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The local Jacquet-Langlands correspondence via Fourier analysis

Jared Weinstein (2010)

Journal de Théorie des Nombres de Bordeaux

Let F be a locally compact non-Archimedean field, and let B / F be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations π of B × of dimension > 1 and irreducible cuspidal representations of GL 2 ( F ) . We present a new construction of this bijection in which the preservation of epsilon factors is automatic. This is done by constructing a family of pairs ( , ρ ) , where M 2 ( F ) × B is an order and ρ is a finite-dimensional representation of a certain...

The module of vector-valued modular forms is Cohen-Macaulay

Richard Gottesman (2020)

Czechoslovak Mathematical Journal

Let H denote a finite index subgroup of the modular group Γ and let ρ denote a finite-dimensional complex representation of H . Let M ( ρ ) denote the collection of holomorphic vector-valued modular forms for ρ and let M ( H ) denote the collection of modular forms on H . Then M ( ρ ) is a -graded M ( H ) -module. It has been proven that M ( ρ ) may not be projective as a M ( H ) -module. We prove that M ( ρ ) is Cohen-Macaulay as a M ( H ) -module. We also explain how to apply this result to prove that if M ( H ) is a polynomial ring, then M ( ρ ) is a free...

The new properties of the theta functions

Stefan Czekalski (2013)

Annales mathématiques Blaise Pascal

It is shown, that the function H ( x ) = k = - e - k 2 x satisfies the relation H ( x ) = n = 0 ( 2 π ) 2 n ( 2 n ) ! H ( n ) ( x ) .

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