Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On the irreducibility of Hecke polynomials


Author: Scott Ahlgren
Journal: Math. Comp. 77 (2008), 1725-1731
MSC (2000): Primary 11F11
DOI: https://doi.org/10.1090/S0025-5718-08-02078-4
Published electronically: February 1, 2008
MathSciNet review: 2398790
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • 1. S. Baba and M. R. Murty, Irreducibility of Hecke polynomials, Math. Res. Lett. 10 (2003), no. 5-6, 709-715. MR 2024727 (2005g:11064)
  • 2. K. Buzzard, On the eigenvalues of the Hecke operator $ T\sb 2$, J. Number Theory 57 (1996), no. 1, 130-132. MR 1378578 (96m:11033)
  • 3. J. B. Conrey, D. W. Farmer and P. J. Wallace, Factoring Hecke polynomials modulo a prime, Pacific J. Math. 196 (2000), no. 1, 123-130. MR 1797238 (2001k:11072)
  • 4. D. W. Farmer and K. James, The irreducibility of some level 1 Hecke polynomials, Math. Comp. 71 (2002), no. 239, 1263-1270. MR 1898755 (2003e:11046)
  • 5. H. Hida and Y. Maeda, Non-abelian base change for totally real fields, Pacific J. Math. 1997, Special Issue, 189-217. MR 1610859 (99f:11068)
  • 6. K. James and K. Ono, A note on the irreducibility of Hecke polynomials, J. Number Theory 73 (1998), no. 2, 527-532. MR 1658012 (2000a:11063)
  • 7. S. Lang, Introduction to modular forms, Corrected reprint of the 1976 original, Springer, Berlin, 1995. MR 1363488 (96g:11037)
  • 8. H. P. F. Swinnerton-Dyer, On $ l$-adic representations and congruences for coefficients of modular forms, in Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), 1-55. Lecture Notes in Math., 350, Springer, Berlin. MR 0406931 (53:10717a)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11F11

Retrieve articles in all journals with MSC (2000): 11F11


Additional Information

Scott Ahlgren
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: ahlgren@math.uiuc.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02078-4
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.

American Mathematical Society