What is p-adic geometry?

Transcription

1 What is p-adic geometry? Marta Panizzut Chow Lectures Leipzig November 5, 2018

2 Non-Archimedean valued fields Berkovich s theory Huber s theory

3 Non-Archimedean valued fields Berkovich s theory Huber s theory

4 Non-Archimedean valued fields A valued field is a field K with an absolute value : K R 0 satisfying a = 0 if and only if a = 0, ab = a b, a + b a + b, If a+b max{ a, b }, then K is a non-archimedean valued field. This condition is stronger than the triangle inequality. The absolute value is called discrete if the value group K is discrete in R >0.

5 Let s construct an example Let p be a prime number. We write x Q as x = p k a b, where p does not divide a and b, and k Z. v p : Q R { } 0 x k

6 Let s construct an example Let p be a prime number. We write x Q as x = p k a b, where p does not divide a and b, and k Z. v p : Q R { } 0 x k v p : Q R { } satisfies the following property: v p (x) = if and only if x = 0. v p (xy) = v p (x) + v p (y). v p (x + y) min{v p (x), v p (y)}. Such a map v p is called a valuation.

7 p-adic numbers We define the p-adic non-archimedean absolute value on Q, x p = p vp(x). Example: = = 1 25 The field of p-adic numbers Q p is the completion of the field of rational numbers Q under the p-adic absolute value. Completion: absolute value p uniquely extends to Q p. Q is dense in Q p. Q p is complete.

8 Ostrowski s Theorem Remark: The field of real numbers R is the completion of Q with respect to the (Archimedean) absolute value. The absolute values, p and q, with p q, are not equivalent (just think of the sequence {p n } n ). Theorem (Ostrowski 1916) Every non-trivial absolute value on Q is equivalent to or p for some prime number p. Theorem (Hasse Minkowski Theorem 1921) A quadratic form Q with coefficients in Q has a non-trivial solution in Q if and only if Q has a non-trivial solution in R and in Q p for every p.

9 More Examples Field K with trivial absolute value. The field C{{t}} of Puiseux series with complex coefficients: x = c 1 t a 1 + c 2 t a 2 + c 3 t a , where a 1 < a 2 < a 3 < are rational numbers that have a common denominator and c 1 0.

10 More Examples Field K with trivial absolute value. The field C{{t}} of Puiseux series with complex coefficients: x = c 1 t a 1 + c 2 t a 2 + c 3 t a , where a 1 < a 2 < a 3 < are rational numbers that have a common denominator and c 1 0. Number fields, i.e., finite field extensions of Q.

11 More Examples Field K with trivial absolute value. The field C{{t}} of Puiseux series with complex coefficients: x = c 1 t a 1 + c 2 t a 2 + c 3 t a , where a 1 < a 2 < a 3 < are rational numbers that have a common denominator and c 1 0. Number fields, i.e., finite field extensions of Q. The completion of Q p (p 1 p ) = n 1 Q p (p 1 p n ) (... perfectoid field, wait for the next lecture!).

12 Valuation ring and residue field Let K be a non-archimedean field. The valuation ring is R = { x K } x 1. It is a local ring with maximal ideal M = { x R x < 1 }. The residue field k is the quotient R/M. Example: K = Q p, then k = Z/pZ. K = C{{t}}, then k = C.

13 Topological properties of NA fields Remark that if a b, then a + b = max{ a, b }. D r (a) = {x Q p : x a r} D r (a) = {x Q p : x a < r} C r (a) = {x Q p : x a = r}

14 Topological properties of NA fields Remark that if a b, then a + b = max{ a, b }. D r (a) = {x Q p : x a r} D r (a) = {x Q p : x a < r} C r (a) = {x Q p : x a = r} We have the following properties: D r (b) C r (a) for every b C r (a). Closed disks are open: D r (a) = D r (a) C r (a). Every point in a disk is a center: D r (x) = D r (y) for every y D r (x). The field K is Hausdorff and locally compact, but totally disconnected.

15 What goes wrong Let K be a complete non-trivial NA valued field. A K-manifold is a space that looks like patches of K n. K is totally disconnected, then X is totally disconnected. A function is analytic if it can be given locally by convergent power series. K is totally disconnected, there are too many analytic functions. We want a better space!

16 Non-Archimedean valued fields Berkovich s theory Huber s theory

17 Berkovich affine line Let A be a ring. A multiplicative seminorm on A is a multiplicative function : A R 0 such that 0 = 0 and 1 = 1, f + g f + g

18 Berkovich affine line Let A be a ring. A multiplicative seminorm on A is a multiplicative function : A R 0 such that 0 = 0 and 1 = 1, f + g f + g Let K be a valued field. We define A 1,Berk K as follows: points seminorms on K[T ] which extend the absolute value on K topology weakest topology such that for every f K[T ] the map f is continuous.

19 Points on A 1,Berk K Multiplicative seminorm x gives a point x A 1,Berk K. K = C The seminorms on C[T ] are of the form f f (z), for some z C. So A 1,Berk C is homeomorphic to C. Seminorms correspond to maximal ideals of C[T ].

20 Points on A 1,Berk K Multiplicative seminorm x gives a point x A 1,Berk K. K = C The seminorms on C[T ] are of the form f f (z), for some z C. So A 1,Berk C is homeomorphic to C. Seminorms correspond to maximal ideals of C[T ]. K an algebraically closed complete non-trivial NA field. Seminorms f f (x), for x K.

21 Points on A 1,Berk K Multiplicative seminorm x gives a point x A 1,Berk K. K = C The seminorms on C[T ] are of the form f f (z), for some z C. So A 1,Berk C is homeomorphic to C. Seminorms correspond to maximal ideals of C[T ]. K an algebraically closed complete non-trivial NA field. Seminorms f f (x), for x K. We have more seminorms: f Dr (a) = sup f (x), with D r (a) = {x K : x a r}. x D r (a)

22 f Dr (a) = sup f (x), with D r (a) = {x K : x a r}. x D r (a) D 0 (x) D r (x) D x y (x) = D x y (y) D r (y) D 0 (y)

23 Berkovich s Classification Theorem Theorem (Berkovich s Classification Theorem) Let K be an algebraically closed, complete non-archimedean field. Every point x A 1,Berk K corresponds to a decreasing nested sequence {D ri (a i )} of closed discs. Classification is done according to D = i D ri (a i ). TYPE I: D = D 0 (a). TYPE II: D = D r (a), with r K. TYPE III: D = D r (a), with r K. TYPE IV: D =.

24 A 1,Berk K Points of Type I are dense in A 1,Berk K. X 1,Berk K is uniquely path connected. Type II points are branching points.

25 Analytification of affine algebraic varieties Let X = SpecK[T 1,..., T n ]/I. The analytification XK Berk is the set of semimorms on K[T 1,..., T n ]/I extending the absolute value on K. X connected iff X Berk path connected X separated iff X Berk Hausdorff X proper iff X Berk compact same dimensions...

26 A tropical break Let I be an ideal in K[x 1 ±,..., x n ± ] and X its variety on the algebraic torus T n. The tropical variety Trop(X ) is defined as { } the closure in R n of (val(x 1 ),..., val(x n )) (x 1, x 2,..., x n ) X. The tropical variety Trop(X ) is a pure rational polyhedral complex. Enumerative geometry Brill-Noether theory of curves Mirror symmetry Optimization...

27 Analytification and Tropicalization Let X be an affine algebraic variety over K, and ϕ : X A m an affine embedding. The tropicalization Trop(X, ϕ) of X depends on ϕ.

28 Analytification and Tropicalization Let X be an affine algebraic variety over K, and ϕ : X A m an affine embedding. The tropicalization Trop(X, ϕ) of X depends on ϕ. Theorem (Payne 09) The analytification X an is homeomorphic to the limit of all tropicalizations Trop(X, ϕ).

29 Non-Archimedean valued fields Berkovich s theory Huber s theory

30 Higher rank valuations Let A be a ring. A valuation is a map to a totally ordered abelian group Γ, : A Γ {0}, satisfying the following properties: 0 = 0, 1 = 1, x + y max{ x, y }.

31 Higher rank valuations Let A be a ring. A valuation is a map to a totally ordered abelian group Γ, : A Γ {0}, satisfying the following properties: 0 = 0, 1 = 1, x + y max{ x, y }. Example: A = K T = { a c n 0 nt an} the ring of convergent power series on T 1. A + = R T the subspace of power series in the valuation ring R. We define X = Spa(A, A + ) as { : A Γ {0} continuous valuation on A f 1, f A +}

32 Points on Spa(A, A + ) TYPE I: f f (a), a 1.

33 Points on Spa(A, A + ) TYPE I: f f (a), a 1. TYPE II and III: f sup y Dr (a) f (y), a 1 and 0 < r 1. TYPE IV: as before for not spherically complete field

34 Points on Spa(A, A + ) TYPE I: f f (a), a 1. TYPE II and III: f sup y Dr (a) f (y), a 1 and 0 < r 1. TYPE IV: as before for not spherically complete field TYPE V: Let a 1 and 0 < r 1. Let γ be a number infinitesimally smaller or bigger than r and Γ = R >0 γ Z. f = c n (T a) n max c n γ n. If r K, then these points are equivalent to point III.

35 Literature M. Baker, An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves, p-adic geoemtry Univ. Lecture Ser., vol. 45, Amer. Math. Soc., 2008, pp V. Berkovich, Spectral theory and analytic geometry over non-archimedean fields, Mathematical Surverys and Monographs, vol. 33, American Mathematical Society, Z.I. Borevich, I.R. Shafarevich, Number Theory, Academic press, London etc S. Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett. 16, no. 3, , P. Scholze, Perfectoid spaces, Publ. math. de l IHÉS 116 (2012), no. 1, J. Weinstein, Arizona Winter School 2017: Adic Spaces, Lecture Notes.