The Sato–Tate Conjecture for the Ramanujan τ -Function
Identifieur interne : 000038 ( Main/Exploration ); précédent : 000037; suivant : 000039The Sato–Tate Conjecture for the Ramanujan τ -Function
Auteurs : M. Ram Murty [Canada] ; V. Kumar Murty [Canada]Source :
Abstract
Abstract: Ramanujan’s 1916 conjecture that |τ(p)|≤2p 11/2 was proved in 1974 by P. Deligne, as a consequence of his work on the Weil conjectures. Serre, and later Langlands, discussed the possible distribution of the τ(p)/2p 11/2 in the interval [−1,1] as p varies over the prime numbers. Inspired by the Sato–Tate conjecture in the theory of elliptic curves, Serre predicted an identical distribution law (the “semi-circular” law). This conjecture was proved recently by Barnet-Lamb, Geraghty, Harris, and Taylor. In this chapter, we give a sketch of how their proof works. We also indicate some lines of future development.
Url:
DOI: 10.1007/978-81-322-0770-2_12
Affiliations:
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<front><div type="abstract" xml:lang="en">Abstract: Ramanujan’s 1916 conjecture that |τ(p)|≤2p 11/2 was proved in 1974 by P. Deligne, as a consequence of his work on the Weil conjectures. Serre, and later Langlands, discussed the possible distribution of the τ(p)/2p 11/2 in the interval [−1,1] as p varies over the prime numbers. Inspired by the Sato–Tate conjecture in the theory of elliptic curves, Serre predicted an identical distribution law (the “semi-circular” law). This conjecture was proved recently by Barnet-Lamb, Geraghty, Harris, and Taylor. In this chapter, we give a sketch of how their proof works. We also indicate some lines of future development.</div>
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