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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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The Igusa local zeta functions of elliptic curves
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by Diane Meuser and Margaret Robinson PDF
Math. Comp. 71 (2002), 815-823 Request permission

Abstract:

We determine the explicit form of the Igusa local zeta function associated to an elliptic curve. The denominator is known to be trivial. Here we determine the possible numerators and classify them according to the Kodaira–Néron classification of the special fibers of elliptic curves as determined by Tate’s algorithm.
References
  • Jun-ichi Igusa, A stationary phase formula for $p$-adic integrals and its applications, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 175–194. MR 1272029
  • D. Meuser, On the poles of a local zeta function for curves, Invent. Math. 73 (1983), no. 3, 445–465. MR 718941, DOI 10.1007/BF01388439
  • Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
  • J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039
  • W. Veys, On the poles of Igusa’s local zeta function for curves, J. London Math. Soc. (2) 41 (1990), no. 1, 27–32. MR 1063539, DOI 10.1112/jlms/s2-41.1.27
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Additional Information
  • Diane Meuser
  • Affiliation: Boston University, Boston, Massachusetts 02215
  • Email: dmm@math.bu.edu
  • Margaret Robinson
  • Affiliation: Mount Holyoke College, South Hadley, Massachusetts 01075
  • Email: robinson@mtholyoke.edu
  • Received by editor(s): July 10, 2000
  • Published electronically: September 17, 2001
  • Additional Notes: This work was supported by National Science Foundation Grant No. DMS-9732228
  • © Copyright 2001 American Mathematical Society
  • Journal: Math. Comp. 71 (2002), 815-823
  • MSC (2000): Primary 11S40, 11G07
  • DOI: https://doi.org/10.1090/S0025-5718-01-01396-5
  • MathSciNet review: 1885630