# REU 2002: Number Theory

## Student participants

(The picture above is of both REU groups.)

From left, Donovan McFeron, Sarah Zubairy, John Gonzalez, David Clark, Alexandra Zuser, Thomas Wright, Manjari Goenka, Benjamin Marko, Annalee Wiswell, Craig Phillips, Virginia Peterson, and Ramla Gabriel

The student participants in the number theory group were:
• Ramla Gabriel (Mount Holyoke Collge, '03), funded by NSF.
• Manjari Goenka (Mount Holyoke College, '04), funded by MHC.
• John Gonzalez (MIT,'04), funded by NSF.
• Benjamin Marko (University of Akron, '03), funded by NSF.
• Annalee Wiswell(Scripps College, '03), funded by NSF.
• Thomas Wright (Bowdoin Collge, '03), funded by NSF.
• Sarah Zubairy (University of Rochester, '04), funded by NSF.

## Brief Description

This summer the seven students listed above worked on four different projects.

Benjamin Marko and Annalee Wiswell wrote the preprint "Igusa local zeta function of the cubic polynomial $x_1^3+x_2^3+...+x_n^3$." They computed the Igusa local zeta function for this cubic when $p \equiv 2 \bmod 3$ and in a special case when $p \equiv 1 \bmod 3$. Marko and Wiswell used a method first introduced in 1965 by Andre Weil. They computed the generalized exponential sum $F^*(i^*)$ associated with their polynomial directly using $p$-adic analysis. From this formula, they could compute the local singular series $F(i)$ (or the inverse Fourier transform of $F^*(i^*)$). Finally they took the Mellin transform of $F(i)$ to get the Igusa local zeta function $Z_\chi(s)$. Both these students will continue work on this project as honors theses during their senior year in college.

Manjari Goenka, John Gonzalez, and Sarah Zubairy wrote the preprint "The natural boundary of the Euler product of the local zeta function associated with the polynomial $f(x,y)=y^2-x^3$." This group studied a method of Marcus du Sautoy's for finding the natural boundary (a boundary in the complex plane beyond which a complex function cannot be analytically continued) of an Euler product of local zeta functions. Following du Sautoy, the group studied the well-known Igusa local zeta function for $f(x,y)=y^2-x^3$ for all $p$ and formed its Euler product $\prod_p Z_p(t)$. They showed that this infinite product converges for Re$(s)>-1/2$ and used du Sautoy's method to analytically continue the product to Re$(s)>-4/5$. However, they were unable to show that Re$(s)=-4/5$ was the natural boundary for the Euler product and can only conjecture that it is. These students are all beginning their junior year in college.

Ramla Gabriel wrote a program in MAPLE to compute the Igusa local zeta function for a diagonal polynomial in a reasonable number of variables using the $p$-adic stationary phase formula (SPF) recursively. She used her program to make conjectures about the number of iterations of SPF necessary to compute the local zeta function for diagonal polynomials. Ramla continued her work during her senior year in college.

Thomas Wright wrote the preprint \lq\lq A stationary phase formula for generalized exponential sums." In this paper, Wright gives a formula for computing generalized exponential sums that is analogous to Igusa's stationary phase formula for local zeta functions. Wright uses his formula to compute the exponential sum in two cases.

The students spoke that summer at the MAA section meeting on June 21-22 2002 at Williams College. Wiswell, Gonzalez, and Wright each presented their summer work at Mathfest in Burlington that August. Wiswell, Zubairy, and Wright spoke or presented posters at the joint meetings in Baltimore, January 2003. Annalee Wiswell received an honorable mention for her summer work from the Alice Shafer Prize Committee of the AWM.

## Reports

The group produced the following papers:

Benjamin Marko and Annalee Wiswell, Igusa Local Zeta Function of the Cubic Polynomial $% f(x)=x_{1}^{3}+\cdot \cdot \cdot +x_{n}^{3}$ (pdf)

Abstract: The purpose of this paper is to compute the Igusa Local Zeta Function of the cubic polynomial $f(x)=x_{1}^{3}+\cdot \cdot \cdot +x_{n}^{3}$. We solve this problem in general for primes congruent to 2 modulo 3 and in a special case for primes congruent to 1 modulo 3. This work was completed as part of the Mount Holyoke Summer Mathematics Institute, an NSF funded REU Program.

Manjari Goenka, John Gonzalez, Sarah Zubairy, The Natural Boundary of the Euler Product of Local Zeta Function associated with the polynomial $f(x,y)=x^2-y^3$

Abstract: We find the Igusa local zeta function for the polynomial $f(x,y)=x^2-y^3$ using the stationary phase formula. Next we investigated but were unsuccessful in finding the natural boundary for the Euler product of this zeta function. This work was completed under the supervision of Professor Margaret Robinson as part of the Mount Holyoke Summer Mathematics Institute, an NSF funded REU Program.

Thomas Wright, A stationary phase formula for generalized exponential sums (pdf)

Abstract: This paper gives a formula for Generalized Exponential sums which depends entirely upon the singularities mod $p$, similar to the purpose of the Stationary Phase Formula for Igusa Local Zeta Functions. This new formula is then applied to two examples: strongly non-degenerate homogeneous polynomials and general quadratic polynomials. This work was completed as part of the Mount Holyoke Summer Mathematics Institute, an NSF funded REU Program.