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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 formsColeman’s Spectral Halo
tilting equivalences
p-adic modular forms
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
p-adic 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
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
p-adic 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
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
p-adic 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
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
p-adic 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
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
p-adic 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
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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!
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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!
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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!
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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!
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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).
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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).
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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).
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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).
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.]
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.]
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
Coleman’s Spectral Halo
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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 )).
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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 )).
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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 )).
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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:
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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:
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
tilting equivalences
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
punchline of this talk
One can do the same thing forperfectoid modular forms!⇒ Hence the name.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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.
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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?
Ben Heuer Perfectoid modular forms
p-adic modular formsColeman’s Spectral Halo
tilting equivalences
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?
Ben Heuer Perfectoid modular forms