Hilbert modular forms and the Ramanujan conjecture

By D. Blasius. 2005

114 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d19f8bbaf2afedb46633e71ca4fbc32ea4ed2499

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In Algebra and Number Theory

1. Alexander Gorodnik
2. A. Nevo
3. Nicolas Templier

In Geometry and Topology

1. A. Nevo
2. Alexander Gorodnik
3. François Maucourant

In Mathematical Physics

1. A. Nevo
2. Alexander Gorodnik
3. Peter Sarnak
4. H. Oh
5. Nicolas Templier
6. François Maucourant

Citations in the corpus

• Quantitative ergodic theorems and their number-theoretic applications (2013, background): “Deligne [D71] on the classical Ramanujan conjecture (see [C] for a comprehensive treatment) combined with the Jacquet–Langlands correspondence [JL70].”