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I recently attended a number theory seminar on Evan's paper "A fundamental region for Hecke's modular group". In the seminar the speaker mentioned that this is an automorphic form.

Since then I've been trying to find out what an automorphic form is but a search for a book on the material usually yields results such as: p-adic automorphic forms on Shimura varieties, automorphic forms on GL (2), Spectral methods of automorphic forms, etc.

My question is what is an automorphic form? Is there a book on it I can take a look at that doesn't talk about it with respect to some other object?

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Some ingredients in the general definition are close to being purely technical and I agree that it's hard to find a straightforward account. Sakellaridis give some some nice talks on this at a Fields workshop a few months ago, and he has notes up here which ought to be pretty good. – Dylan Moreland May 23 '12 at 6:54
There's also this relatively recent book by Bump called Automorphic Forms and Representations. I imagine that he spells these things out there. – Dylan Moreland May 23 '12 at 6:58
It is my belief that automorphic forms are generalisations of modular forms to actions of other discrete subgroups of Lie groups. I am no expert though. – fretty May 23 '12 at 7:34
up vote 3 down vote accepted

I happen to have very recently started digging into automorphic forms. I would recommend against Bump's book for a first look at automorphic forms. I adore the book in general, but if you're going to get a first look at that level (which is very advanced), I recommend Goldfeld's Automorphic forms and L-functions for GL(n,R). Despite the title, it has a relatively easy to read introduction on automorphic forms on $SL_2(\mathbb{Z})$, which are the simplest examples.

What I suppose this really says is that modular forms are sort of the place to start, and the analogy to keep in mind. Automorphic forms are sort of deeply tied to lie groups like $GL_n(\mathbb{R})$, so you'll likely not get too far away from them. If you have access to a library, I recommend Analytic Number Theory by Iwaniec and Kowalski (it has a hefty price tag, so it's not for casual buying perhaps). Chapters 5 and 14 give good intros to classical automorphic forms (though modular forms, again).

Now let's try to answer your question a bit: what is an automorphic form? In a sense, they are generalization of periodic functions $f: \mathbb{R} \to \mathbb{C}$ with $f(x+1) = f(x)$. But we can say a bit more.

So let's just jump in. This is an incomplete picture, and sort of lends itself towards a modular form-type explanation - let's put that out there first. But say that $G$ is an algebraic group over $\mathbb{R}$, most commonly $GL_n(\mathbb{R})$ or something like it. Let $K$ be a maximal compact subgroup of $G$, and $\Gamma$ a discrete subgroup of $G$. Then a function $f \in C^{\infty} (G)$ is a $K$-finite automorphic form for $\Gamma$ if

  1. The automorphic condition: $f(\gamma g) = f(g) \quad \forall \gamma \in \Gamma$

  2. K-finiteness: The vector space $\langle f_k(g) := f(gk)| k \in K\rangle$ is finite dimensional in $C_\infty(G)$

  3. Holomorphic at Cusps: Something 'analogous' to the statement that $f$ is holomorphic on the quotient space $G/K$, and can be interpreted as a certain differential condition guaranteeing sufficiently nice behavior at most places (I blur the details the most here)

  4. Moderate Growth: Something 'equivalent' to the statement that $f$ is of 'moderate growth' - really that $f$ is holomorphic at the cusps (if this has any meaning). Ultimately, this can be interpreted as saying that $f$ doesn't explode too quickly near specific special points

Somehow, any classical modular form corresponds to a classical automorphic forms. If you have a chance to read Tate's Thesis, you might consider it an exposition on automorphic forms on $GL_1$. Other than that, I'd recommend starting with modular forms.

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Thanks. I checked out the books you recommended and they were helpful. – Eugene May 24 '12 at 0:56

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