Pseudodifferential Operators with Automorphic Symbols

1. Front Matter

   Pages i-xvi

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2. Basic modular distributions

   • André Unterberger

   Pages 7-26

3. From the plane to the half-plane

   • André Unterberger

   Pages 27-49

4. A short introduction to the Weyl calculus

   • André Unterberger

   Pages 51-81

5. Composition of joint eigenfunctions of \(\varepsilon\; \mathrm{and}\ \xi\frac{\partial}{\partial x}\)

   • André Unterberger

   Pages 83-122

6. The sharp composition of modular distributions

   • André Unterberger

   Pages 123-167

7. The operator with symbol \(\mathfrak{E}_{\upsilon}\)

   • André Unterberger

   Pages 169-182

8. From non-holomorphic to holomorphic modular forms

   • André Unterberger

   Pages 183-195

9. Back Matter

   Pages 197-202

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The main results of this book combine pseudo differential analysis with modular form theory. The methods rely for the most part on explicit spectral theory and the extended use of special functions. The starting point is a notion of modular distribution in the plane, which will be new to most readers and relates under the Radon transformation to the classical one of modular form of the non-holomorphic type. Modular forms of the holomorphic type are addressed too in a more concise way, within a general scheme dealing with quantization theory and elementary, but novel, representation-theoretic concepts.

• Weil calculus
• automorphic distributions
• holomorphic modular forms
• microlocal analysis
• modular distributions

“The book under review is devoted to developing a satisfactory Weyl calculus on the class of tempered automorphic distributions under the group SL(2, Z). … The book is … readable and is an interesting introduction to the subject. The main ideas are explained in detail and illustrated by exact computations.” (Đỗ Ngọc Diệp, Mathematical Reviews, December, 2015)

• Université de Reims Laboratoire de Mathématiques, Reims, France

  André Unterberger

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