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# Icosahedral Fibres of the Symmetric Cube and Algebraicity

Ramakrishnan, Dinakar (2011) Icosahedral Fibres of the Symmetric Cube and Algebraicity. In: On certain L-functions: conference in honor of Freydoon Shahidi on certain L-functions. Clay Mathematics Proceedings (13). American Mathematical Society , Providence, RI, pp. 483-499. ISBN 0-8218-5204-3 http://resolver.caltech.edu/CaltechAUTHORS:20110902-114949812

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## Abstract

For any number field F, call a cusp form π = π_∞⊗πf on GL(2)/F special icosahedral, or just s-icosahedral for short, if π is not solvable polyhedral, and for a suitable “conjugate” cusp form π' on GL(2)/F, sym^3(π) is isomorphic to sym^3(π'), and the symmetric fifth power L-series of π equals the Rankin-Selberg L-function L(s, sym^2(π') × π) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let π be s-icosahedral (of trivial central character). Then π f is algebraic without local components of Steinberg type, π ∞ is of Galois type, and π_v is tempered every-where. Moreover, if π' is also of trivial central character, it is s-icosahedral, and the field of rationality Q(πf) (of πf) is K := Q[√5], with π' _f being the Galois conjugate of πf under the non-trivial automorphism of K. There is an analogue in the case of non-trivial central character ω, with the conclusion that π is algebraic when ω is, and when ω has finite order, Q(πf) is contained in a cyclotomic field.

Item Type:Book Section
Additional Information:© 2011 Dinakar Ramakrishnan. Partly supported by the NSF grant DMS-0701089.
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Funding AgencyGrant Number
NSFDMS-0701089
Classification Code:2010 MSC: Primary: 11F70; Secondary: 11F80, 22E55
Record Number:CaltechAUTHORS:20110902-114949812
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20110902-114949812
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Official Citation:Icosahedral Fibres of the Symmetric Cube and Algebraicity Dinakar Ramakrishnan pp.483-499
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:25212
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:02 Sep 2011 21:52