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 GabrielThe student participants in the number theory group were:
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.
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