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The Igusa local zeta functions of elliptic curves

Author(s): Diane Meuser; Margaret Robinson.
Journal: Math. Comp. 71 (2002), 815-823.
MSC (2000): Primary 11S40, 11G07
Posted: September 17, 2001
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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:

[1]: J.-I. Igusa, A stationary phase formula for -adic integrals and its applications, Algebraic geometry and its applications, Springer-Verlag, (1994), 175-194. MR 95a:11104
[2]: D. Meuser, On the poles of a local zeta function for curves, Invent. Math. 73 (1983), 445-465. MR 85i:14014
[3]: J. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151, Springer-Verlag (1994). MR 96b:11074
[4]: J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable IV, Lecture Notes in Math. 476, B.J. Birch and W. Kuyk, eds., Springer-Verlag, Berlin (1975), 33-52. MR 52:13850
[5]: W. Veys, On the poles of Igusa's local zeta function for curves, J. London Math. Soc. 41 (1990), 27-32. MR 92j:11142


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