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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)?

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These questions should all be answered somewhere in Stein's Modular Forms: A Computational Approach (wstein.org/books/modform/modform). In particular there are dimension formulas at wstein.org/books/modform/modform/dimension_formulas.html . –  Qiaochu Yuan Feb 27 '13 at 3:16
    
Thanks for the reference, though it's not clear to me what $c_0(n)$ is. –  Eric Gregor Feb 27 '13 at 5:53
    
I think $c_0(N)$ is the number of cusps of $X_0(N)$. There's also a formula for it in the link. –  Qiaochu Yuan Feb 27 '13 at 6:22
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up vote 3 down vote accepted

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.

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This is a really nice answer. I'd greatly appreciate your recommendation for a good book on this stuff to go along with the previous suggestion. Maybe something to feel the big picture as well as the little details. –  Eric Gregor Mar 1 '13 at 4:25
    
@EricGregor Indeed, the subject lends itself to getting off on interesting tangents... which, indeed, affects much writing about the subject. Gunning's old orange Princeton series "Intro to modular forms" is perhaps singular in its brevity and simplicity. Bump's relatively recent "Automorphic forms..." and Iwaniec' two books with "automorphic" in the titles do give many more details. The intro to "classical" modular forms by Kowalski in the Bernstein-Gelbart volume "Intro to the Langlands program" quickly covers the basics. –  paul garrett Mar 1 '13 at 14:43
    
Adding to @paulgarrett's suggestions, I think A First Course in Modular Forms by Diamond and Shurman is an excellent reference source –  Brent J Mar 30 '13 at 21:34
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