Perfectoid modular forms - nms.kcl.ac.uk filep-adic modular forms Coleman’s Spectral Halo tilting...

p-adic modular formsColeman’s Spectral Halo

tilting equivalences

Perfectoid modular forms

Ben Heuer

King’s College London/LSGNT

9 November 2018

Ben Heuer Perfectoid modular forms



p-adic modular forms




Classical picture over C: Let k ,N be positive integers, k ≥ 2.

Definition

A modular form of weight k , level Γ(N) is a function f : H→ C s.t.

f (γz) = (cz + d)k f (z) for all γ =(a bc d

)∈ Γ(N)

[plus some condition about extending to the cusps].

Where does this definition come from?On Y = Γ(N)\H one has an automorphic line bundle ω.Modular forms = sections of ω⊗k over compactification X of Y .

q∗ω H

ω Γ(N)\H

q

Point is: ω has canonical trivialisation when we pull it back to H.





Definition



)∈ Γ(N)


Where does this definition come from?

On Y = Γ(N)\H one has an automorphic line bundle ω.Modular forms = sections of ω⊗k over compactification X of Y .

q∗ω H

ω Γ(N)\H

q

Point is: ω has canonical trivialisation when we pull it back to H.





Definition



)∈ Γ(N)


Where does this definition come from?On Y = Γ(N)\H one has an automorphic line bundle ω.Modular forms = sections of ω⊗k over compactification X of Y .

q∗ω H

ω Γ(N)\H

q

Point is: ω has canonical trivialisation when we pull it back to H.Ben Heuer Perfectoid modular forms




Let p be a prime. Then p-adic modular forms arise from p-adicallyinterpolating modular forms defined over Q.

Example: The Eisenstein series

G ∗k = ζ∗(1− k) +∞∑n=1

σ∗k−1(n)qn, where σ∗k(n) =∑d |n

(d ,p)=1

dk

Interpolate: Replace x 7→ xk by any continuous grouphomomorphism κ : Z×p → Z×p . In fact, let’s take κ : Z×p → R× forany top. Zp-algebra R. We obtain the p-adic Eisenstein family

G ∗κ = ζ∗(1−κ)+∞∑n=1

σ∗κ−1(n)qn, where σ∗κ−1(n) =∑d |n

(d ,p)=1

κ(d)d−1





Let p be a prime. Then p-adic modular forms arise from p-adicallyinterpolating modular forms defined over Q.Example: The Eisenstein series

G ∗k = ζ∗(1− k) +∞∑n=1


(d ,p)=1

dk


G ∗κ = ζ∗(1−κ)+∞∑n=1


(d ,p)=1

κ(d)d−1






G ∗k = ζ∗(1− k) +∞∑n=1


(d ,p)=1

dk

Interpolate: Replace x 7→ xk by any continuous grouphomomorphism κ : Z×p → Z×p .

In fact, let’s take κ : Z×p → R× forany top. Zp-algebra R. We obtain the p-adic Eisenstein family

G ∗κ = ζ∗(1−κ)+∞∑n=1


(d ,p)=1

κ(d)d−1






G ∗k = ζ∗(1− k) +∞∑n=1


(d ,p)=1

dk

Interpolate: Replace x 7→ xk by any continuous grouphomomorphism κ : Z×p → Z×p . In fact, let’s take κ : Z×p → R× forany top. Zp-algebra R.

We obtain the p-adic Eisenstein family

G ∗κ = ζ∗(1−κ)+∞∑n=1


(d ,p)=1

κ(d)d−1






G ∗k = ζ∗(1− k) +∞∑n=1


(d ,p)=1

dk


G ∗κ = ζ∗(1−κ)+∞∑n=1


(d ,p)=1

κ(d)d−1




p-adic modular curves

Let N ≥ 3 be an integer coprime to p.Let C be an algebraically closed complete extension of Qp.Let X be the compactified modular curve of level Γ(N) over C ,considered as a rigid analytic space.

Let X (0) ⊆ X be the ordinary locus, an open rigid subspace.For any n ≥ 0, have compactified modular curves X ∗Γ0(pn) → X

∗.

Fact

The projection XΓ0(pn)(0)→ X (0) has a canonical section and

XΓ0(pn)(0) = X (0) t XΓ0(pn)(0)a

where XΓ0(pn)(0)a ⊆ X ∗Γ0(pn)(0) is called the anticanonical locus.

Let’s consider the automorphic bundle ω on XΓ0(p)(0)a. We wantto use it to define p-adic modular forms.





Let N ≥ 3 be an integer coprime to p.Let C be an algebraically closed complete extension of Qp.Let X be the compactified modular curve of level Γ(N) over C ,considered as a rigid analytic space.Let X (0) ⊆ X be the ordinary locus, an open rigid subspace.

For any n ≥ 0, have compactified modular curves X ∗Γ0(pn) → X∗.

Fact


XΓ0(pn)(0) = X (0) t XΓ0(pn)(0)a







Let N ≥ 3 be an integer coprime to p.Let C be an algebraically closed complete extension of Qp.Let X be the compactified modular curve of level Γ(N) over C ,considered as a rigid analytic space.Let X (0) ⊆ X be the ordinary locus, an open rigid subspace.For any n ≥ 0, have compactified modular curves X ∗Γ0(pn) → X

∗.

Fact


XΓ0(pn)(0) = X (0) t XΓ0(pn)(0)a






the modular curve of infinite level

By work of Scholze, there’s a perfectoid space XΓ(p∞) which wesee as an analogue of the upper half plane H for the p-adic world.

q∗ω XΓ(p∞)(0)a A1 ⊆ P1

ω XΓ0(p)(0)a

q

πHT

Turns out the bundle q∗ω is trivial! [Reason: π∗HTO(1) = q∗ω]Moreover, the map q is a torsor for the group

K0(p) = {( ∗ ∗c ∗ ) ∈ GL2(Zp) | c ∈ pZp}

[Only away from the boundary, terms and conditions apply]⇒ Can use it to define p-adic modular forms!






q∗ω XΓ(p∞)(0)a A1 ⊆ P1

ω XΓ0(p)(0)a

q

πHT

Turns out the bundle q∗ω is trivial! [Reason: π∗HTO(1) = q∗ω]

Moreover, the map q is a torsor for the group

K0(p) = {( ∗ ∗c ∗ ) ∈ GL2(Zp) | c ∈ pZp}







q∗ω XΓ(p∞)(0)a A1 ⊆ P1

ω XΓ0(p)(0)a

q

πHT


K0(p) = {( ∗ ∗c ∗ ) ∈ GL2(Zp) | c ∈ pZp}

[Only away from the boundary, terms and conditions apply]

⇒ Can use it to define p-adic modular forms!






q∗ω XΓ(p∞)(0)a A1 ⊆ P1

ω XΓ0(p)(0)a

q

πHT


K0(p) = {( ∗ ∗c ∗ ) ∈ GL2(Zp) | c ∈ pZp}





p-adic modular forms after Chojecki-Hansen-Johannson

Let z be the canonical parameter on A1 ⊆ P1.

Let z be its pullback via πHT : XΓ(p∞)(0)a → A1 ⊆ P1.

Definition (

Chojecki-Hansen-Johannson

)

Let κ : Z×p → C× be a continuous character. We define a sheaf ωκ

on the rigid space XΓ0(p)(0)a by setting

ωκ = {f ∈ OXΓ(p∞)(0)a |γ∗f = κ−1(cz + d)f for all γ ∈ K0(p)}.

Theorem

The sheaf ωκ is a line bundle.

Definition

The space of p-adic modular forms of weight κ is Mκ = Γ(ωκ).




integral modular forms

On defines on Mκ a Hecke action as usual. An eigenvalue for theHecke algebra is called an eigenform.

Turns out eigenvalues are all p-adically integral.Reason: There is a good notion of integral modular forms:

Definition

We define a sheaf ωκ,+ on the rigid space XΓ0(p)(0)a by setting

ωκ,+ = {f ∈ O+XΓ(p∞)(0)a

|γ∗f = κ−1(cz + d)f for all γ ∈ K0(p)}.

The sections M+κ = Γ(ωκ,+) are the integral p-adic modular forms

of weight κ.This space is preserved by the Hecke action.





On defines on Mκ a Hecke action as usual. An eigenvalue for theHecke algebra is called an eigenform.Turns out eigenvalues are all p-adically integral.

Reason: There is a good notion of integral modular forms:

Definition


ωκ,+ = {f ∈ O+XΓ(p∞)(0)a








On defines on Mκ a Hecke action as usual. An eigenvalue for theHecke algebra is called an eigenform.Turns out eigenvalues are all p-adically integral.Reason: There is a good notion of integral modular forms:

Definition


ωκ,+ = {f ∈ O+XΓ(p∞)(0)a



of weight κ.

This space is preserved by the Hecke action.





On defines on Mκ a Hecke action as usual. An eigenvalue for theHecke algebra is called an eigenform.Turns out eigenvalues are all p-adically integral.Reason: There is a good notion of integral modular forms:

Definition


ωκ,+ = {f ∈ O+XΓ(p∞)(0)a







perfectoid p-adic modular forms

Consider the subgroup of K0(p) defined by

K0(p∞) = {( ∗ ∗0 ∗ ) ∈ GL2(Zp)}.

Definition (perfectoid p-adic modular forms)

ωκ,perf = {f ∈ OXΓ(p∞)(0)a |γ∗f = κ−1(d)f for all γ ∈ K0(p∞)}.

The space of perfectoid p-adic modular forms of weight κ isMperfκ = Γ(ωκ,perf). As before, one defines Mperf,+

κ using O+.

[Conceptual picture: There is an intermediate perfectoid space

XΓ(p∞)(0)a → XΓ0(p∞)(0)a → XΓ0(p)(0)a

where the first map is a K0(p∞)-torsor (this time for real!). Turnsout ωκ,perf is a line bundle on XΓ0(p∞)(0)a.]




What happened so far.

What happened so far:

Have a space XΓ(p∞)(0)a which is a p-adic analogue of H.

It happens to be perfectoid.

Can use it to give a nice definition of p-adic modular forms Mκ

Get a notion of integral modular forms for free.

By a slightly modifying the definition, we get a larger space ofperfectoid modular forms.

What happens next:

What are they good for

Why are they called perfectoid modular forms




Coleman’s Spectral Halo




the eigencurve

For applications of p-adic modular forms, one is particularlyinterested in p-adic families of modular eigenforms.Turns out: There is a geometric object parametrising such families.

Definition

Let W be the rigid space Spf(Zp[[Z×p ]])rigη . This is the rigid spacerepresenting p-adic weights: For any p-adic field K we have

W(K ) = Homcts(Z×p ,K×).

Theorem (Coleman-Mazur, 98’)

There is a rigid space E → W such that for any p-adic field K , theK -points over any κ ∈ W(K ), κ : Z×p → K× correspond tooverconvergent p-adic modular eigenforms of finite slope (ap 6= 0).The map E → W is locally finite flat.




Coleman’s Spectral Halo Conjecture

The geometry of the eigencurve encodes deep questions aboutcongruences of eigenforms and Galois representations.Despite a lot of work and progress, many aspects are stillmysterious (e.g. what are all its smooth points?)

However, a certain chunk of it is conjecturally very simple:

Conjecture (Coleman’s Spectral Halo)

There is 1 < r such that over the boundary annulus Wr of weightswith |κ| ≥ r , the eigencurve decomposes as

Er =∞⊔i=0

Er ,i →Wr

where each Er ,i →Wr is finite flat.




Coleman’s Spectral Halo Conjecture

The geometry of the eigencurve encodes deep questions aboutcongruences of eigenforms and Galois representations.Despite a lot of work and progress, many aspects are stillmysterious (e.g. what are all its smooth points?)However, a certain chunk of it is conjecturally very simple:

Conjecture (Coleman’s Spectral Halo)

There is 1 < r such that over the boundary annulus Wr of weightswith |κ| ≥ r , the eigencurve decomposes as

Er =∞⊔i=0

Er ,i →Wr

where each Er ,i →Wr is finite flat.




Andreatta-Iovita-Pilloni: adic weight space

Really nice idea (Coleman, Andreatta-Iovita-Pilloni):Study the boundary of the eigencurve by adding boundary pointsin characteristic p: Replace weight space by the adic space

Wad = Spa(Zp[[Z×p ]],Zp[[Z×p ]])an

Crucial difference to W: For each connected component of Wad ,this also has a point corresponding to

κ : Z×p → Fp((T ))×,

and we picture this as being at the boundary of W.In fact: Can take Z×p → L× for L complete extension of Fp((T )).








κ : Z×p → Fp((T ))×,

and we picture this as being at the boundary of W.

In fact: Can take Z×p → L× for L complete extension of Fp((T )).








κ : Z×p → Fp((T ))×,

and we picture this as being at the boundary of W.In fact: Can take Z×p → L× for L complete extension of Fp((T )).




Andreatta-Iovita-Pilloni: t-adic modular forms

Turns out: One can extend the eigencurve to adic weight space!

Theorem

The eigencurve extends to a locally finite flat adic space

Ead →Wad .

For any boundary point κ ∈ Wad , there is an Fp((t))-vector spaceof t-adic modular forms Mκ such that the points of Ead over κcorrespond to t-adic overconvergent eigenforms of finite slope.

Definition of Mκ is similar to the case of p-adic modular forms.As before, one can define integral forms and perfectoid forms.Idea: There should be some sort of symmetry in the space Mκ thatexplains the Spectral Halo and related conjectures. Suggestion:




C as a perfectoid field

Recall that we denote by C an algebraically closed completeextension of Qp. One can ”tilt” C and obtain a field ofcharacteristic p as follows:

Let OC be the ring of integers of C .

Set OC [ := lim←−FOC/p where F sends x 7→ xp.

Let t ∈ OC [ be the element defined by (. . . , p1/p2, p1/p, p)

Set C [ = OC [ [1/t]. This turns out to be a field.




tilting perfectoid algebras

This works in much greater generality: There is a certain kind ofC -Banach algebras called perfectoid C -algebras. Given any suchperfectoid C -algebra R, we obtain a C [-algebra R[:

Let R+ be the ring of power-bounded elements.

Set R[+ := lim←−FR+/p where F sends x 7→ xp.

Set R[ = R[+[1/t]. This turns out to be a perfect C [-algebra.




punchline of this talk

One can do the same thing forperfectoid modular forms!⇒ Hence the name.




Tilting equivalence of perfectoid modular forms

Theorem (2019)

Let (κn)n∈N be a sequence of p-adic weights κn : Z×p → O×Cconverging to the boundary of weight space such that for all n ∈ N

κpn+1 ≡ κn mod p.

Via O×C [ = lim←−x 7→xp

(OC/p)×, this defines a weight

κ[ : Z×p → O×C [ , x 7→ (κn(x))n∈N.

Then there is an isomorphism of OC [-modules

M+,perfκ[

= lim←−F

M+,perfκn /p.




tilting equivalence of modular forms

One deduces a similar statement for usual p-adic modular forms:

Definition

For any f ∈ M+κ with q-expansion f =

∑anq

n ∈ OC [[q]], set

f (p) :=∑

apnqn ∈ OC [[q]]

Then OC [ = lim←−FOC/p induces OC [ [[q]] = lim←−

f 7→f (p)

OC/p[[q]]

Theorem (2019)

In the situation of the previous theorem, there is also aHecke-equivariant isomorphism of OC [-submodules

M+κ[

= lim←−f 7→f (p)

M+κn/p.




Conclusion

One can study the boundary of the eigencurve by using aspace of modular forms in characteristic p.

One can describe the transition from characterstic 0 tocharacteristic p by way of a tilting equivalence.

The tilting equivalence arises from perfectoid modular forms.

Cliffhanger

What does this tell us about the Spectral Halo Conjecture?


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