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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?

A Z/p-cover Y

π

− → P1

k (characteristic of k is p > 0)

yp − y = f(x) f(x) ∈ k(x), unique up to adding an element of the form ap − a, a ∈ k(x). Example 1 y5 − y = 1 x5 − 1 (x − 1)2 Isomorphic to y5 − y = 1 x5 − 1 (x − 1)2 + −1 x 5 − −1 x

• = 1

x − 1 (x − 1)2

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: {P1, . . . , Pr} on P1

k.

Let dj be the order of the pole of f(x) at Pj, p ∤ dj. Let ej := dj + 1, the conductor at Pj. Lemma 2 The genus of Y is g =

• r
• j=1

ej − 2

• d

(p − 1)/2. We say Y has branching datum [e1, e2, . . . , er]⊤. Example 3 The cover y5 − y =

1 x5 − 1 (x−1)2 ∼ 1 x − 1 (x−1)2 has branching datum [2, 3]⊤

Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 3 / 17

Deformation of Artin-Schreier covers

C

φ

− → P1

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 P1

k

P1

A

Spec k Spec A,

φ Φ

(1)

Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 4 / 17

Equal characteristic deformation

Suppose φ : Y − → P1

k is a Z/5-cover defined by

y5 − y = 1 x6 and A = k[[t]]. τ : Y − → P1

k[[t]] is defined by

y5 − y = x + 2t5 x5(x − t5)2 ⇒ The special fiber is birational equivalent to y5 − y =

1 x6 .

⇒ The generic fiber (where t = 0) branches at two points x = 0 and x = t5, with conductors 4 and 3, respectively. ⇒ τ is a flat deformation of type [7] − → [4, 3]⊤.

Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 5 / 17

Mixed characteristic deformation

φ : Y − → P1

k is a Z/2-cover given by

y2 − y = 1 x3 . It deforms to (lifts to) characteristic 0 by Z2 = 1 + 4 X3 . (2) If Z = 1 − 2Y , 4Y 2 − 4Y + 1 = 1 + 4 X3 Y 2 − Y = 1 X3 (3) The generic fiber is a Kummer cover that branches at 4 points 0, −22/3, −22/3ξ3, and −22/3ξ2

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). πet

1 (C)p = πet 1 (Cη)p

Can calculate πet

1 (Cη) 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 ASg the moduli space of Artin-Schreier covers of genus g. ASg 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 AS14 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 ASg

Let p = 5 and g = 14. {9} {5, 4} {7, 2} {3, 3, 3} {4, 3, 2} {5, 2, 2} {3, 2, 2, 2} = ⇒ AS14 is connected.

Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 9 / 17

Connectedness of ASg

Theorem 5 (D.) ASg is connected when g is sufficiently large. When p = 2, 3, the moduli space ASg is always connected. When p = 5, ASg is connected for any g ≥ 14 and g = 0, 2. It is disconnected otherwise. When p > 5, ASg is connected if g ≥ (p3−2p2+p−8)(p−1)

8

and g ≤ p−1

2 .

It is disconnected if p−1

2

< g ≤ (p − 1)(p − 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]⊤

• ver 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 yp − y =

1 xe−1 , and

{e1, . . . , er} 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 [e1, . . . , er]⊤. 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]]. R{X} :=

i aiXi | limi→∞|ai| = 0

• .

Closed unit disc ↔ R{X} . Boundary of a disc ↔ R[[X−1]]{X}. Disc of radius r ↔ R[[a−1X]] 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, t5, and t10

• t

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/pn-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 {e1, e2, . . . , em} is a partition of {e}. Then there is a deformation

• f type [e] −

→ [e1, . . . , em]⊤ if and only if there exists an exact Hurwitz tree

• f type [e] −

→ [e1, . . . , em]⊤.

Huy Dang (University of Virginia) Hurwitz trees and deformations of Artin-Schreier covers May 16, 2020 15 / 17

Equal characteristic deformation of Z/pn-covers

℘(Y1, Y2) =

• 1

x2(x − t4), 1 x3(x − t4)2(x − t2)2

• .

(4)

• 12,

dx x2(x−1)2

• 0, 1

x3

• ǫe1 = 1
• 6, dx

x4

• 0 : 2

t4 : 2 0 : 3 t4 : 3 t2 : 2

• 24,

dx x3(x−1)3

• 14,

dx x6(x−1)2

• 0, 1

x7

• ǫe0 = 1

e1 e0 e1 e0

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 Y Y X X Spec k Spec A

φ2 φ Φ2 Φ φ1 Φ1

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