General questions about Eisenstein series and modular forms

Question

I am in a situation where I am trying to get a feel for modular forms type stuff, but don't have anyone to talk to about it (I'm not in academia at the moment). I would like to test my understanding of Eisenstein series and modular forms by asking a few related questions. I'm not asking for in depth analytical answers, but just getting a feel for the bit picture.

Let $M_k$ be the ambient space of modular forms of weight $k$.

1. It is a fact that $M_k$ is the direct sum of the subspace of cusp forms, $S_k$, and the subspace of Eisenstein series, $E_k$. My question is, is the dimension of $E_k$ equal to the number of cusps (i.e. equivalence classes of $\mathbb{Q}\cup \{\infty\}$).

EDIT: This has been answered.

1. I say the previous, because my understanding is that each "basis" Eisenstein series is associated uniquely to some cusp. For example the classic guy $$\mathcal{E}(z):=\sum_{(c,d)\neq(0,0)} (cz+d)^{-k}$$is associated, I understand, to the cusp containing $\infty$, since $$\mathcal{E}(\infty)=\sum_{d\neq 0} (d)^{-k}\neq 0.$$ Is there an intuitive idea of what it means that this Eisenstein series is "associated" to the cusp at $\infty$? Is it just the non-vanishing? That doesn't seem quite right, since otherwise a linear combination of Eisenstein series would produce a cusp form, which is impossible.

2. Is there a similar "off the cuff" fact about the dimension of $S_k$? Or is this more subtle?

3. Is it possible for an Eistenstein series to be zero at one of the cusps (obviously it can't be zero at all)?

Answer 1

The general idea is that, given a subgroup $\Gamma$ of finite index inside $SL_2(\mathbb Z)$ (though also we'd often require that this subgroup be definable by "congruence conditions" on the entries), the quotient $X=\Gamma\backslash{\mathfrak H}$ of the upper half-plane $\mathfrak H$ by $\Gamma$ needs finitely-many points added to it to "compactify" it. These are the "cusps". With fixed $\Gamma$ and fixed "weight" $k$, the cuspforms are the holomorphic weight-$k$ modular forms "vanishing at all cusps". (Since holomorphic modular forms are not actually invariant by $\Gamma$, this notion of vanishing includes some technicalities...) For even weight $2k>2$, the dimension of the space of weight-$2k$ holomorphic modular forms modulo cuspforms is equal to the number of cusps. (For odd weight $2k+1$, depending on $\Gamma$, some cusps can be "irregular", or some other modifier, in the sense of admitting no non-vanishing holomorphic modular form...) Thus, at least for even weight, relative to fixed $\Gamma$, there is an Eisenstein series attached to each cusp, which takes non-zero value there, and value $0$ at all other cusps.

How to exhibit/construct these? The action of $\Gamma$ extends to the compactification, and the isotropy (=stabilizer) subgroup $\Gamma_\sigma$ of a given cusp $\sigma$ makes sense. The corresponding Eisenstein series is a sum over $\Gamma_\sigma\backslash \Gamma$... For $\Gamma=SL_2(\mathbb Z)$ there is a single (equivalence class of) cusp, $i\infty$, and the expression written in the question is one formulaic version of the corresponding formation of Eisenstein series as sum over a coset space of this type.

The dimensions of spaces of holomorphic cuspforms are computable via Riemann-Roch, in effect. This is subtler than computing the dimensions of spaces of holomorphic Eisenstein series.