NON-ARCHIMEDEAN GEOMETRY JOHANNES NICAISE Abstract. These are the lecture notes for an LSGNT/TCC course in the Winter term of 2017. Please report any typos or other errors you find! 1. What is non-archimedean geometry? 1.1. Non-archimedean fields. (1.1.1) Let k be a field. An absolute value on k is a multiplicative norm, that is, a function j · j : k ! R≥0 : x 7! jxj that satisfies the following axioms: (1) j0j = 0 and j1j = 1; (2) jxyj = jxj · jyj for all x; y in k; (3) triangle inequality: jx + yj ≤ jxj + jyj for all x; y in k. We say that j · j is non-archimedean if a stronger version of (3) holds: (3'): ultrametric triangle inequality: jx+yj ≤ maxfjxj; jyjg for all x; y in k. We say that j · j is archimedean if it is not non-archimedean. Exercise 1.1.2. Show that the inequality in (30) is always an equality if jxj 6= jyj. Exercise 1.1.3. Show that an absolute value j · j on k is archimedean if and only if it satisfies the archimedean property: for every real number r there exists an integer n such that jnj ≥ r. Thus j · j is non-archimedean if and only if j · j is uniformly bounded on the image of Z in k. Hint: consider the binomial expansion of (x + y)m and send m to 1. Definition 1.1.4. A non-archimedean field is a field k endowed with a non- archimedean absolute value j · j such that k is complete with respect to the metric induced by j · j. (1.1.5) Let (k; j · j) be a non-archimedean field. We will use the following notation (due to Berkovich): ko = fx 2 k j jxj ≤ 1g koo = fx 2 k j jxj < 1g ek = ko=koo 1 2 JOHANNES NICAISE Exercise 1.1.6. Show that ko is a valuation ring in k, that is, it is a subring such that for every non-zero element x of k, either x or x−1 belongs to ko. Show that koo is the unique maximal ideal in ko; thus ek is a field. We call ko the valuation ring of k and ek the residue field of k. The set ∗ jk j of absolute values of non-zero elements in k is a subgroup of (R>0; ·), called the value group of k. Example 1.1.7. (1) On any field k we can consider the trivial absolute value j·j0, defined ∗ by j0j0 and jxj0 = 1 for all x in k . The pair (k; j · j0) is a non- archimedean field, with k = ko = ek and koo = (0). Although this construction seems empty, analytic geometry over the trivally valued field actually has some deep and surprising applications! (2) Let F be any field and set k = F ((t)), the field of Laurent series over F . We endow it with a t-adic absolute value, given by j0j = 0 ordt(x) and jxj = " where ordt(x) is the t-adic valuation of x (the smallest exponent of t with non-zero coefficient in the Laurent series expansion). Here " is a fixed value in (0; 1). As " varies all these absolute values are equivalent in the sense that they give rise to the same metric topology on k, but they are different as absolute values. We have ko = F [[t]], koo = (t) and ek = F . (3) Let p be a prime number and let k = Qp be the field of p-adic numbers. We endow it with a p-adic absolute value, given by ordp(x) j0j = 0 and jxj = " where ordp(x) is the p-adic valuation of x (the smallest exponent of p with non-zero coefficient in the p-adic expansion). Here " is a fixed value in (0; 1). In number theory, it is customary to take " = 1=p (for instance to get a so-called product formula). As " varies all these absolute values are equivalent in the sense that they give rise to the same metric topology on k, but they o oo are different as absolute values. We have k = Zp, k = (p) and ek = Fp, the field with p elements. (4) If (k; j · j) is a non-archimedean field, then j · j extends uniquely to an absolute value on any algebraic extension k0 of k (here we use the assumption that k is complete). The field k0 is complete if it is a finite extension of k but not in general; its completion is denoted by kb0 and is again a non-archimedean field. If k0 is an algebraic closure of k, then kb0 is still algebraically closed. These considerations allow us to produce new examples of non-archimedean fields. For instance, if k = C((t)) then a [ 1=n k = C((t )) n>0 is an algebraic closure of k, called the field of complex Puiseux series. Exercise: describe the extensions of the t-adic absolute values to ka a a and determine the completion of k . If k = Qp and k is an algebraic a closure, the completion kc is denoted by Cp and called the field of complex p-adic numbers. It is difficult to describe this field explicitly NON-ARCHIMEDEAN GEOMETRY 3 because the extensions of Qp of degree divisible by p are hard to classify. (1.1.8) The absolute values on Q have been completely classified by Ostrowski (1916). r • Archimedean: j · j1 where j · j1 denotes the usual absolute value and r is an element of (0; 1] (note that the triangle inequality is violated for r > 1). These are all equivalent and the completion of Q with respect to any of these is R. • Non-archimedean: the trivial absolute value j · j0 and the p-adic absolute values, for all primes p and bases " 2 (0; 1). For fixed p, the p-adic absolute values are all equivalent and give rise to the completion Qp. 1.2. Why do we care about non-archimedean fields? Non- archimedean fields pop up in the \local" study of families of objects in number theory and algebraic geometry. Let us discuss some representative examples. (1.2.1) Number theory: diophantine problems (finding solutions of polynomial equations) over Q are hard! It is often useful to consider the problem over R and all the p-adic fields Qp (where we can exploit the analytic structure given by the absolute value) and then try to glue these \local" data together to an answer over Q. This is called the \local-to- global principle". The archetypal example is the Hasse-Minkowski theorem for quadratic forms: two quadratic forms over Q are equivalent (i.e., they coincide up to a linear coordinate change) if and only if they are equivalent over R and all the p-adic fields Qp. These \local" equivalences can be checked explicitly by computing some basic invariants. One can deduce that a quadratic form over Q has a non-trivial zero if and only if this holds over R and all the Qp. (1.2.2) Algebraic geometry: suppose that we want to understand the geometry of a smooth projective variety X over C. A standard technique consists in placing X in a family that degenerates into a union of irreducible components with a simpler structure; for instance, we can try to let a curve of positive genus degenerate into a union of projective lines. We try to find a variety X together with a morphism 1 f : X ! AC = Spec C[t] −1 −1 such that f (1) is isomorphic to the curve X and X0 = f (0) is a union of projective lines. In order to draw conclusions about the shape of X we need to understand what happens to the fibers of f as t approaches 0. The correct way to do this in algebraic geometry is to consider the base change X ×C[t] C((t)). This is a projective variety over the non-archimedean field k = C((t)) that can be thought of as an infinitesimal punctured tubular neighbourhood around X0. We can exploit the analytic structure given by the absolute value on k to analyze the geometry of our degeneration (this is closely related to tropical geometry). 4 JOHANNES NICAISE 1.3. What is wrong with k-analytic manifolds? (1.3.1) Non-archimedean geometry is the theory of analytic spaces (manifolds) over non-archimedean fields. Let us first decide what to expect from such a theory by looking at the classical setting over the complex numbers. Let X be a smooth complex algebraic variety. Then, through a process of analytification, we can turn X into a complex manifold Xan (we will recall the construction below). The important points about this process are the following: (1) We can apply new tools to Xan compared to X, namely, those coming from complex analysis, differential geometry and algebraic topology (for instance, Hodge theory). (2) If X is proper, we do not lose information by passing to Xan. This is guaranteed by Serre's GAGA principle (GAGA=g´eom´etrie alg´ebriqueet g´eom´etrieanalytique). It says, among other things, that two smooth and proper complex varieties X and Y are isomorphic if and only if their analytifications are isomorphic as complex manifolds. Thus we can reconstruct X from Xan, at least in principle. Beware that this fails if X is not proper. Point (2) is essential and we will see below that it fails miserably if we adopt the na¨ıve definition of analytic manifold over a non-archimedean field.