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What is a good reference to learn about Hilbert modular forms both in the classical and adelic settings? For example, why usually one considers only parallel weight forms? Thanks.


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2 Answers 2

It's not at all the case that one only considers parallel weight forms. For various reasons they can be technically simpler, but to get a more precise answer to this aspect of your question, it would be best if you were to give an example of a particular piece of literature that considers only parallel weigh forms, and then ask why in that particular context the author restricts to parallel weight forms.

As for the question of classical and adelice settings, and the relationship between them: a good general reference for automorphic forms/representations on $GL_n$ of number fields (Hilbert modular forms are the case of $GL_2$ of a totally real field) is Clozel's Ann Arbor article (available here at Jim Milne's web-page).

But before learning the adelic picture for Hilbert modular forms, you may want to be sure that you understand it well in the case of classical modular forms. Once you do, the Hilbert modular case is a fairly straightforward adaptation.

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I know of two pretty good books on Hilbert modular forms: Garrett's "Holomorphic Hilbert Modular Forms" and Freitag's Hilbert Modular forms. The latter only covers classical Hilbert modular forms whereas the former introduces both classical and adelic Hilbert modular forms.

Shimura's article "The special values of the zeta function associated with Hilbert modular forms" (Duke Math. Journal) is something of a classic and carefully builds up the theory of Hilbert modular forms (beginning with the classical theory and building up to the adelic theory). The reason that one constructs adelic Hilbert modular forms is to gain invariance under the full Hecke algebra (which is not automatic in the case of classical Hilbert modular forms over a totally real field of strict ideal class number greater than $1$). The Hecke operators play an extremely important role in Shimura's paper, so he spends quite a bit of time developing them. He also proves a number of other basic properties of Hilbert modular forms.

As Emerton suggested in his response, it is important to have a good grip on the classical case before moving on to the adelic case. This is especially important in Shimura's article, as the space of adelic Hilbert modular forms has two well-known and important direct sum decompositions. One is a direct sum over representatives of the strict ideal classes and has as its summands spaces of classical Hilbert modular forms. The other is a direct sum over Hecke characters extending a character $\psi: (\mathcal{O}_K/\mathfrak{N})^\times\rightarrow \mathbb C^\times$. Moving between all of these spaces is a bit subtle.

If you are looking for the quickest way to learn the basics of classical / adelic Hilbert modular forms I would probably suggest starting by reading the preliminary sections of a few papers. I say this because Hilbert modular forms can get VERY technical very quickly, to the point that one has trouble seeing the forrest for the trees. There have been a number of papers written within the last year or two on the computation of systems of Hecke eigenvalues of spaces of Hilbert modular forms. Most of these papers contain very good expositions on the basics of classical / adelic / quaternionic Hilbert modular forms. John Voight has written a few of these papers and has them available on his website. Another person active in this area is Lassina Dembele.

I do not quite agree with your assertion that people only (or even usually) consider the case of parallel weight. In fact, one often has good reason to consider Hilbert modular forms of non-parallel weight (as Shimura notes in the introduction of the aforementioned article). Much of the theory of classical (elliptic) modular forms generalizes rather straight-forwardly to the Hilbert modular case without restrictions on the weight vector. For example, in their paper "Twists of Newforms", Shemanske and Walling study the newform theory of arbitrary weight Hilbert modular cusp forms and show that for the most part, it is exactly as one would hope.

That said, there is one very good reason for restricting to the case of parallel weight: this is the only case in which Eisenstein series exist. Said differently, if the weight is not parallel then all Hilbert modular forms are cuspidal.

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Ugh. Apparently I read Emerton's response a bit too quickly. He was not suggesting that the OP learn the theory of classical Hilbert modular forms before learning adelic Hilbert modular forms but rather urging the OP to learn classical modular forms before learning Hilbert modular forms. Sorry for the misrepresentation. –  Ben Linowitz Feb 13 '11 at 4:29
Could you explain the last point a bit precisely, or give some reference about it? Why only cuspidal for not parallel weights? Thanks –  user4245 Feb 13 '11 at 6:37
Shimura shows this on page 640 ( just above Prop. 1.1) of the article I mentioned in my answer. –  Ben Linowitz Feb 13 '11 at 7:03

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