GVF NOTES/ ITAÏ BEN YAACOV, E.H. Contents 1. Continuous logic 2 1.1. Ultraproducts 2 1.2. Formulas 2 2. Measure theory 3 2.7. Positive linear functionals 5 3. Valued fields 6 3.1. Generically stable types 6 3.2. Comparison of V“ with V an 6 3.4. The linear part of a generically stable type 6 3.6. valuative volumes and mixed volumes 7 3.9. Minkowski’s mixed volumes 9 4. Multiply valued fields 10 4.6. Topology on V ALF 12 4.7. Admissible measures 12 4.11. Induced structure on a subfield 13 4.12. Renormalization 13 4.19. Proper subvarieties 15 4.21. Extending valuations on subrings 15 4.26. Symmetric extensions 17 4.29. Heights 18 4.34. A criterion for globalization 18 4.35. GVF’s as continuous logic structures 18 4.38. Archimedean valuations 19 5. Classical GVF structures. 21 6. Conjectures 22 a 7. Gm-fullness and the GVF structure on K . 23 7.1. Axiomatizability of Gm-fullness. 24 8. Adelic canonical amalgamation 25 9. One variable 32 10. 1-types of height zero 33 10.1. Local case 34 10.2. The purely non-archimedean case 34 References 39 Date: January 26, 2016. 1 2 GVF NOTES/ ITAÏ BEN YAACOV, E.H. 1. Continuous logic A language consists of function and relation symbols as usual. A structure is a set (or a family of sets in the many-sorted situation) with function symbols and terms treated as usual. But the interpretation of a relation symbol R(x1; : : : ; xn) n is a function A ! R. In addition, we are given a map h : A ! R≥0, the gauge. This can be viewed as a continuous division into sorts; only sets of bounded ’gauge’ are treated seriously, but no specific sharp bound is specified. We assume, for each function symbol F , that h(F (a1; : : : ; an)) ≤ cF (h(a1); : : : ; h(an)) for some continuous cF ; and R(a1; : : : ; an) ≤ cR(h(a1); : : : ; h(an)). 1.1. Ultraproducts. Let I be an index set, u an ultrafilter on I, L a language, Ai real-valued L-structures. We define the ultraproduct. The universe consists of the sequences (ai)i2I with ai 2 Ai and with h(ai) bounded; modulo the equivalence relation: a ∼u b () fi : a(i) = b(i) 2 ug. The function symbols are interpreted pointwise. The relation symbols: R(a1; : : : ; an) = lim R(a1(i); : : : ; an(i)) i!u 1.2. Formulas. Formulas are formed using connectives and quantifiers. We de- note their truth value (in some given structure) by [φ] and explain how to compute it. Connectives: Any continuous function c : Rn ! R can be viewed as a connec- tive c; the truth value [c(φ1; : : : ; φn)] equals c([φ1];:::; [φn]). Quantifiers: let c : R ! R be a continuous function with compact support. Then we have a formula (Supcx)φ whose interpretation is supxc(h(x))[φ]. Remark 1.3. (1) We could allow more general quantifiers for each continuous c : Rn+1 ! R whose support projects properly to n R ; and form (Supcx)φ(x; y1; : : : ; yn) whose interpretation is supxc(h(x); h(y1); : : : ; h(yn))[φ]. (2) We could more generally allow truth values in any space Y with a proper map p : Y ! R≥0; in this case, for each relation symbol R, a rule: p([R(x)]) ≤ cR(h(x)) is imposed. In particular if Y is compact, no gauge is needed. (3) In situations where we are concerned with effectivity, it makes sense to restrict the continuous functions c to some nice, uniformly dense on com- pacts class. For the moment, this is irrelevant. Proposition 1.4 (Los). For any formula φ, any ultraproduct I; u; Ai; ai 2 Ai with h(ai) bounded, φ(a1; : : : ; an) = lim φ(a1(i); : : : ; an(i)) i!u GVF NOTES/ ITAÏ BEN YAACOV, E.H. 3 2. Measure theory We consider locally compact spaces X; a measure means: a regular Borel measure; it is not necessarily finite, but compact sets have finite measure. In fact µ may just be identified with the compatible collection of finite measures µjU, where U is a compact subset of X. If π : Y ! X is a proper map, and ν a measure on Y , then π∗ν is the measure −1 on X with (π∗ν)(U) = µ(π (U)). Lemma 2.1. Let Y be a locally compact space, G a finite group acting on Y , X = Y=G the quotient, π : Y ! X the natural map. Let µ be a measure on X. Then there exists a unique G-invariant measure ν on Y with π∗ν = µ. Proof. (1) This reduces to the case that Y is compact (lift µjU for each compact subset U of X, note that they are compatible using the uniqueness statement.) (2) In the compact case, we translate this to positive linear functionals on C(X), using Riesz. We have a positive linear functional µ : C(X) ! R, and must show there is a unique extension to a positive linear functional ν : C(Y ) ! R which is invariant. Note that G acts on C(Y ), and let tr(f) = Pfσ(f): σ 2 Gg. Note that tr(f) = F ◦ π for a unique function F ; since tr(f) 2 C(Y ), we have 1 F 2 C(X). Let us denote T r(f) = G F . For any invariant extension ν, since ν(f) = ν(σ(f)) for σ 2 G, we have: µ(F ) = ν(tr(f)) = jGjν(f) On the other hand it is clear that f 7! µ(T r(f)) is a positive linear functional on C(Y ). The displayed formula in the proof extends to R T r(f)(x)dµ(x) = R f(y)dν(y) for continuous functions f; or: Z Z (1) f(y)dν(y) = T r(f)(x)dµ(x) Remark 2.2. (1) if N is a normal subgroup of G, Y 0 = Y=N so that we have: 0 Y !π1 Y !π2 Y=G = X 0 0 with π = π2π1, and ν is the extension to Y obtained in the same way, 0 0 then (π1)∗ν = ν . This follows from uniqueness, since ν is the unique G=N-invariant extension of µ, and clearly (π1)∗ν is also G=N-invariant. 1 P (2) (1) and the lemma go through for G compact, replacing G g2G f(g(x)) by the Haar integral. Theorem 2.3. (Caratheodory’s extension theorem) Let R be a ring on a subset S of P(Ω) and let µ be a pre-measure on R. Then there exists an extension µ0 of µ to a measure on the σ-algebra generated by R. 4 GVF NOTES/ ITAÏ BEN YAACOV, E.H. Remark 2.4. Note that if R is a semi-ring, then the following condition is suffi- cient in order for R to be a ring: If (Bi) is an increasing sequence of elements in R and [Bi 2 R then (Bi) is eventually constant. In particular, it follows from the compactness theorem that if R is a semi-ring consisting of definable subsets of some structure M then R is also a ring. We will also need to know certain consequences of the stability of probability theory, viewed as a continuous-logic theory; see [3], [4]. Here is a statement in explicit terms of a key property of Morley sequences in stable theories, namely they are universal witness for dividing. Lemma 2.5. Let µ be a probability measure on X, ν a probability measure on Y . Let θ be a probability measure on on XN×Y , projecting to the product measure µN, and also projecting to some fixed measure θ2 on X × Y via any of the coordinate projections. Then θ2 = µ × ν. Proof. (With thanks to David Kazhdan.) We have to show that θ2(U × V ) = µ(U)ν(V ), for any Borel U ⊂ X; V ⊂ Y . Fix V , and for any Borel W ⊂ XN, 0 θ(W ×V ) 0 R N let µ (W ) = ν(V ) . By Radon-Nykodim, µ (W ) = W gdµ for some g 2 1 N L (X ).Let πi be the i’th projection, and let h : X ! R be any Borel function; we R N R 0 R have g(h◦πi)dµ = h◦πidµ = ( hdµ)ν(V ), so this quantity does not depend on the projection i. We may assume X is compact and totally disconnected; then g can be approximated by a continuous step function g0; R jg − g0j < . Say R g0dµ = 1 − 0; so j0j ≤ . Now g0 depends on only finitely many coordinates. 0 Take i bigger than any of them. Then since g and 1U ◦ πi depend on disjoint coordinates, θ (U × V ) θ(π−1(U) × V ) Z Z 2 = i = gdµN ∼ g0dµN = µ(U)(1−0) ∼ µ(U) −1 −1 ν(V ) ν(V ) πi (U) πi (U) (Here ∼ denotes equality up to an error of .) Letting ! 0 we obtain the required equality. We will also use a modularity - type phenomenon that belongs to the theory RCF of real closed fields, or more precisely to the randomization of RCF, or any o-minimal theory; for Abelian varieties of bad reduction, the relevant o- minimal theory will be that of the value group. We will see that this component induces modularity on certain global types; the results are due to Zhang, Bilu, Gubler, Chambert-Loir, Yamaki, Baker, Rumely, and related to Bogomolov’s conjecture and its descendents.