REU 1997: P-adic Analysis and Local Zeta Functions (M. Robinson)

The student participants were:

• Natalia Burova, Mount Holyoke College '98
• Christine Carracino, University of Virginia '98
• David Jao, MIT '98
• Jamie Kawabata, Rose-Hulman Institute of Technology '98
• Steven Spallone, University of Pennsylvannia '98

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The group produced the following reports:

Natalia Burova, Classification of cubic curves and their Igusa local zeta functions

Abstract: The purpose of this paper is to look at the classification of cubic curves of the form $f(x,y)=y^2+ax^3+bx^2+cx+d$ and give general form for the Igusa'a local zeta function $z(t)$ for such curves. The main result that we are looking for is the form of the numerators of the Iguza local zeta function, $z(s)$, for the cubics. We are hoping that we can find some pattern to them. This paper is an attempt to find a somewhat general formula for $z(s)$ for certain cases of the cubics, more precisely, polynomials of the form $$f(x,y)=y^2+p^{j}x^3+p^{k}x^2+p^{l}x+p^{m}$$. (gzipped postscript, to unzip execute "gunzip")

Christine Carracino and Steven Spallone, Methods in finding $p$-adic integrals

Abstract: This purpose of this paper is mainly to study the Resolution of Singularities method of computing Igusa Local Zeta Functions. Some simple examples are presented, a few techniques of applying the method are given, and its connection to the Stationary Phase Formula is explored. There is also a small section on computing terms of the Poincar\'{e} series. (gzipped postscript, to unzip execute "gunzip")

David Jao, $P$-adic points on algebraic curves

Abstract: In this paper we present some techniques for counting the $p$-adic points lying on an algebraic variety and calculating the generating functions associated with the number of congruence classes which lift to $p$-adic points on the variety. (gzipped postscript, to unzip execute "gunzip")

Jamie Kawabata, Elliptic Curve Classification 9

Abstract: In this paper, cubic curves are classified based on whether and which coefficients are divisible by powers of $p$. For each case, a method is given to reduce the integral to an easier case, or at least something that will lead to recursion. Taken together, these cases comprise a program to integrate any cubic. A few examples of using this program are given. The classification into cases is then refined so that closed form formulas can be found for more of the cases. (gzipped postscript, to unzip execute "gunzip")