Nicholas M. Katz
Princeton University
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Primary Section: 11, Mathematics Membership Type:
Member
(elected 2004)
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Research Interests
Given a polynomial equation with integer ("whole number") coefficients in several variables, e.g., say f(X, Y) := Y^2 + X^3 - X + 1 = 0, we want to understand its "solutions", in all possible senses of that word. The fundamental problem in number theory is to understand solutions in integers, but quite often we cannot attack this problem directly. Instead, for each prime number p, we study solutions "mod p", meaning that we look for integers n and m such that the integer f(n, m) is divisible by p (instead of being 0, which is what we require for an integer solution). This study can sometimes show that an equation has no integer solutions, since any integer solution is a mod p solution for every p. For instance, the equation above has no mod 3 solutions, and hence it has no integer solutions. Given an equation f, we count the number of its mod p solutions when n and m are both between 0 and p-1: call this number N(p,f). The basic theme of my research is to understand how N(p,f) varies when we vary f, but fix p, or when we fix f but vary p. For instance, if we take a second equation g, we can look at N(p,f) - N(p,g), and ask several questions about it: 1) Archimedean question: how big is this difference? 2) p-adic question: what is the biggest power of p that divides this difference? 3) l-adic question: for each prime number l other than p, what is the biggest power of l that divides this difference? Or we can vary g, and ask about the statistics of the answers we get to each of these questions about N(p,f) - N(p,g). Answering such questions involves apparently unrelated areas of mathematics, such as random matrix theory, representation theory, and theory of linear differential equations and their monodromy groups.
