Details of Hecke Summation for weight 2 Eisenstein Series

Question

In reference to https://math.stackexchange.com/a/51467/356443

I don't understand the details of the Hecke summation trick in order to define an Eisenstein Series of weight $2$.

For convergence issues, we can't just define $$ E(z) = \sum_{c, d}'' \dfrac{1}{(cz+d)^2}. \qquad (z \in \mathbb{H}) $$ The trick is to consider $$ E(z,s) = \sum_{c, d}'' \dfrac{1}{(cz+d)^2 |cz+d|^{2s}}, \qquad (z\in\mathbb{H}; \Re(s) > 0) $$ and try to find an analytic continuation for $s$. This justifies setting $s=0$ to get a holomorphic function that should satisfy the right transformation rule.

It is noted that it is not always possible to continue $E(z,s)$ analytically. In the cases that such an analytic continuation does exist, what is the strategy for the computation to construct such a continuation?

Answer 1

In response to your question in the last paragraph, there are at least two strategies in the literature to obtain the analytic continuation of $E(z,s)$ (which, by the way, really should read $E_2(z,s)$ to remind oneself of the exponent of $2$ in the series, the weight) to $s=0$. One is by writing $E_2(z,s)$ as the Mellin transform of a suitable theta function - this is analogous to Riemann's famous proof of the analytic continuation (and functional equation!) of the Riemann zeta function, and the key technical tool is the Poisson summation formula. As pointed out in the comments, this is explained, e.g., in Diamond--Shurman, 4.10, and at least in the weight zero case, is Exercise 1.6.2 in Bump's "Automorphic forms and representations".

The second approach would be to compute the Fourier expansion of $E_2(z,s)$ and to verify the analytic continuation as well as the functional equation on the level of Fourier coefficients. Again, the Poisson summation formula is key to this. This is explained in great detail, but again in the weight zero case only, in Bump, 1.6, and in Zagier's Introduction to modular forms, 3.A, and probably in many other places as well.

However, if, as it would seem, you really are only interested in understanding what the value at $s=0$ of $E_2(z,s)$ is, you neither need to know Mellin transforms, nor Poisson summation, indeed not even analytic continuation at all. Instead, in that case Hecke's trick boils down to a skilful (but elementary!) computation of $\lim_{s\to 0^+}E_2(z,s)$ in terms of the (non-modular but holomorphic) $G_2(z)$. See Zagier's Elliptic modular forms and their applications, 2.3.

Answer 2

So far as I can tell, the original "Hecke summation", meromorphic continuation in an auxiliary variable "$s$" and then evaluation at a special point (often normalized to $s=0$) occurs in

E. Hecke, Theorie der Eisensteinschen Reihen h"oheren Stufe und ihre Anwendung auf Funktiontheorie und Arithmetik, Abh. Math. Sem. Hamburg 5 (1927), 199-224; collected works, 461-486.

Papers of G. Shimura that use this and related ideas are

On some arithmetic properties of modular forms of one and several variables, Ann. Math. 102 (1975) no. 3, 491-515.

The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976) no. 6, 783-804.

Looking superficially at those two papers, Shimura does not redo Hecke's determination that for weight 2 a non-trivial condition must be met so that the meromorphically-continued-and-evaluated-at-$s=0$ thing is holomorphic in the upper-half-plane coordinate. This fails for weight $k=2$ (over $\mathbb Q$) as we know. Hecke computes the Fourier expansion fairly directly to see this: the constant term has a leftover not-holomorphic summand.

I first saw this in lectures of Shimura about such things in 1975-6, where he did prove such things, using Fourier expansions for a proof of meromorphic continuation, and also to see what happens at the special point. I don't know whether he later put this into his books. Similar, but more complicated, computations make sense to attempt for higher-rank groups admitting holomorphic principal series.

My old book on Hilbert modular forms reproduces essentially Hecke's argument, with supplements from Shimura's lectures referring to Hilbert modular forms as well.

The germ of a higher-level explanation for the phenomenon is already hinted-at in some remarks of Shimura's in those papers, concerning the differential operators often called "Maass-Shimura" operators. Namely, the holomorphic discrete series repns of weight $2k$ occur as subreps of principal series with natural parameter $s=k$. Shimura's differential operators are "raising" operators inside the family of principal series, and increase the "weight" by $2$. There are also "lowering" operators acting, and the "holomorphic" vector in the weight-$2k$ discrete series is annihilated by the lowering operator. Shimura does not speak in these terms.

In particular (because formation of Eisenstein series is an intertwining operator) the lowering operator maps $E_{2k,s}$ to $(s-k)\cdot E_{2k-2,s}$. (The normalization here is more natural, but less elementary: the "holomorphic point" is at $s=k$, not $s=0$.) For $2k>2$ the Eisenstein series $E_{2k-2,s}$ is holomorphic at the special point, so in that case the lowering operator annihilates $E_{2k,k}$. As it happens (though there are reasons...), the lowering operator is essentially the Cauchy-Riemann operator, so this gives holomorphy.

For $2k=2$, the lowering operator maps $E_{2,s}$ to $(s-1)E_{0,s}$, and the special point is $s=1$. In this case, the weight-zero Eisenstein series has a pole at $s=1$. This is cancelled by the factor $s-1$, picking up the residue, which is a constant. Thus, the lowering operator does not quite annihilate $E_{2,1}$.

Analogous, more complicated, things happen on higher-rank groups admitting holomorphic discrete series.