Hurwitz trees and deformations of Artin-Schreier covers Huy Dang - PowerPoint PPT Presentation

Hurwitz trees and deformations of Artin-Schreier covers Huy Dang University of Virginia May 16, 2020 Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 1 / 17

What is an Artin-Schreier curve? π → P 1 A Z /p -cover Y − k (characteristic of k is p > 0 ) y p − y = f ( x ) f ( x ) ∈ k ( x ) , unique up to adding an element of the form a p − a , a ∈ k ( x ) . Example 1 y 5 − y = 1 1 x 5 − ( x − 1) 2 Isomorphic to � − 1 � 5 � − 1 � y 5 − y = 1 1 = 1 1 x 5 − ( x − 1) 2 + − x − ( x − 1) 2 x x Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 2 / 17

Ramification Suppose f ( x ) has r poles: { P 1 , . . . , P r } on P 1 k . Let d j be the order of the pole of f ( x ) at P j , p ∤ d j . Let e j := d j + 1 , the conductor at P j . Lemma 2 r � � � The genus of Y is g = e j − 2 ( p − 1) / 2 . j =1 � �� � d We say Y has branching datum [ e 1 , e 2 , . . . , e r ] ⊤ . Example 3 The cover y 5 − y = 1 ( x − 1) 2 ∼ 1 1 ( x − 1) 2 has branching datum [2 , 3] ⊤ 1 x 5 − x − Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 3 / 17

Deformation of Artin-Schreier covers φ → P 1 C − k is a G -Galois cover. A is a Noetherian complete k -algebra with residue field k . A deformation of φ over A C C φ Φ P 1 P 1 (1) k A Spec k Spec A, Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 4 / 17

Equal characteristic deformation → P 1 Suppose φ : Y − k is a Z / 5 -cover defined by y 5 − y = 1 x 6 → P 1 and A = k [[ t ]] . τ : Y − k [[ t ]] is defined by x + 2 t 5 y 5 − y = x 5 ( x − t 5 ) 2 ⇒ The special fiber is birational equivalent to y 5 − y = 1 x 6 . ⇒ The generic fiber (where t � = 0 ) branches at two points x = 0 and x = t 5 , with conductors 4 and 3 , respectively. → [4 , 3] ⊤ . ⇒ τ is a flat deformation of type [7] − Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 5 / 17

Mixed characteristic deformation → P 1 φ : Y − k is a Z / 2 -cover given by y 2 − y = 1 x 3 . It deforms to (lifts to) characteristic 0 by Z 2 = 1 + 4 X 3 . (2) If Z = 1 − 2 Y , 4 Y 2 − 4 Y + 1 = 1 + 4 X 3 (3) 1 Y 2 − Y = X 3 The generic fiber is a Kummer cover that branches at 4 points 0 , − 2 2 / 3 , − 2 2 / 3 ξ 3 , and − 2 2 / 3 ξ 2 3 . Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 6 / 17

Motivation C curve over k , char ( k ) = p Grothendieck = = = = = = = ⇒ C deforms to a curve C over W ( k ) (char W ( k ) = 0 ). 1 ( C ) � p = π et π et 1 ( C η ) � p Can calculate π et 1 ( C η ) using topology. We say C lifts to characteristic 0 . Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 7 / 17

Moduli space of Artin-Schreier covers Denote by AS g the moduli space of Artin-Schreier covers of genus g . AS g can be partitioned into irreducible strata containing Artin-Schreier curves with the same branching data. Example 4 Suppose p = 5 and g = 14 . Then the sum of conductor is 9 , and the strata of AS 14 correspond to the partitions of 9 : { 9 } , { 7 , 2 } , { 5 , 4 } , { 5 , 2 , 2 } , { 4 , 3 , 2 } , { 3 , 3 , 3 } , and { 3 , 2 , 2 , 2 } . Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 8 / 17

Geometry of AS g Let p = 5 and g = 14 . { 9 } { 5 , 4 } { 3 , 3 , 3 } { 7 , 2 } { 4 , 3 , 2 } { 5 , 2 , 2 } { 3 , 2 , 2 , 2 } = ⇒ AS 14 is connected. Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 9 / 17

Connectedness of AS g Theorem 5 (D.) AS g is connected when g is sufficiently large. When p = 2 , 3 , the moduli space AS g is always connected. When p = 5 , AS g is connected for any g ≥ 14 and g = 0 , 2 . It is disconnected otherwise. When p > 5 , AS g is connected if g ≥ ( p 3 − 2 p 2 + p − 8)( p − 1) and g ≤ p − 1 2 . 8 It is disconnected if p − 1 < g ≤ ( p − 1)( p − 2) . 2 Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 10 / 17

Local equicharacteristic deformation problem Suppose p = 5 . Can we deform a curve with branching datum to [7 , 4] ⊤ over R := k [[ t ]] to one with branching datum [4 , 3 , 2 , 2] ⊤ ? By a local-global principle, it is enough to answer the “local deformation problem”. Question 5.1 (Local equicharacteristic deformation problem) Suppose φ is a Z /p -extension k [[ z ]] /k [[ x ]] given by y p − y = 1 x e − 1 , and { e 1 , . . . , e r } is a partition of e . Does there exists a Z /p -deformation R [[ Z ]] /R [[ X ]] of φ over R := k [[ t ]] whose generic fiber has branching datum [ e 1 , . . . , e r ] ⊤ . R [[ X ]] can be thought of as an open unit disc! Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 11 / 17

Disc and annuli R complete discrete valuation ring. Open unit disc ↔ R [[ X ]] . � � � i a i X i | lim i →∞ | a i | = 0 R { X } := . Closed unit disc ↔ R { X } . Boundary of a disc ↔ R [[ X − 1 ]] { X } . Disc of radius r ↔ R [[ a − 1 X ]] where a ∈ K, | a | = r . Open annulus of thickness ǫ ↔ R [[ X, U ]] / ( XU − a ) , where v ( a ) = ǫ . Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 12 / 17

Another example of a tree If a cover branches at 0 , t 5 , and t 10 t o t Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 13 / 17

Z /p -action on a boundary Theorem 6 A Z /p -cover of a boundary Spec R [[ X − 1 ]] { X } is determined by its depth and its boundary Swan conductors. Remark Not true for Z /p n -covers ( n > 1 ). May be able to generalize to Z /p ⋊ Z /m ! Strategy: Construct desired covers for subdiscs and annuli, then glue them together along their boundaries. Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 14 / 17

Deformation is determined by exact differential forms Theorem 7 (D.) Suppose { e 1 , e 2 , . . . , e m } is a partition of { e } . Then there is a deformation → [ e 1 , . . . , e m ] ⊤ if and only if there exists an exact Hurwitz tree of type [ e ] − → [ e 1 , . . . , e m ] ⊤ . of type [ e ] − Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 15 / 17

Equal characteristic deformation of Z /p n -covers � � 1 1 ℘ ( Y 1 , Y 2 ) = x 2 ( x − t 4 ) , . (4) x 3 ( x − t 4 ) 2 ( x − t 2 ) 2 t 4 : 2 t 4 : 3 0 : 2 0 : 3 � � dx � � 12 , dx 24 , x 2 ( x − 1) 2 x 3 ( x − 1) 3 e 1 e 1 ǫ e 1 = 1 � � 6 , dx � � dx 14 , x 4 t 2 : 2 x 6 ( x − 1) 2 e 0 e 0 ǫ e 0 = 1 � � � � 0 , 1 0 , 1 x 3 x 7 Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 16 / 17

Deforming cyclic covers in towers Theorem 8 (D.) Suppose φ : Z − → X is a cyclic G -Galois cover of curves over k , and A is a complete discrete valuation ring of equal characteristic p over k . Z Z φ 2 Φ 2 Y φ Y Φ φ 1 Φ 1 X X Spec k Spec A Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 17 / 17