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The irreducibility of some level 1 Hecke polynomials

Authors: D. W. Farmer and K. James
Journal: Math. Comp. 71 (2002), 1263-1270
MSC (2000): Primary 11F11
Published electronically: June 22, 2001
MathSciNet review: 1898755
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Abstract | References | Similar Articles | Additional Information


Let $T_{p,k}(x)$ be the characteristic polynomial of the Hecke operator $T_{p}$ acting on the space of level 1 cusp forms $S_{k}(1)$. We show that $T_{p,k}(x)$is irreducible and has full Galois group over  $\mathbf{Q}$for $k\le 2000$ and $p<2000$, $p$ prime.

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

  • [A] T. Apostol, Modular functions and Dirichlet series in number theory, GTM 41, Springer-Verlag, (1990). MR 90j:11001
  • [B] K. Buzzard, On the eigenvalues of the Hecke operator $T\sb 2$, J. Number Theory 57 (1996), no. 1, 130-132. MR 96m:11033
  • [Ch] P. Chiu, Transforms, finite fields, and fast multiplication, Math. Mag. 63 (1990), no. 5, 330-336. MR 93c:11113
  • [Co] H. Cohen, A course in computational algebraic number theory, Springer-Verlag. (1993). MR 94i:11105
  • [CF] J.B. Conrey and D.W. Farmer, Hecke operators and the nonvanishing of $L$-functions, Topics in number theory (University Park, PA, 1997), 143-150, Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 1999. MR 2000f:11055
  • [CFW] J.B. Conrey, D.W. Farmer, and P.J. Wallace, Factoring Hecke polynomials modulo a prime, Pacific J. Math. 196 (2000), 123-130. CMP 2001:01
  • [HM] H. Hida and Y. Maeda, Non-abelian base change for totally real fields, Pacific J. Math., special issue (1997), 189-217. MR 99f:11068
  • [JO] K. James and K. Ono, A note on the irreducibility of Hecke polynomials. J. Number Theory 73 (1998), no. 2, 527-532. MR 2000a:11063
  • [Ko] N. Koblitz, Introduction to elliptic curves and modular forms, second edition, GTM 97, Springer-Verlag, (1993). MR 94a:11078
  • [KZ] W. Kohnen and D. Zagier, Values of $L$-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175-198. MR 83b:10029
  • [M] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory. vol IT-15 (1969), no. 1, 122-127. MR 39:3887
  • [W] D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory. vol IT-32 (1986), no. 1, 54-62. MR 87g:11166

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Additional Information

D. W. Farmer
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

K. James
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975

Received by editor(s): January 6, 2000
Received by editor(s) in revised form: September 4, 2000
Published electronically: June 22, 2001
Additional Notes: The research of the first author was supported in part by the American Institute of Mathematics. We thank the referee for many helpful comments
Article copyright: © Copyright 2001 American Mathematical Society

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