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I've tried several times to read up on modular curves, and I've despaired every time. It seems that there are several competing narratives, that all get enfused and conspire to befuddle me. There are modular forms, automorphic forms, Hecke theory, L-functions, Galois representations, hyperbolic space, primitive forms, Fuchsian groups, the Langlands program and so forth.

It is somewhat of an introductory question, but I would be very grateful if one of you could put things in some sort of order, and separate the narratives. (What generalizes what, what is parrallel to what, and what implies what)

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The baby case is the following: modular curves are (the completion of) quotients of the upper half plane (an hyperbolic space) by congruence subgroups (special Fuchsian groups). Its differential forms correspond to weight 2 modular forms. Inside the space of modular forms, there are Hecke operators acting, and the eigenvalues for them play a crucial role in the theory. They are in correspondence with Galois representations and geometric objects (elliptic curves or abelian varieties of GL2-type for weight 2). This generalizes in all directions and so do the correspondence (conjecturally). –  A. Pacetti Dec 1 '10 at 21:38
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Just curious: have you looked at Diamond / Shurman "A First Course in Modular Forms"? –  B R Dec 1 '10 at 23:39
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Minor correction to above comment: Global differentials correspond to weight 2 cusp forms, and differentials with at most log singularities at cusps correspond to weight 2 modular forms. The shift from vanishing of a function to regularity of a differential is basically due to an exponential coordinate change. –  S. Carnahan Dec 2 '10 at 8:16

3 Answers 3

I recommend reading

  • Modular points, modular curves, modular surfaces and modular forms

by Don Zagier.

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I'll be straying outside of my expertise here, so the following probably contains some errors. I'm just adding my perspective as someone who has (at one time at least) thought about some of this stuff. Hopefully this will help foster some discussion.

1) Modular curves are locally symmetric spaces, meaning they are of the form $\Gamma\backslash G/K$ where $G$ is (for simplicity) a semi-simple Lie group (e.g. $SL_2({\bf R}$), $K$ a maximal compact subgroup (e.g. $SO(2)$), and $\Gamma$ a discrete subgroup of finite index in $G({\bf Z})$ (e.g. $SL_2({\bf Z})$) (an "arithmetic" subgroup). Margulis' work on rigidity of lattices in Lie groups implies that unless $G$ is $SO(1,n)$ or $SU(1,n)$, $\Gamma$ is a congruence subgroup of $G({\bf Z})$ (a congruence subgroup is an arithmetic subgroup containing the kernel of the reduction map $G({\bf Z})\rightarrow G({\bf Z}/N{\bf Z})$). Note that $SO(1,2)\simeq SL_2({\bf R})$, $SO(1,3)\simeq SL_2({\bf C})$.

In my experience, people typically only talk about hyperbolic (or symmetric) spaces and fuchsian groups in passing because there is more information available in explicitly using $G$ and $\Gamma$ (So they may start out discussing e.g. a quotient of hyperbolic n-space by a discrete subgroup, but they prove things using that $H^n$ is really $SO(n,1)/SO(n)$ and $\Gamma$ is really a subgroup of $SO_{n,1}({\bf Z})$).

When $\Gamma$ is a congruence subgroup of $G({\bf Z})$, you can think of $\Gamma \backslash G/K$ as $G({\bf Q})\backslash G({\bf A})/K\cdot K_f$, where $K_f$ is a compact open subgroup of $G({\bf A_{\rm f}})$ This tends to be simpler to work with, since $G({\bf Q})$ has a simpler structure than $\Gamma$ (algebraic groups over fields instead of rings). Note that strange things can happen with noncongruence subgroups (e.g. there might not be any cusp forms).

Automorphic forms are certain functions on $\Gamma\backslash G$. Modular forms are classically defined on $G/K$, but you can do a little transform-and-lift to get them as certain "holomorphic" automorphic forms. With automorphic forms, the power of representation theory enters, and you have automorphic $L$-functions (and Hecke operators).

2) Modular curves are Shimura varieties, meaning (kind of) that $G/K$ has a complex structure, and so (after some work) $\Gamma \backslash G/K$ is an algebraic variety. More work shows that the Shimura variety is defined over a number field (the canonical model of the variety over the reflex field).

Thus you can attach to it a Hasse-Weil zeta function, which naturally factors as an alternating product of $L$-functions attached to the cohomology groups of the Shimura variety. These are supposed to be automorphic. In the modular case, the Eichler-Shimura relation makes this connection pretty simple to prove, but it is not known in general, and is really hard. (I'm not sure what the state of the art is. I'm pretty sure Hilbert modular varieties ($GL_2$ over a totally real field), Picard modular surfaces ($SU(2,1)$), and the next-simplest Siegel modular variety ($GSp_4$, thinking of $GL_2$ as $GSp_2$) are known. I think more general unitary groups are known, but I can't pin down exact statements).

The cohomology groups carry an action by $G({\bf A}_f)$, which gives rise to an action via Hecke operators (thinking of Hecke operators as members of the group algebra for $G({\bf A}_f)$). There are simpler ways to see this. The etale cohomology groups also carry an action by the absolute Galois group of the reflex field, which gives rise to $\ell$-adic Galois representations.

Modular/automorphic forms are sections of ("automorphic") vector bundles on $\Gamma \backslash G/K$. The algebraic structure on the Shimura variety has consequences for automorphic forms e.g. in terms of rationality of Fourier coefficients and special values of $L$-functions.

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just a small comment: every shimura variety has a canonical model, not only the abelian ones. see for example this overview by Milne jmilne.org/math/articles/2003a.pdf –  user5831 Dec 2 '10 at 13:15
    
Thanks for confirming that. I really should have doubled checked Milne's notes (given that that's where I learned everything I know about them)! (I ended up deleting the offending remark, as I was cleaning up the answer) –  B R Dec 2 '10 at 17:11
    
Just a pedantic remark: your description of $\Gamma \backslash G / K$ as $G(\mathbf{Q}) \backslash G(\mathbf{A}) / K K_f$ only works if $\Gamma$ is a congruence subgroup. –  David Loeffler Dec 2 '10 at 20:09
    
Thanks. I meant to write "finite index subgroup of $G({\bf Z})$" (an arithmetic subgroup). And this is still only correct up to the exceptions in Margulis' theorem (so even if I wrote what I meant, it still would have needed correction). I'm going to clarify this in the post. I primarily work adelically, so I have a blind spot to these issues (among others!). –  B R Dec 2 '10 at 20:58

"Moduli" are parameters that algebraic varieties depend on - continuous invariants if you like, as opposed to discrete invariants. Or the same for complex structures on a given topological manifold, if you like to think that way. A modular curve is the simplest actual case of the phenomenon, the case of a single variable. This was first noticed when the varieties were elliptic curves. There was an elaborate working-out of the theory throughout the nineteenth century: the usual language was of modular equations (in two variables), which after Riemann were probably read by geometers typically as defining Riemann surfaces. There is a lot of organisation for all the possible modular curves coming from classical elliptic function theory.

And in the twentieth century a number of successive points of view were found explaining all this theory again. A sensible place to start is probably Hecke's theory; partly that's a matter of taste, some people might prefer the work of Hurwitz which might wrongly be considered subsumed by now. You don't need the generality of all Fuchsian groups, or all automorphic forms, or all Galois representations, or all L-functions. Modular curve theory can indicate how all of these fit in, as classical examples.

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