THE WEIL ZETA FUNCTION AND THE IGUSA LOCAL ZETA FUNCTION MARGARET M. ROBINSON Abstract. These are a set of notes and problem sets written for the Un- dergraduate Lecture Series section of ”Zeta Functions all the Way,” the 2006 topic of the Mentoring Program for Women in Mathematics at the Institute for Advanced Study. 1. Introduction In the next four lectures I want to discuss two zeta functions the Weil zeta function and the Igusa local zeta function which, in comparison to the zeta functions we just studied, are more algebraic. These zeta functions are both associated with a polynomial f(x1, x2, ..., xn) in n variables with coefficients, for simplicity, in the integers. First I am going to introduce the two generating functions, often called Poincar´eSeries, which are associated with these zeta functions. Then I will define each zeta function and compute examples of them for different polynomials. I hope that this way we can compare the two zeta functions easily. I will give you some problems to work on in the afternoons which are supposed to make the zeta functions come to life for you. I will also discuss what is known about the functions and what is not known about then. Finally, I hope to give you a feeling for how these two functions are related. I also want to thank Diane Meuser from Boston University for reading these notes and making many very helpful suggestions, e 1.1. Poincar´eSeries. Consider the unique finite field Fpe with p elements. To understand this Poincar´eseries and the associated Weil zeta function, one really needs to understand a lot about finite fields. So we will talk a lot more about finite fields. However, given a polynomial f(x1, x2, ..., xn) in n variables in the ring of polynomials Z[x1, x2, ..., xn] with coefficients in the integers, fix a prime p and consider f with coefficients reduced modulo p, so that it now has coefficients in the finite field Fp. To this polynomial, we associate the generating function ∞ X e PW (T ) = |Ne| T e=1 (n) where Ne = {(x1, x2, ..., xn) ∈ Fpe |f(x1, x2, ..., xn) = 0 in Fpe } for e > 0 and | · | denotes set cardinality. The generating function P (T ) turns out to be a rational function of T . The first part of the celebrated 1949 Weil conjectures (now theorems) was that the Weil zeta function should be a rational function. The rationality was proved by Bernie Dwork in 1960 using p-adic analysis and by showing that there is a nice relation between the |Ne|. The rationality of the Poincar´eseries follows right from the rationality of the zeta function. 1 2 MARGARET M. ROBINSON For the same polynomial we can also associate another generating function to it as ∞ X e PI (T ) = |Ne| T e=1 e (n) e where Ne = {(x1, x2, ..., xn) ∈ (Z/p Z) |f(x1, x2, ..., xn) ≡ 0 mod p } for e > 0 and an integer. The generating function PI (T ) is also a rational function of T . Its rationality was conjectured in problem 9 of the exercises at the end of Chapter 1, section 5 in the textbook Number Theory by Borevich and Shafarevich published in Russian in 1964 and in English in 1966. The rationality was settled by Igusa in 1974 when he showed that the associated Igusa local zeta function was rational. Igusa proved that the zeta function was rational using p-adic analysis and Hironaka’s resolution of singularities in characteristic 0. Hironaka won the Field’s Medal in 1964 for this result. en Note: Both |Ne| and |Ne| are less than or equal to p . Sometimes you will see P∞ −en e the series (almost always, in the Igusa case) defined as PW (T ) = e=0 |Ne|p T P∞ −en e or PI (T ) = e=0 |Ne|p T with |N0| = |N0| = 1. Using this definition, each series begins with the constant 1 for e = 0 and converges for |T | < 1. However, we are not really interested in the convergence properties of the series because they are both rational functions of T and are thus meromorphic functions on the whole complex plane. Example 1. If we take the simplest non-constant polynomial f(x) = x. We find Ne = Ne = {0} so that |Ne| = |Ne| = 1 and the two Poincar´eseries are identical: ∞ X T P (T ) = P (T ) = T e = . W I 1 − T e=1 e Example 2. For f(x1, x2) = x1x2, we can show that |Ne| = {p roots where e e x1 = 0 and x2 is free to be any of the p elements in Fpe } + {p − 1 roots where e (2) x2 = 0 and x1 is free to be anything besides 0} = 2p − 1 total roots in Fpe . The Weil Poincar´eseries for this polynomial is: ∞ X e e PW (T ) = (2p − 1)T e=1 ∞ ∞ X X = 2(pT )e − T e e=1 e=1 2pT T = − 1 − pT 1 − T (2p − 1)T − pT 2 = . (1 − T )(1 − pT ) On the other hand, when we compute |Ne| things are a little more complicated: |N1| = 2p − 1, |N2| = {p(|N1| − 1) roots where x1 and x2 are not both divisible by 2 2 e−1 p} + {p roots where they are } = 3p − 2p, so that |Ne| = {p (|N1| − 1) roots 2 where x1 and x2 not both congruent to 0 modulo p}+{p (|Ne−2|) roots where they e e−1 are }. With careful counting we get that |Ne| = (e + 1)p − ep . THE WEIL ZETA FUNCTION AND THE IGUSA LOCAL ZETA FUNCTION 3 The Igusa Poincar´eseries for this polynomial is: ∞ X e e−1 e PI (T ) = [(e + 1)p − ep ]T e=1 ∞ X = (e + 1)(pT )e − ep−1(pT )e e=1 ∞ ∞ X X = e(1 − p−1)(pT )e + (pT )e e=1 e=1 (1 − p−1)(pT ) pT = + (1 − pT )2 (1 − pT ) (2p − 1)T − p2T 2 = (1 − pT )2 Note: We express the Igusa series in this way so that the comparison with the Weil series is more direct. The usual form of the Igusa Poincar´eseries, as mentioned before, would be to begin the series at e = 0 with |N0| = 1 and to replace T above with p−2T so that the series converges for |T | < 1. Thus, later we will write the Igusa Poincar´eseries for this example as 1 − p−2T P (T ) = . I (1 − p−1T )2 Questions: Now, of course, in these two examples we were able to find a formula for |Ne| and |Ne| without too much trouble. 1) However, it is not at all clear how formulas for |N1| and |N1| depend on p. We know that for any p we could compute these values but finding a formula in p is not easy. See papers Serre [12] , Koblitz [7], and Koblitz [8]. 2) The Local-Global Principle, first articulated by H. Hasse in his PhD thesis, asks whether information about solutions in Fp or really in Qp, the p-adic numbers, for all primes p can be put together in some way to say something about solutions in the rational numbers. Results like the proof that there are infinitely many primes by considering the Euler product expansion of the Riemann zeta function. 3) What is involved in finding a formula for |Ne| and |Ne| as e grows? What can we say about |Ne| − |Ne|? 4) The fact that the Poincar´eseries are rational means that |Ne| and |Ne| have nice recursion formulas. These nice formulas are tied to underlying geometric char- acteristics of the polynomial which are encapsulated in the Weil Conjectures. 2. Weil zeta function For any polynomial with coefficients in the integers, fix a prime p and consider f(x1, x2, ..., xn) with coefficients reduced modulo p, so that it now has coefficients in Fp. We define the Weil zeta function by the following awkward looking expression: ∞ e X |Ne|T Z (T ) = exp( ). W e e=1 One of my friends says that he is always embarrassed when he has to explain why there is an exponential function in the formula for ZW (T ). But in addition 4 MARGARET M. ROBINSON to reasons of tradition, we will see that the zeta function takes on a particularly e simple form when this definition is used. For example, if |Ne| = α then X (αT )e Z (T ) = exp( ) W e e≥1 = exp(− log(1 − αT )) 1 = 1 − αT where we make use of the identity that ∞ X xe − log(1 − x) = . e e=1 The exponential function also allows the zeta function to have a nice multiplicative property with respect to the values of |Ne|. Suppose that |Ne| = Ae + Be (as happens when f(x) = g(x)h(x) where g and h have no common roots), then ∞ e ∞ e X AeT X BeT Z (T ) = exp( ) · exp( ). W e e e=1 e=1 e e e e e e So that if |Ne| = α1 +α2 +···+αm −β1 −β2 −· · ·−βn the properties of logarithms and exponenials mean that (1 − β1T )(1 − β2T ) ··· (1 − βnT ) ZW (T ) = . (1 − α1T )(1 − α2T ) ··· (1 − αmT ) We will spend most of our time investigating the zeta function as defined above but before we continue with that I want to mention that there is always a product form of the Weil zeta function which is analogous to the Euler product of the Riemann zeta function. Although in general we just know that the Weil zeta function is rational, for a projectively non-singular polynomial (whatever that is) the Weil conjectures tell us a lot more about the form of the rational function.