# What is Deligne's article “Forms modulaires and representations de GL(2)” about?

Does anyone have a review of the article?

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See MR0347738 (50 #240) if you have access to Mathscinet or to a library. –  Chandan Singh Dalawat Mar 7 '11 at 6:15
Might it be about Modular forms and representations of GL(2)? –  Captain Oates Mar 7 '11 at 6:30
While in view of the answer this is not relevant anymore for this precise request, it might be useful in the future: Zentralblatt Math. zentralblatt-math.org , on the site 'Database ZMATH', (which is very similar to Math. Reviews/MathSciNet) gives limited access without subscription. The limitation is that for every query only three answers are displayed; yet if the query is precise enough, or you make it more preicse in the process, this is sufficient. (Note it is 'formes' not 'forms'.) Also, if you need very old articles (say 1920s), 'Database JFM' is useful. –  quid Mar 7 '11 at 10:16

This is a fairly detailed summary, with references, of the approach to modular or automorphic forms through the theory of group representations, in the case of $\text{GL}_2(Q)$. It has three parts. In the first two sections the author carefully explains the connection between representation theory and the classical point of view. Section 1 sets things up. Admissible representations are defined, coordinates are introduced, automorphic forms are defined at length, with attention to the relation between the classical and adelic formulations. Cusp forms and the Petersson inner product are introduced. Section 2 interprets general results on automorphic forms in the representation-theoretic context. The author explains how holomorphic automorphic forms may be viewed as certain vectors in certain $\text{GL}_2({\bf R})$-modules. Whittaker and Kirillov models are discussed, and are used to interpret the "new forms'' of Atkin and Lehner as certain vectors in certain $\text{GL}_2(A_f)$-modules ($A_f$ denotes the finite adèles). In this, Hecke operators are interpreted as $\text{GL}_2(A_f)$-convolution operators. On this basis, it is shown how to make a basis for the (holomorphic) automorphic forms out of new forms, and it is shown that an eigen-form for the Hecke operators is determined by "almost all'' its eigenvalues.
The third paragraph treats the results in local class field theory that have come out of the representation-theoretic approach. The 2-dimensional "semisimple'' representations of the Weil group of a local field, and then (in odd residual characteristic) the "dictionaries'' matching these with representations of GL$_{\text 2}$ or quaternion algebras, along with their essential properties, are described.