resolution of singularities on the tower of modular...

Semistable Models of Curves

Resolution of singularities on the tower of modular

curves

Jared Weinstein

IAS

Stanford Number Theory Seminar, Oct. 22, 2010

Jared Weinstein Resolution of singularities on the tower of modular curves


Semistable models: Definition

Let R be a complete DVR with fraction field K and residue field k

(e.g., R = Zp, K = Qp , k = Fp).



Semistable models: Definition

Let R be a complete DVR with fraction field K and residue field k

(e.g., R = Zp, K = Qp , k = Fp).

Definition

Let X/K be a smooth proper curve. A semistable model of X is aproper curve X/R such that

X⊗R K = X , and

X⊗R k has ordinary double points as singularities.

That is: The singularities of X are as mild as possible.



Semistable models: Dual Graphs

If X⊗R k looks like: Then its dual graph Γ is:



Semistable models: Dual Graphs

If X⊗R k looks like: Then its dual graph Γ is:

If a semistable model of X is known, then the cohomology of Xcan be computed readily.



Semistable models: Cohomological consequence

Suppose a smooth proper curve X/K has semistable model X/Rwith dual graph Γ.

Theorem

We have an exact sequence of Q`[Gal(K/K )]-modules

0 → H1(X⊗R k ,Q`) → H1(X ⊗K K ,Q`) → H1(Γ,Q`)(−1) → 0

In particular, if Γ is a tree, thenH1(X⊗R k ,Q`) → H1(X ⊗K K ,Q`) is an isomorphism.(Consequence of Rapoport-Zink’s computation of “vanishingcycles.”)



Semistable models: Cohomological consequence

Suppose a smooth proper curve X/K has semistable model X/Rwith dual graph Γ.

Theorem

We have an exact sequence of Q`[Gal(K/K )]-modules

0 → H1(X⊗R k ,Q`) → H1(X ⊗K K ,Q`) → H1(Γ,Q`)(−1) → 0

In particular, if Γ is a tree, thenH1(X⊗R k ,Q`) → H1(X ⊗K K ,Q`) is an isomorphism.(Consequence of Rapoport-Zink’s computation of “vanishingcycles.”)Semistable models make cohomology a combinatorial matter!



Semistable models: Existence

Given X/K , there might not exist a semistable model X. However:

Theorem (Deligne-Mumford)

There exists a finite extension L/K for which XL admits a

semistable model XL.



Semistable models: Existence

Given X/K , there might not exist a semistable model X. However:

Theorem (Deligne-Mumford)

There exists a finite extension L/K for which XL admits a

semistable model XL.

Theorem (Coleman)

Given a finite morphism X → Y , there exists a finite extension

L/K and a finite morphism XL → YL of semistable models which

extends X → Y .

The above theorems have an interpretation in the language of rigidanalysis.



Rigid analysis: Basic Concepts: Affinoids

In rigid analysis, the basic objects are the affinoids.

Example

The affinoid unit disk (over Qp) is:

= MaxSpecQp〈T 〉

Qp〈T 〉 is the ring of power series∑

anTn with an → 0.

Generally, a one-dimensional affinoid algebra is a finite extension ofQp〈T 〉. This is always a Banach algebra under the sup norm.A one-dimensional affinoid is the MaxSpec of an affinoid algebra.A rigid curve is obtained by gluing affinoids.



Rigid analysis: Basic Concepts: The reduction of an

affinoid

Let Z = MaxSpecA be an affinoid. We have the following rings atour disposal:

A◦ = {f ∈ A : |f (z)| ≤ 1, all z ∈ Z}

A◦◦ = {f ∈ A : |f (z)| < 1, all z ∈ Z}

A = A◦/A◦◦.

The reduction of Z is red(Z ) = SpecA.




affinoid


A◦ = {f ∈ A : |f (z)| ≤ 1, all z ∈ Z}

A◦◦ = {f ∈ A : |f (z)| < 1, all z ∈ Z}

A = A◦/A◦◦.

The reduction of Z is red(Z ) = SpecA.If Z is the affinoid unit disk, then A = Qp〈T 〉, A◦ = Zp〈T 〉, andA◦◦ = pA◦. Thus, red(Z ) = SpecFp[T ] = A1/Fp.




affinoid


A◦ = {f ∈ A : |f (z)| ≤ 1, all z ∈ Z}

A◦◦ = {f ∈ A : |f (z)| < 1, all z ∈ Z}

A = A◦/A◦◦.

The reduction of Z is red(Z ) = SpecA.If Z is the affinoid unit disk, then A = Qp〈T 〉, A◦ = Zp〈T 〉, andA◦◦ = pA◦. Thus, red(Z ) = SpecFp[T ] = A1/Fp.If Z is the “circle” {z : |z | = 1}, then A = Qp〈T ,T−1〉 andred(Z ) = SpecFp[T ,T−1] = Gm/Fp .



Rigid Analysis: Coleman’s “wide opens”

Goal: Given a smooth proper curve X/K , compute a semistablemodel X/R .




Goal: Given a smooth proper curve X/K , compute a semistablemodel X/R .Method: Chop up the rigidification X rig into simple building blocks(“wide opens”) which obey certain intersection rules; each wideopen corresponds to a component in X⊗R k .




Goal: Given a smooth proper curve X/K , compute a semistablemodel X/R .Method: Chop up the rigidification X rig into simple building blocks(“wide opens”) which obey certain intersection rules; each wideopen corresponds to a component in X⊗R k .A wide open is a rigid-analytic curve W containing an affinoid Z

such that W \Z is a disjoint union of open annuli:

W{ Z

Here, Z is an underlying affinoid of W .



Semistable models and semistable coverings

Let X be a rigid analytic curve. A semistable covering is a family{Wi} of wide opens, such that

1 X =⋃

i Wi .

2 For i 6= j , Wi ∩Wj is a disjoint union of open annuli.

3 No three of the Wi may mutually intersect.

4 For each i , the complement Zi = Wi\⋃

j 6=i Wj is anunderlying affinoid of Wi , with good reduction.



Semistable models and semistable coverings

Let X be a rigid analytic curve. A semistable covering is a family{Wi} of wide opens, such that

1 X =⋃

i Wi .

2 For i 6= j , Wi ∩Wj is a disjoint union of open annuli.

3 No three of the Wi may mutually intersect.

4 For each i , the complement Zi = Wi\⋃

j 6=i Wj is anunderlying affinoid of Wi , with good reduction.

If X/K is a smooth proper curve, then semistable models X of Xcorrespond to semistable coverings {Ui} of X rig. The componentsof X⊗R k will be projective models of the reductions red(Zi).



Semistable covering: Example

Let q ∈ Q×p have v(q) = 3, and let X be the “Tate curve” Q

×

p /qZ.





×

p /qZ.

(X is the rigidification of an elliptic curve with Tate parameter q.)

<

<

|p|3 ≤ |z | ≤ 1

=

|p|3 < |z | < |p|

|z | = |p|2

∪

|p|2 < |z | < 1

|z | = |p|

∪

|p| < |z | < |p|−1

|z | = 1





×

p /qZ.


<

<

|p|3 ≤ |z | ≤ 1

=

|p|3 < |z | < |p|

|z | = |p|2

∪

|p|2 < |z | < 1

|z | = |p|

∪

|p| < |z | < |p|−1

|z | = 1

Each underlying affinoid has reduction Gm.





×

p /qZ.


<

<

|p|3 ≤ |z | ≤ 1

=

|p|3 < |z | < |p|

|z | = |p|2

∪

|p|2 < |z | < 1

|z | = |p|

∪

|p| < |z | < |p|−1

|z | = 1

Each underlying affinoid has reduction Gm.We get a semistable model of X whose special fiber has dual graph

Each vertex is a P1.



Semistable reduction of modular curves

Let Xn = X (Γ(pn) ∩ Γ1(N)). The Katz-Mazur model of Xn overZnrp [ζpn ] has reduction

Ig

Ig

Ig

After base extension, want to find a semistable model, whosereduction will look like...



Semistable reduction of modular curves

? ? ? ?

Ig

Ig

IgEach “?” is the reduction of a semistable model forXn,x = Spec OXn,x , where x ∈ X1(N)(Fp) is supersingular. Thecohomology H1(?,Q`) encodes the local Langlands correspondencefor irreducible representations of Gal(Qp/Qp)!



Semistable models of modular curves: previous work

1 The Deligne-Rapoport model of X0(p) is already semistable.

2 Edixhoven found a semistable model of X0(p2).

3 Coleman-McMurdy found a semistable model of X0(p3).

4 Wewers found equations for curves which should appear in thesemistable reduction of X (pn).



Semistable models of modular curves

Recall Xn,x = Spec OXn,x is the completion of Xn at asupersingular point x .





Using the “wide opens” method, we found semistable modelsXn,x over Zp for which Xn+1,x → Xn,x is finite.






In the tower, · · · → Xn+1,x ⊗ Fp → Xn,x ⊗ Fp → · · · , suppose· · · → Cn+1 → Cn → · · · is a chain of irreducible components.Then the morphisms are purely inseparable for n large enough!






In the tower, · · · → Xn+1,x ⊗ Fp → Xn,x ⊗ Fp → · · · , suppose· · · → Cn+1 → Cn → · · · is a chain of irreducible components.Then the morphisms are purely inseparable for n large enough!

We describe the inverse limit of the dual graphs of theXn,x ⊗ Fp. Each vertex corresponds to one of the chains· · · → Cn+1 → Cn → · · · as above. Each vertex is labeledwith the equation of Cn for n large enough.

The stabilizer of any particular vertex is a group which arisesin Bushnell-Kutzko’s theory of types for classifyingrepresentations of GL2(Qp).



Structure of the semistable reduction of the tower of

modular curves: The “tame” part C tame

Legend

• xyp − xpy = 1• P1





C tame

Center vertex is curveX : xyp − xpy = 1 over Fp





C tame


Its stabilizer in GL2(Qp) is SL2(Zp)





C tame


Its stabilizer in GL2(Qp) is SL2(Zp)

SL2(Zp) acts on X by(

a b

c d

)(

x

y

)

=

(

ax + by

cx + dy

)

H1(C tame,Q`) encodes LLC forsupercuspidal π of conductor p2.




modular curves: Wild components

Legend

• xyp − xpy = 1• P1





F0

F1 F2





F0

F1 F2

Legend

• xyp − xpy = 1• yp + y = xp+1

• yp − y = x2

• P1

Fi/Qp are quad.extns.

F0/Qp isunramified.

A vertex labeled Fi↔ a Gal. rep. ofthe form IndFi/Qp

θ.




modular curves: Complete picture

Legend

• xyp − xpy = 1• yp + y = xp+1

• yp − y = x2

• P1


resolution of singularities on the tower of modular...

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