# REU 2007: Number Theory

(The picture above is of both 2007 REU groups and is thanks to Harriet Pollatsek. )
From left, Vicki Foss, Eleanor Birrell, Lisa Dudley, Mike Fink, Hillary Sackett, Ryan Eberhart, Margaret Robinson, Sohini Mahapatra, Sam Ruth, Zeb Engberg (up above Sam), Neal Lima, Jillian McLeod

## Our work this summer resulted in a joint paper available below and several individual papers and presentations also below.

This summer the number theory REU group studied a double Poincare series associated to a polynomial
over the p-adic numbers. This function of two variables is
a natural way to combine the Weil zeta function and Igusa zeta
function. This summer the group answered questions regarding the rationality of this
double Poincare series, specifically for the case of elliptic curves.
This paper extends the work of Diane Meuser in 1986 on a
related function and relies on the findings of the 1999 Mount
Holyoke College REU in number theory.

Poincare Series of Weil-Igusa Type For Elliptic Curves (paper)

## The student participants in the number theory group were:

- Zebediah Engberg (Hampshire College, '09), funded by NSF.

Rationality of the Weil-Igusa Type Poincare Series (paper)

- Neal Lima (University of Connecticut, '08), funded by NSF.

Igusa Zeta functions in arbitrary p-adic Fields (talk)

- Sohini Mahapatra (University of Michigan,'09), funded by NSF.

Computing the Cardinalities for Reduction Type II (paper)

- Sam Ruth (Northwestern, '08), funded by NSF.

The Case I0* (paper)

- Hillary Sackett (Smith College, '08), funded by NSF.

On Zeta Functions of Weil and Igusa Type (talk)

This work was produced under NSF grant number DMS-0353700.

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