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Zeta Functions in Algebraic Geometry Mircea Mustat˘A

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Zeta Functions in Algebraic Geometry Mircea Mustat˘A Zeta functions in algebraic geometry Mircea Mustat¸˘a Contents Foreword vii Chapter 1. Introduction: An overview of zeta functions1 1.1. The Hasse-Weil zeta function1 1.2. The zeta function of an arithmetic variety2 1.3. The Igusa zeta function3 1.4. Motivic versions of the above (local) zeta functions4 1.5. Zeta functions in group theory5 Chapter 2. Basics of Hasse-Weil zeta functions7 2.1. Review of finite fields7 2.2. Preliminaries: varieties over finite fields7 2.3. The Hasse-Weil zeta function 10 2.4. The statements of the Weil conjectures 12 2.5. Comments on the conjectures 13 2.6. Two examples: computing the Betti numbers for Grassmannians and full flag varieties 14 Chapter 3. The Weil conjectures for curves 17 3.1. Rationality of the zeta function 18 3.2. The functional equation 19 3.3. The analogue of the Riemann hypothesis 21 Chapter 4. Weil cohomology theories and the Weil conjectures 23 4.1. Weil cohomology theories 23 4.2. Rationality and the functional equation via Weil cohomology 29 4.3. A brief introduction to `-adic cohomology 32 Chapter 5. Fulton's trace formula for coherent sheaf cohomology 39 5.1. The statement of the main theorem 39 5.2. The proof of the Localization Theorem 42 5.3. Supersingular Calabi-Yau hypersurfaces 45 Chapter 6. The Lang-Weil estimate and the zeta function of an arithmetic scheme 49 6.1. The Chow variety 49 6.2. The Lang-Weil estimate 50 6.3. Estimating the number of points on arbitrary varieties 54 6.4. Review of Dirichlet series 55 6.5. The zeta function of an arithmetic scheme 62 v vi CONTENTS Chapter 7. The Grothendieck ring of varieties and Kapranov's motivic zeta function 69 7.1. The Grothendieck ring of algebraic varieties 69 7.2. Symmetric product and Kapranov's motivic zeta function 75 7.3. Rationality of the Kapranov zeta function for curves 80 7.4. Kapranov zeta function of complex surfaces 81 Chapter 8. Dwork's proof of rationality of zeta functions 87 8.1. A formula for the number of Fq-points on a hypersurface 87 8.2. The construction of Θ 89 8.3. Traces of certain linear maps on rings of formal power series 92 8.4. The rationality of the zeta function 98 Appendix A. Quotients by finite groups and ground field extensions 101 A.1. The general construction 101 A.2. Ground field extension for algebraic varieties 104 A.3. Radicial morphisms 106 A.4. Quotients of locally closed subschemes 108 Appendix B. Basics of p-adic fields 111 B.1. Finite extensions of Qp 111 B.2. Unramified extensions of Qp and Teichm¨ullerlifts 113 B.3. The field Cp 116 B.4. Convergent power series over complete non-Archimedean fields 118 B.5. Examples of analytic functions 121 Bibliography 125 Foreword These are lecture notes from a graduate course I taught in Spring 2011. They cover some elementary topics related to Hasse-Weil zeta functions. I hope to add in the near future a second part, largely independent of the first one, covering p-adic and motivic zeta functions. vii CHAPTER 1 Introduction: An overview of zeta functions Zeta functions encode the counting of certain objects of geometric, algebraic, or arithmetic behavior. What distinguishes them from other generating series are special analytic or algebraic properties. Zeta functions come up in a lot of area of mathematics. The ones we will deal with come in two flavors: local and global. Here local means relative to a prime p in Z, or in some ring of integers in a number field. In this case, one expects the zeta function to be a rational function, in a suitable variable. By a global zeta function we mean an object that takes into account all primes. In this case one expects to have a product formula in terms of local factors. The basic example is the well-known factorization of the Riemann zeta function: −1 Y 1 ζ(s) = 1 − : ps p prime A good understanding of the local factors of the zeta function can be used to show that the global zeta function is defined in some region fs 2 C j Re(s) > ηg, and then there are fundamental questions regarding analytic continuation and the ex- istence of a functional equation. Again, the model is provided by the Riemann zeta function. However, very little is known in a more general setting. The gen- eral philosophy is that the analytic properties of the zeta function encode a lot of information about the geometric/arithmetic/algebraic of the object that is studied. In what follows we give an overview of the types of zeta functions that we will discuss in the following chapters. In all this discussion, we restrict to the simplest possible setting. 1.1. The Hasse-Weil zeta function This is one of the most famous zeta functions, and it played an important role in the development of algebraic geometry in the twentieth century. It is attached n to a variety over a finite field, say k = Fq. Suppose, for simplicity, that X ⊂ Ak is a closed subvariety defined by the equations f1; : : : ; fd. For every m ≤ 1, let n Nm := jfu 2 X(Fqm )j = jfu 2 Fqm j fi(u) = 0 for all igj: The Hasse-Weil zeta function of X is 0 1 X Nm Z(X; t) := exp tm 2 Q[[t]]: @ m A m≥1 A fundamental result is that Z(X; t) is a rational function. This was conjectured by Weil in [We2], who also proved it for curves and abelian varieties in [We1]. 1 2 1. INTRODUCTION: AN OVERVIEW OF ZETA FUNCTIONS The general case was proved by Dwork in [Dwo]. Another proof in the case of smooth projective varieties was later given by Grothendieck and its school using ´etalecohomology, see [Gro]. Both the methods of Grothendieck and of Dwork have been extremely influential for the development of arithmetic geometry. When X is a smooth projective variety, ZX (t) satisfies • The functional equation. • A connection with the Betti numbers defined over C. • An analogue of the Riemann hypothesis. These three properties, together with the rationality mentioned above, form the Weil conjectures [We2], now a theorem of Grothendieck [Gro] and Deligne [Del3]. See x2.4 for the precise statements. 1.2. The zeta function of an arithmetic variety n Suppose now that X ⊂ AZ is defined by the ideal (f1; : : : ; fd) ⊆ Z[x1; : : : ; xn]. For every prime p, we may consider f ;:::; f 2 F [x ; : : : ; x ] defining X ⊆ An , 1 d p 1 n p Fp and the corresponding Z(Xp; t). One then defines Y s LX (s) := Z(Xp; 1=p ): p6 ja n If X ⊂ AQ, then we may assume that the equations defining X have coefficients in some localization Z[1=a], where a is a positive integer. In this case we may still define Xp when p does not divide a, and we obtain LX as above, by taking the product over those p that do not divide a (this definition on the choice of a, but for us this is not important). Let us consider the case X = Spec Q, when we may take Xp = Spec Fp for every prime p. Note that 0 1 X te Z(X ; t) = exp = exp(−log(1 − t)) = (1 − t)−1: p @ e A e≥1 −1 Q 1 Therefore LX (s) = p 1 − ps = ζ(s) is the Riemann zeta function. In general, it is not hard to see that LX is defined in some half-plane fs 2 C j Re(s) > ηg (we will discuss this in Chapter6, giving a precise value for η, as a consequence of the Lang-Weil estimates, which in turn follow from the Weil conjectures for curves). It is conjectured that if X is a smooth projective variety over Q, then LX has an analytic continuation that is a meromorphic function. One also expects that after a suitable normalization (necessary for taking into account the infinite prime and the primes of bad reduction) LX satisfies a functional equation. Very little is know in this direction. Both properties are known for Pn and related varieties (such as toric varieties or flag varieties). The case of elliptic curves is known as a consequence of the Taniyama-Shimura conjecture (proved by Wiles [Wil], Taylor- Wiles [TW] and Breuil-Conrad-Diamond-Taylor [BCDT]), which implies that in this case LX can be described as the L-function attached to a modular form. 1.3. THE IGUSA ZETA FUNCTION 3 1.3. The Igusa zeta function Suppose now, for simplicity, that p is a prime in Z, and X,! An is defined Zp by f 2 Zp[x1; : : : ; xn]. The Igusa zeta function of f is defined by Z s Zf (s) := jf(x)jpdx: n Zp This is defined using the p-adic absolute value j·jp and the Haar measure on Zp. It is easy to see using the definition that Zf is analytic in the half-plane fs j Re(s) > 0g. Let us give some motivation for this definition. 1.3.1. The Archimedean analogue of Zf . The following analogue in the Archimedean setting (over R or C) appeared before Igusa's zeta function, in the setting of complex powers. Suppose, for example, that f 2 R[x1; : : : ; xn], and we want to define jf(x)js for s 2 C as a distribution. Given a test function Φ, consider the map Z s ! jf(x)jsΦ(x)dx: Rn It is not hard to see that this is well-defined and analytic in the half-space fs 2 C j Re(s) > 0g.
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