The Berkovich Ramification Locus: Structure and Applications

Transcription

1 The Berkovich Ramification Locus: Structure and Applications Xander Faber University of Hawai i at Mānoa ICERM Workshop on Complex and p-adic Dynamics xander/lectures/icerm BerkTalk.pdf February 14, 2012

2 Itinerary Analytic P 1 and Rational Functions The Berkovich Ramification Locus Examples of Ramification Loci Structure Theorems Examples Redux Applications Final Thoughts...

3 Notation C p completion of Q p, normalized with p =1/p F p D(a, r) D(a, r) residue field of C p = {x C p : x a r} closed disk = {x C p : x a <r} open disk" P 1 Berkovich C p -analytic space associated to P 1 Note: When p is large, you may think of Puiseux series instead i.e., the completion of an algebraic closure of C((t)).

4 The Analytic Space P 1 Points: Type 1 elements of P 1 (C p ) (disks of radius 0) Type 2 ζ a,r disks D(a, r) with a C p,r p Q = C p Type 3 ζ a,r disks D(a, r) with a C p,r p Q Type 4 descending chains D(a 1,r 1 ) D(a 2,r 2 ) such that i D(a i,r i )= Think: Points of P 1 are disks in P 1 (C p ) P 1 is a compact Hausdorff tree P 1 (C p ) P 1 dense Branch points are all of Type 2 {branches at ζ a,r } P 1 (F p ) ζ c,r ζ b, b c = ζ c, b c Segments chains of disks; e.g., c D(c, r ) D(c, b c ) D(c, r) metric : ρ(ζ c,r,ζ c,r )=log p (r/r ) x ζ c,r a b c

5 Rational Functions ϕ : P 1 P 1 nonconstant rational function defined over C p ϕ : P 1 P 1 functorial extension to P 1 On Type 1 Points: agrees with ϕ : P 1 (C p ) P 1 (C p ) On Type 2, 3 Points: If ϕ( D(a, r))=d(b, s), then ϕ(ζ a,r )=ζ b,s. If ϕ( D(a, r))=p 1 (C p ) D(b, s), then ϕ(ζ a,r )=ζ b,s. Not a complete description... can also have ϕ( D(a, r))=p 1 (C p ) On Type 4 Points: Take limits of Type 2,3 points. Think: If points of P 1 are disks in P 1 (C p ), then ϕ keeps track of mapping properties of disks.

6 The Berkovich Ramification Locus If points of P 1 are disks in P 1 (C p ), then ϕ keeps track of mapping properties of disks. D(a, r) ϕ D(b, s) is m-to-1 for some m 1 = ϕ is m-to-1 in a small neighborhood of ζ a,r. Write deg(ϕ) =d>0. The local multiplicity m ϕ : P 1 {1,...,d} is m ϕ (x) :=rk OP 1,ϕ(x) ( OP1,x). Definition. The Berkovich ramification locus is the set R ϕ = {x P 1 : m ϕ (x) 2}. On Type 1 Points: m ϕ (x) is the usual algebraic multiplicity, so it detects ramification Crit(ϕ) On Type 2, 3, 4 Points: O P 1,x is a field, so m ϕ (x) detects the residue extension

7 Examples of R ϕ Definition. The Berkovich ramification locus is the set R ϕ = {x P 1 : m ϕ (x) 2}. 1. ϕ(z) =z 2 /C p p =2: 2. ϕ(z) =z 3 3z /C2 Crit(ϕ) ={±1,, } p>2:

8 More Examples of R ϕ Definition. The Berkovich ramification locus is the set R ϕ = {x P 1 : m ϕ (x) 2}. 3. ϕ(z) = ( 2 1)z 3 +(3 2 2)z 2 (3 2)z 1 /C 2 0 Every point of R ϕ has multiplicity 2 except the indicated point x. m ϕ (x) =3 4. ϕ /Cp cubic, p> ϕ(z) = z 3 z 3 z+1 z 3 +3z 2 z+1 z 3 +pz 2 z+1 z 3 +(p+1)z 2 z+1

9 Remark: Critical Points Do Not Determine R ϕ Theorem. (Goldberg) Given distinct points P 1,...,P 2d 2 P 1 (C p ), there exist ( at most Cat(d) = 1 2d 2 ) d d 1 rational functions ϕ Cp (z) of degree d such that Crit(ϕ) ={P 1,...,P 2d 2 },uptopgl 2 (C p )-postcomposition. Example. (d =3,p>3) Let c D(1, 1). Then there are exactly Cat(3) = 2 cubic rational functions ϕ, ψ C p (z) such that Crit(ϕ) =Crit(ψ) ={0, 1,,c}. Moreover, exactly one of R ϕ and R ψ is a connected set. R ϕ R ψ

10 Structure of R ϕ Theorem. (Favre / Rivera-Letelier) The multiplicity m ϕ is upper semicontinuous. If deg(ϕ) > 1, then R ϕ is a nonempty perfect subset of P 1. Theorem. (XF) If ϕ is defined over C p, then R ϕ has at most deg(ϕ) 1 connected components. Proof Idea: Let X be a connected component of R ϕ, x X. = At least 2m ϕ (x) 2 2 critical points in X = deg(ϕ) 1 connected components (Hurwitz formula) This result is best possible: given integers 1 n<d, there exists a rational function ϕ C p (z) such that R ϕ has n connected components. (This is a very special construction though.)

11 Structure of R ϕ Theorem. (XF) R ϕ has at most deg(ϕ) 1 connected components. Properties: Definition. The visible ramification is the function τ ϕ : R ϕ Crit(ϕ) R 0 defined by τ ϕ (x) = lim sup v T P 1 x tubular ρ-radius in direction v Continuous piecewise affine for ρ-coordinate on any finite graph Determined by its values on an explicitly computable finite tree Constant as x tends to a critical point along Hull(Crit(ϕ)) τ ϕ (x) 1 for x Hull(Crit(ϕ)) p 1 [Hard] Theorem. (XF) If ϕ C p (z) is nonconstant, then R ϕ Hull(Crit(ϕ)) + 1 p 1.

12 Examples Redux Graphs of τ ϕ Theorem. (XF) R ϕ Hull(Crit(ϕ)) + 1 p 1 ϕ(z) =z 3 3z /C2 ϕ(z) = ( 2 1)z 3 +(3 2 2)z 2 (3 2)z 1 /C 2

13 A Dynamical Application with M. Manes and B. Viray Question: Given ϕ, ψ Q(z), are they conjugate over Q? For a field k, write Conj ϕ,ψ (k) ={s Aut(P 1 k ):s ϕ s 1 = ψ}. Theorem. Fix p>3 a prime of good reduction for ϕ and ψ. The reduction map red : Conj ϕ,ψ (Q) Conj ϕ,ψ (F p ) is well-defined and injective. Corollary. Conj ϕ,ψ (F p )= for p 1 = ϕ is not conjugate to ψ. Sketch: ϕ has good reduction ϕ 1 (ζ 0,1 )=ζ 0,1. s Conj ϕ,ψ (C p ) = s 1 (ζ 0,1 ) is totally ramified and fixed for ϕ. = s 1 (ζ 0,1 )=ζ 0,1 by ramification study = s has good reduction. red(s) =red(t) st 1 Aut ϕ (C p ) has trivial reduction order p. p =2or 3 by classification of finite subgroups of PGL 2 (Q).

14 Application: Rolle s Theorem for Rational Functions Theorem. (XF) ϕ C p (z) nonconstant. If ϕ has at least two distinct zeros in D(a, r), then it has a critical point in D ( a, rp 1/(p 1)). Classical (due to A. Robert) unless ϕ (D(a, r)) = P 1 (C p ) Idea: 2 zeros D(a, r) R ϕ = D ( a, rp 1/(p 1)) Hull(Crit(ϕ)) Corollary. (XF) ϕ C p (z) nonconstant. If ϕ (D(a, r)) = P 1 (C p ), then there is a critical point in D ( a, rp 1/(p 1)). Question: Can the latter be refined to D ( a, rp 1/(p+1))?

15 Final Thoughts Question: Extend to higher genus curves? (Baldassarri) Uses p-adic differential equations case P 1 P 1 is the most delicate Project: Implement an algorithm for computing R ϕ τ ϕ is effectively computable The Future: R ϕ appears to be isometric to a union of ramification loci of lower degree functions. Could this give a decomposition of ϕ into lower degree maps? ϕ(z) = 5z3 9z 2 3z +1 ψ(z) = z 2 /C 2 η(z) = 5z 4 (z 1) 2 0 R ϕ = R ψ R η 3 5 1