $p = 7$ we thus obtain $$ \eta^4(\sqrt{-7}) = \frac{\Gamma\left(\frac{1}{7}\right) \Gamma\left(\frac{2}{7}\right) \Gamma\left(\frac{4}{7}\right)} {56\,\pi\,\Gamma\left(\frac{3}{7}\right) \Gamma\left(\frac{5}{7}\right) \Gamma\left(\frac{6}{7}\right)} = \frac{\left(\Gamma\left(\frac{1}{7}\right) \Gamma\left(\frac{2}{7}\right) \Gamma\left(\frac{4}{7}\right)\right)^2} {64\,\pi^4 \sqrt{7}}$$ and combining this with the values for $\gamma_2$ and $\gamma_3$ we get $$\begin{align} \operatorname{E}_4 &= \frac{255\left(\Gamma\left(\frac{1}{7}\right) \Gamma\left(\frac{2}{7}\right) \Gamma\left(\frac{4}{7}\right)\right)^4} {28672\,\pi^8} \\ \operatorname{E}_6 &= \frac{1539\left(\Gamma\left(\frac{1}{7}\right) \Gamma\left(\frac{2}{7}\right) \Gamma\left(\frac{4}{7}\right)\right)^6} {1835008\,\pi^{12}} \end{align}$$ as you have mentioned.

For $h(-n)>1$, the remaining problem was to isolate individual eta values from the product. Steady progress has been made to overcome the intrinsic limitations of earlier methods. Let me refer you to [Har04] or [CH05] for a method that builds on results by Williams et al., van der Poorten, Chapman, and Hart from around 2000. I have not looked into it, so I cannot tell whether this method also improves subtask 1, or uses it as a building block, or both.

References

[BB87] J. M. Borwein and P. B. Borwein: Pi and the AGM, Wiley 1987, ISBN 0-471-83138-7.

[CH05] R. Chapman and W.B. Hart: Evaluation of the Dedekind eta function. In: Canadian Mathematical Bulletin 49 (2006), pp. 21-35. DOI: 10.4153/CMB-2006-003-1.

[CS67] S. Chowla and A. Selberg: On Epstein's zeta function. In: Crelles Journal für die reine und angewandte Mathematik 227 (1967), pp. 86-110. Available online.

[Har04] W. B. Hart: Evaluation of the Dedekind Eta Function. PhD thesis 2004, Macquarie University, Sydney.

[Web08] H. Weber: Lehrbuch der Algebra, Vol. III. In german. AMS Chelsea Publishing, 3rd edition 1961, ISBN 0-8218-2971-8. Reprinted 2001, 1st edition 1908.

[Zag08] Don Zagier: Elliptic modular forms and their applications. In: Kristian Ranestad (ed.): The 1-2-3 of modular forms. Springer 2008, DOI: 10.1007/978-3-540-74119-0.

Solution 2:

These sorts of formulas are known, I think. Particular cases go back to Hurwitz (in the late 1800's) and perhaps even further back, to Gauss (maybe), Eisenstein, and others. The general case, though, is due (as far as I know) to Damerell.

Actually, what I am now going to discuss involves just getting the transcendental factors correct, rather than pinning the number down precisely. But I'm pretty sure that, at least in particular cases like $\sqrt{-7}$, this "pinning down" would have been known, although perhaps not well-known.

Damerell's paper (his thesis, I think) is here. I find it pretty hard to read, though, since it doesn't use modern notation and terminology for modular forms.

I learnt about Damerell's results from the papers of Katz, in which he constructs his (so-called) two-variable $p$-adic $L$-functions for Hecke characters. He has a beautiful paper called "$p$-adic $L$-functions via moduli" which I couldn't find on line. But here is his 1978 ICM address, with Damerell's result stated on the second page: if you take $r = 0$ in $A(k,r)$, then you have a sum of powers of elements in the ring of integers of an imaginary quadratic field, which is exactly the value of an Eisenstein series. (In the case where of $\sqrt{-7}$, I guess your value of $E_4$ is summing over elements of $\mathbb Z[\sqrt{-7}]$ rather than $\mathbb Z[(1 + \sqrt{-7})/2]$ so there is a slight difference, which I think should be easy to sort out. I also think Damerell's paper itself actually handle non-maximal orders directly.)

Of course you need to know the value of $\Omega$, the relevant period: but this is given by the Chowla--Selberg formula. What you can see, then, is that Damerell's result gives a description of the Eisenstein values you are interested in terms of a product of $\Gamma$-values and powers of $\pi$ with an algebraic number which (at least in the form of the result stated by Katz) is not pinned down.

This formula is related to special values of certain $L$-functions, and so Damerell's result is a special case of a statement conjectured by Deligne. Pinning down the algebraic factor in general is the topic of the Bloch--Kato conjecture. In this particular case, I don't know what is known about it. Here is a paper discussing these kind of rationality/algebraicity results, connecting the particular case of CM elliptic curves (or CM abelian varieties, more generally) to Deligne's general framework.

You might also want to look at Weil's book on Elliptic functions according to Eisenstein and Kronecker, which might have some relevant classical information.

In conclusion, I would compare your method with known proofs of the Chowla--Selberg formula, of Damerell's theorem, of Lerch's formula, and of other related results, to see whether you have rediscovered a known approach. If not, i.e. if you've found a new approach to this circle of ideas, that's definitely interesting. (Even if it's not new, it's still interesting! --- but just not new.)