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Hilbert modular forms and the Ramanujan conjecture

  • Don Blasius
Part of the Aspects of Mathematics book series (ASMA)

Abstract

This paper completes the proof, at all finite places, of the Ramanujan Conjecture for motivic holomorphic Hilbert modular forms which belong to the discrete series at the infinite places. In addition, the Weight-Monodromy Conjecture of Deligne is proven for the Shimura varieties attached to GL(2) and its inner forms, and the conjecture of Langlands, often today called the local-global compatibility, is established at all places for these varieties. This latter conjecture gives, for a finite place v of the field of definition F, an automorphic description of the action of decomposition group Γv of the Galois group Gal \( (\bar F/F) \) on the l-adic cohomology of the the variety, at least if l is distinct from the residue characteristic of v. In particular, the Hasse-Weil zeta functions of these varieties are computed at all places.

Keywords

Zeta Function Galois Representation Discrete Series Cuspidal Representation Open Compact Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

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  • Don Blasius

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