REU 1999: The Igusa Local Zeta Function and Elliptic Curves (M. Robinson)

From left, back: Mark Peterson, Lubomira Ivanova, Kim Schneider, Michael Joyce, Jason Slemons, Emily Clark, Aaron Koll. From left, front: Anushka Krishnachander, Jenny Ross, Mariana Campbell, Yanir Rubinstein, Ed Dubois, Margaret Robinson.

• Mariana Campbell (UC San Diego '00), University of Pennsylvania
• Ed DuBois (Pomona '00),
• Michael Joyce (Tulane '00), Brown University
• Anushka Krishnachander (Amherst '01)
• Kim Schneider (Bowdoin '00), Penn State University
• Jason Slemons (UC Berkeley '00), University of Washington

The group produced the following preliminary report which computes the Igusa local zeta function for six of the eight reduction types given by the Kodaira-Neron classification of an elliptic curve :

The Igusa local zeta function for the different reduction types of the special fiber of an elliptic curve

The following paper by Diane Meuser and Margaret Robinson published in Mathematics of Computation cites the work of the 1999 REU group and completes the calculation of the Igusa local zeta function for the two infinite families of elliptic curve in the Kodaira-Neron Classification:

The Igusa local zeta functions of an elliptic curve, Math. Comp., 71 (2001), no. 238, 815-823.

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-Neron classification of the special fibers of elliptic curves as determined by Tate's algorithm.