The Linnik conjecture. A local approach.
Let be a locally compact non-Archimedean field, and let be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations of of dimension and irreducible cuspidal representations of . 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 , where is an order and is a finite-dimensional representation of a certain...
Let denote a finite index subgroup of the modular group and let denote a finite-dimensional complex representation of Let denote the collection of holomorphic vector-valued modular forms for and let denote the collection of modular forms on . Then is a -graded -module. It has been proven that may not be projective as a -module. We prove that is Cohen-Macaulay as a -module. We also explain how to apply this result to prove that if is a polynomial ring, then is a free...
It is shown, that the function