# Eisenstein Series and cusps of $\Gamma_{1}(N)$

Let $q = e^{2\pi i\tau}$, $\operatorname{Im}\tau > 0$ and let $$G(\tau) = 1 - 24\sum_{n = 1}^{\infty}\frac{nq^{n}}{1 - q^{n}}$$ be the weight 2 Eisenstein series for $\Gamma(1)$. Consider the inequivalent cusps of $\Gamma_{1}(N)$. For each cusp, there is an Eisenstein series of weight 2 associated to this cusp. How does one relate this Eisenstein series to a linear combination of $G(k\tau)$'s (for some integers $k$)?

In particular, the case I am considering is when $N = 7$. The inequivalent cusps are $0$, $2/7$, $1/3$, $3/7$, $1/2$, and $\infty$. Consider the cusp $2/7$ and denote by $E_{2/7}$ the associated Eisenstein series. I would like to write $E_{2/7}$ as a finite sum of $G(k\tau)$ for some integers $k$. One of the first issues I'm having is how would I know the $q$-expansions of Eisenstein series of $E_{2/7}$ (which is related to the issue that I can't seem to find a good reference for the definition of a weight 2 Eisenstein series associated to a cusp)? Second, would finding the desired linear combination of $G(k\tau)$'s be just an exercise in matching up finitely many coefficients in the $q$-expansion?

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This is far from answering your whole question, but did you look at the first few sections of Chapter 4 in Diamond and Shurman's A first course in Modular forms? –  M Turgeon Aug 4 '12 at 17:34
Yes I have, while I did see that there was a section about Eisenstein series of weight 2 for $\Gamma_{1}(N)$, there doesn't seem to be any discussion about the specific Eisenstein series associated to a cusp of $\Gamma_{1}(N)$. –  user37221 Aug 5 '12 at 18:37
Fair enough. But they do compute the Fourier series of these Eisenstein series, and so I thought this could be useful –  M Turgeon Aug 5 '12 at 19:12

The question has some implicit hypotheses, possibly not clear to the questioner, and this implicitness and ambiguities about it complicate matters. First, the more natural descriptions of Eisenstein series for GL(2) of weights k>2 are not of the form in the question, but are $\sum_{c,d} 1/(cz+d)^k$. This does not converge for $k=2$, so $k=2$ has to be approached more delicately (via an analytic continuation, Hecke summation, producing _something_like_ the expression in the question). However, at level one, that is, for $SL_2(\mathbb Z)$, there is no truly-holomorphic Eisenstein series of weight $2$. The analytic continuation has an extra term, which one may discard, but then destroying the literal automorphy condition. Maybe that doesn't matter, but one should be careful about "understandings".

Thus, depending what one means, wants, or needs, while at higher levels the meromorphic continuation can produce holomorphic modular forms at level 2. (This positive outcome always occurs for Hilbert modular forms, that is, for ground fields totally real anything other than $\mathbb Q$.) First, whatever description one chooses for "Eisenstein series" (attached to cusps?), a suitable weighted average of the level-7 (for example) such should be level-one. A literal notion of holomorphy tells us there is no level-one, weight-two such. Thus, there are (at most) six linearly independent Eisenstein series at that level, so not quite possible to "attach" one to each cusp. It is not hard to say more.

Then there is the further issue of expressing various Eisenstein series in terms of each other, by the group action. At square-free level, the underlying (!) representation theory is simpler (Iwahori-fixed vectors in principal series are well understood, at least up to a very useful point.)

But, at this point, without knowing more precisely what the questioner wants, or may discover is wanted, there are too many things that can be said to know which to choose to say. :)

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Firstly, I'm not sure exactly what you mean by an Eisenstein series of wt. $2$ associated to a cusp.

Taking constant terms of $q$-expansions (or, more a canonically, taking the residues at the cusps of the associated one-form $E(\tau) d\tau$) gives an isomorphism between the weight $2$ Eisenstein series on $\Gamma_1(N)$ and the subspace of $\mathbb C^{\mathrm{cusps}}$ consisting of $z_i$ such that $\sum_i z_i = 0$. (This follows from the residue theorem.)

I imagine you have some basis for the space of residues in mind, which you are transferring via this isomorphism to a basis for the space of wt. $2$ Eisenstein series, but I'm not sure which basis it is.

In any even, most Eisenstein series on $\Gamma_1(N)$ won't be expressible in terms of $G(\tau)$; it only gives rise to Eisenstein series on $\Gamma_0(N)$.

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