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Mathematics of Computation

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On the irreducibility of Hecke polynomials

Author: Scott Ahlgren
Journal: Math. Comp. 77 (2008), 1725-1731
MSC (2000): Primary 11F11
Published electronically: February 1, 2008
MathSciNet review: 2398790
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Abstract: Let $ T_{n, k}(X)$ be the characteristic polynomial of the $ n$th Hecke operator acting on the space of cusp forms of weight $ k$ for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials $ T_{n, k}(X)$. Using this criterion with some machine computation, we show that if there exists $ n\geq 2$ such that $ T_{n, k}(X)$ is irreducible and has the full symmetric group as Galois group, then the same is true of $ T_{p, k}(X)$ for each prime $ p\leq 4,000,000$.

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Scott Ahlgren
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Received by editor(s): February 21, 2007
Received by editor(s) in revised form: May 31, 2007
Published electronically: February 1, 2008
Additional Notes: The author thanks the National Science Foundation for its support through grant DMS 01-34577. He also thanks the Department of Computing at Macquarie University for its hospitality during part of the time when this research was conducted.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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