Dirichlet character twisted nzhang28@illinois.edu University of - - PowerPoint PPT Presentation

Dirichlet character twisted Eisenstein series and J-spectra

Ningchuan Zhang 张凝川

nzhang28@illinois.edu University of Illinois at Urbana-Champaign International Workshop on Algebraic Topology 2019 August 19, 2019

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Background Twisted J-spectra Relations with twisted Eisenstein series

Background

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Background Twisted J-spectra Relations with twisted Eisenstein series

Bernoulli numbers

Defjnition Bernoulli numbers are defjned to be the coeffjcients in the Taylor expansion: shows up in number theory: shows up in algebraic topology: is equal to the numerator of . Question Is this a coincidence?

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Background Twisted J-spectra Relations with twisted Eisenstein series

Bernoulli numbers

Defjnition Bernoulli numbers Bn are defjned to be the coeffjcients in the Taylor expansion: tet et − 1 =

∞

∑

n=0

Bn tn n!. shows up in number theory: shows up in algebraic topology: is equal to the numerator of . Question Is this a coincidence?

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Background Twisted J-spectra Relations with twisted Eisenstein series

Bernoulli numbers

Defjnition Bernoulli numbers Bn are defjned to be the coeffjcients in the Taylor expansion: tet et − 1 =

∞

∑

n=0

Bn tn n!. Bn shows up in number theory: E2k(q) = 1 − 4k B2k

∞

∑

n=1

σ2k−1(n)qn. shows up in algebraic topology: is equal to the numerator of . Question Is this a coincidence?

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Background Twisted J-spectra Relations with twisted Eisenstein series

Bernoulli numbers

Defjnition Bernoulli numbers Bn are defjned to be the coeffjcients in the Taylor expansion: tet et − 1 =

∞

∑

n=0

Bn tn n!. Bn shows up in number theory: E2k(q) = 1 − 4k B2k

∞

∑

n=1

σ2k−1(n)qn. Bn shows up in algebraic topology: ∣π4k−1(J)∣ is equal to the numerator of 4k/B2k. Question Is this a coincidence?

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Background Twisted J-spectra Relations with twisted Eisenstein series

Bernoulli numbers

Defjnition Bernoulli numbers Bn are defjned to be the coeffjcients in the Taylor expansion: tet et − 1 =

∞

∑

n=0

Bn tn n!. Bn shows up in number theory: E2k(q) = 1 − 4k B2k

∞

∑

n=1

σ2k−1(n)qn. Bn shows up in algebraic topology: ∣π4k−1(J)∣ is equal to the numerator of 4k/B2k. Question Is this a coincidence?

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Background Twisted J-spectra Relations with twisted Eisenstein series

Sketch of the answer

Answer This is not a coincidence.

1 Katz used a Riemann-Hilbert type correspondence to prove

acts trivially on

2

measures congruences of a

• representation

.

3 Chromatic resolution shows

.

4 There is a spectral sequence 5 The image of

completed at each prime is .

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Background Twisted J-spectra Relations with twisted Eisenstein series

Sketch of the answer

Answer This is not a coincidence.

1 Katz used a Riemann-Hilbert type correspondence to prove

acts trivially on

2

measures congruences of a

• representation

.

3 Chromatic resolution shows

.

4 There is a spectral sequence 5 The image of

completed at each prime is .

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Background Twisted J-spectra Relations with twisted Eisenstein series

Sketch of the answer

Answer This is not a coincidence.

1 Katz used a Riemann-Hilbert type correspondence to prove

Ek ≡ 1 mod pm ⇐ ⇒ Z×

p acts trivially on (Zp)⊗k

mod pm.

2

measures congruences of a

• representation

.

3 Chromatic resolution shows

.

4 There is a spectral sequence 5 The image of

completed at each prime is .

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Background Twisted J-spectra Relations with twisted Eisenstein series

Sketch of the answer

Answer This is not a coincidence.

1 Katz used a Riemann-Hilbert type correspondence to prove

Ek ≡ 1 mod pm ⇐ ⇒ Z×

p acts trivially on (Zp)⊗k

mod pm.

2 (M/p∞)Z× p measures congruences of a Z×

p-representation M.

3 Chromatic resolution shows

.

4 There is a spectral sequence 5 The image of

completed at each prime is .

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Background Twisted J-spectra Relations with twisted Eisenstein series

Sketch of the answer

Answer This is not a coincidence.

1 Katz used a Riemann-Hilbert type correspondence to prove

Ek ≡ 1 mod pm ⇐ ⇒ Z×

p acts trivially on (Zp)⊗k

mod pm.

2 (M/p∞)Z× p measures congruences of a Z×

p-representation M.

3 Chromatic resolution shows (M/p∞)Z× p ≃ H1

c (Z× p;M).

4 There is a spectral sequence 5 The image of

completed at each prime is .

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Background Twisted J-spectra Relations with twisted Eisenstein series

Sketch of the answer

Answer This is not a coincidence.

1 Katz used a Riemann-Hilbert type correspondence to prove

Ek ≡ 1 mod pm ⇐ ⇒ Z×

p acts trivially on (Zp)⊗k

mod pm.

2 (M/p∞)Z× p measures congruences of a Z×

p-representation M.

3 Chromatic resolution shows (M/p∞)Z× p ≃ H1

c (Z× p;M).

4 There is a spectral sequence

Es,t

2

= Hs

c(Z× p;πt(K∧ p ))

⇒ πt−s (S0

K(1)).

5 The image of

completed at each prime is .

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Background Twisted J-spectra Relations with twisted Eisenstein series

Sketch of the answer

Answer This is not a coincidence.

1 Katz used a Riemann-Hilbert type correspondence to prove

Ek ≡ 1 mod pm ⇐ ⇒ Z×

p acts trivially on (Zp)⊗k

mod pm.

2 (M/p∞)Z× p measures congruences of a Z×

p-representation M.

3 Chromatic resolution shows (M/p∞)Z× p ≃ H1

c (Z× p;M).

4 There is a spectral sequence

Es,t

2

= Hs

c(Z× p;πt(K∧ p ))

⇒ πt−s (S0

K(1)).

5 The image of J completed at each prime is S0

K(1).

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Background Twisted J-spectra Relations with twisted Eisenstein series

Generalized Bernoulli numbers and Eisenstein series

Let be a primitive Dirichlet character. Defjnitions The generalized Bernoulli numbers associated to are defjned by: The Eisenstein series associated to is defjned by: with the

• expansion of its normalization given by:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Generalized Bernoulli numbers and Eisenstein series

Let χ ∶ (Z/N)× → C× be a primitive Dirichlet character. Defjnitions The generalized Bernoulli numbers associated to are defjned by: The Eisenstein series associated to is defjned by: with the

• expansion of its normalization given by:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Generalized Bernoulli numbers and Eisenstein series

Let χ ∶ (Z/N)× → C× be a primitive Dirichlet character. Defjnitions The generalized Bernoulli numbers Bn,χ associated to χ are defjned by: Fχ(t) =

N

∑

a=1

χ(a)teat eNt − 1 =

∞

∑

n=0

Bn,χ tn n!. The Eisenstein series associated to is defjned by: with the

• expansion of its normalization given by:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Generalized Bernoulli numbers and Eisenstein series

Let χ ∶ (Z/N)× → C× be a primitive Dirichlet character. Defjnitions The generalized Bernoulli numbers Bn,χ associated to χ are defjned by: Fχ(t) =

N

∑

a=1

χ(a)teat eNt − 1 =

∞

∑

n=0

Bn,χ tn n!. The Eisenstein series associated to χ is defjned by: Gk(z;χ) ∶= ∑

(m,n)≠(0,0)

χ(n) (mNz + n)k , with the

• expansion of its normalization given by:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Generalized Bernoulli numbers and Eisenstein series

Let χ ∶ (Z/N)× → C× be a primitive Dirichlet character. Defjnitions The generalized Bernoulli numbers Bn,χ associated to χ are defjned by: Fχ(t) =

N

∑

a=1

χ(a)teat eNt − 1 =

∞

∑

n=0

Bn,χ tn n!. The Eisenstein series associated to χ is defjned by: Gk(z;χ) ∶= ∑

(m,n)≠(0,0)

χ(n) (mNz + n)k , with the q-expansion of its normalization given by: Ek(q;χ) = 1 − 2k Bk,χ−1

∞

∑

n=1

σk−1,χ−1(n)qn.

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Background Twisted J-spectra Relations with twisted Eisenstein series

Automorphic equation

Proposition is a modular form of weight and level .Furthermore, for Proposition The automorphic equation above is equivalent to

• rep

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Background Twisted J-spectra Relations with twisted Eisenstein series

Automorphic equation

Proposition Ek,χ is a modular form of weight k and level Γ1(N). Furthermore, for Proposition The automorphic equation above is equivalent to

• rep

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Background Twisted J-spectra Relations with twisted Eisenstein series

Automorphic equation

Proposition Ek,χ is a modular form of weight k and level Γ1(N).Furthermore, Ek(γ ⋅ z;χ) = χ(a)(cz + d)kEk(z;χ), for γ = (a b c d) ∈ Γ0(N). Proposition The automorphic equation above is equivalent to

• rep

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Background Twisted J-spectra Relations with twisted Eisenstein series

Automorphic equation

Proposition Ek,χ is a modular form of weight k and level Γ1(N).Furthermore, Ek(γ ⋅ z;χ) = χ(a)(cz + d)kEk(z;χ), for γ = (a b c d) ∈ Γ0(N). Proposition The automorphic equation above is equivalent to Ek,χ ∈ Hom(Z/N)×-rep (Cχ,H0(Mell(Γ1(N)),ωk)).

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Background Twisted J-spectra Relations with twisted Eisenstein series

Twisted J-spectra

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Background Twisted J-spectra Relations with twisted Eisenstein series

J-spectra with level structures

Let be the moduli stack over

• f formal groups of

height at all primes with

• level structure, that is

We can construct a sheaf of

• ring spectra
• ver

, whose global section is

• local sphere. From the local structures
• f

, we can defjne Here .

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Background Twisted J-spectra Relations with twisted Eisenstein series

J-spectra with level structures

Let Mmult(N) be the moduli stack over Z of formal groups of height 1 at all primes with µN-level structure, that is (Mmult(N))(R) ∶= {( ̂ G/R,η ∶ µN

∼

• → ̂

G[N])}. We can construct a sheaf of

• ring spectra
• ver

, whose global section is

• local sphere. From the local structures
• f

, we can defjne Here .

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Background Twisted J-spectra Relations with twisted Eisenstein series

J-spectra with level structures

Let Mmult(N) be the moduli stack over Z of formal groups of height 1 at all primes with µN-level structure, that is (Mmult(N))(R) ∶= {( ̂ G/R,η ∶ µN

∼

• → ̂

G[N])}. We can construct a sheaf of E∞-ring spectra Otop over Mmult, whose global section is K-local sphere. From the local structures

• f

, we can defjne Here .

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Background Twisted J-spectra Relations with twisted Eisenstein series

J-spectra with level structures

Let Mmult(N) be the moduli stack over Z of formal groups of height 1 at all primes with µN-level structure, that is (Mmult(N))(R) ∶= {( ̂ G/R,η ∶ µN

∼

• → ̂

G[N])}. We can construct a sheaf of E∞-ring spectra Otop over Mmult, whose global section is K-local sphere. From the local structures

• f µN, we can defjne

Here .

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Background Twisted J-spectra Relations with twisted Eisenstein series

J-spectra with level structures

Let Mmult(N) be the moduli stack over Z of formal groups of height 1 at all primes with µN-level structure, that is (Mmult(N))(R) ∶= {( ̂ G/R,η ∶ µN

∼

• → ̂

G[N])}. We can construct a sheaf of E∞-ring spectra Otop over Mmult, whose global section is K-local sphere. From the local structures

• f µN, we can defjne

J ∶= S0

K

∏p S0

K/p

S0

Q

(∏p S0

K/p) Q

⌟ Here .

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Background Twisted J-spectra Relations with twisted Eisenstein series

J-spectra with level structures

Let Mmult(N) be the moduli stack over Z of formal groups of height 1 at all primes with µN-level structure, that is (Mmult(N))(R) ∶= {( ̂ G/R,η ∶ µN

∼

• → ̂

G[N])}. We can construct a sheaf of E∞-ring spectra Otop over Mmult, whose global section is K-local sphere. From the local structures

• f µN, we can defjne

J ∶= S0

K

∏p S0

K/p

S0

Q

(∏p S0

K/p) Q

⌟ J(N) ∏p S0

K/p (pvp(N))

S0

Q

(∏p S0

K/p (pvp(N))) Q

⌟ Here S0

K/p(pv) ∶= (K∧ p ) h(1+pvZp).

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Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet characters as integral representations

Let be a Dirichlet character of conductor and be the

• subalgebra of

generated by the image of . The Dirichlet character realizes as a

• representation in fjnite free
• modules.

Example When and sending a generator to . Then since . This is a free

• module of rank

with basis . The minimal polynomial of is , from which we deduce the matrix representation of with respect the basis

• f

is:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet characters as integral representations

Let χ be a Dirichlet character of conductor N and be the

• subalgebra of

generated by the image of . The Dirichlet character realizes as a

• representation in fjnite free
• modules.

Example When and sending a generator to . Then since . This is a free

• module of rank

with basis . The minimal polynomial of is , from which we deduce the matrix representation of with respect the basis

• f

is:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet characters as integral representations

Let χ be a Dirichlet character of conductor N and Z[χ] be the Z-subalgebra of C generated by the image of χ. The Dirichlet character χ realizes Z[χ] as a (Z/N)×-representation in fjnite free Z-modules. Example When and sending a generator to . Then since . This is a free

• module of rank

with basis . The minimal polynomial of is , from which we deduce the matrix representation of with respect the basis

• f

is:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet characters as integral representations

Let χ be a Dirichlet character of conductor N and Z[χ] be the Z-subalgebra of C generated by the image of χ. The Dirichlet character χ realizes Z[χ] as a (Z/N)×-representation in fjnite free Z-modules. Example When N = 7 and χ ∶ (Z/7)× → C× sending a generator 3 ∈ (Z/7)× to ζ6 ∈ C×. Then Z[χ] ≃ Z[ζ6] since (Z/7)× ≃ Z/6. This is a free Z-module of rank 2 with basis {1,ζ6}. The minimal polynomial of is , from which we deduce the matrix representation of with respect the basis

• f

is:

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Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet characters as integral representations

Let χ be a Dirichlet character of conductor N and Z[χ] be the Z-subalgebra of C generated by the image of χ. The Dirichlet character χ realizes Z[χ] as a (Z/N)×-representation in fjnite free Z-modules. Example When N = 7 and χ ∶ (Z/7)× → C× sending a generator 3 ∈ (Z/7)× to ζ6 ∈ C×. Then Z[χ] ≃ Z[ζ6] since (Z/7)× ≃ Z/6. This is a free Z-module of rank 2 with basis {1,ζ6}. The minimal polynomial of ζ6 is Φ6(x) = x2 − x + 1, from which we deduce the matrix representation of χ(3) with respect the basis {1,ζ6} of Z[χ] is: χ(3) = (0 1 1 −1).

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Background Twisted J-spectra Relations with twisted Eisenstein series

Replacing a homotopy action with a topological one

Warning The action of

• n

induced by lifts to a homotopy action on the Moore spectrum . It DOES NOT rigidify to a topological action in general, e.g. the previous example. One solution for some . By an obstruction theory of Cooke, the homotopy action of

• n

induced by is equivalent to a topological action. Some good cases When , the homotopy action on induced by is equivalent to a topological one, e.g. when with for being a Fermat prime.

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Background Twisted J-spectra Relations with twisted Eisenstein series

Replacing a homotopy action with a topological one

Warning The action of (Z/N)× on Z[χ] induced by χ lifts to a homotopy action on the Moore spectrum M(Z[χ]). It DOES NOT rigidify to a topological action in general, e.g. the previous example. One solution for some . By an obstruction theory of Cooke, the homotopy action of

• n

induced by is equivalent to a topological action. Some good cases When , the homotopy action on induced by is equivalent to a topological one, e.g. when with for being a Fermat prime.

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Background Twisted J-spectra Relations with twisted Eisenstein series

Replacing a homotopy action with a topological one

Warning The action of (Z/N)× on Z[χ] induced by χ lifts to a homotopy action on the Moore spectrum M(Z[χ]). It DOES NOT rigidify to a topological action in general, e.g. the previous example. One solution for some . By an obstruction theory of Cooke, the homotopy action of

• n

induced by is equivalent to a topological action. Some good cases When , the homotopy action on induced by is equivalent to a topological one, e.g. when with for being a Fermat prime.

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Background Twisted J-spectra Relations with twisted Eisenstein series

Replacing a homotopy action with a topological one

Warning The action of (Z/N)× on Z[χ] induced by χ lifts to a homotopy action on the Moore spectrum M(Z[χ]). It DOES NOT rigidify to a topological action in general, e.g. the previous example. One solution Z[χ] = Z[ζn] for some n. By an obstruction theory of Cooke, the homotopy action of (Z/N)× on M(Z[1/n,ζn]) induced by χ is equivalent to a topological action. Some good cases When , the homotopy action on induced by is equivalent to a topological one, e.g. when with for being a Fermat prime.

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Background Twisted J-spectra Relations with twisted Eisenstein series

Replacing a homotopy action with a topological one

Warning The action of (Z/N)× on Z[χ] induced by χ lifts to a homotopy action on the Moore spectrum M(Z[χ]). It DOES NOT rigidify to a topological action in general, e.g. the previous example. One solution Z[χ] = Z[ζn] for some n. By an obstruction theory of Cooke, the homotopy action of (Z/N)× on M(Z[1/n,ζn]) induced by χ is equivalent to a topological action. Some good cases When Z[χ] = Z[ζ2n], the homotopy action on M(Z[χ]) induced by χ is equivalent to a topological one, e.g. when N = 2l ⋅ p with p = 22m + 1 for 0 ≤ m ≤ 4 being a Fermat prime.

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Background Twisted J-spectra Relations with twisted Eisenstein series

The χ-twisted J-spectrum

Motivated by the Dirichlet equivariance of twisted Eisenstein series, we construct the

• twisted
• spectrum by

Construction Let , defjne Here, acts on the wedge product diagonally. Remark means the homotopy

• eigen-spectrum.

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Background Twisted J-spectra Relations with twisted Eisenstein series

The χ-twisted J-spectrum

Motivated by the Dirichlet equivariance of twisted Eisenstein series, we construct the χ-twisted J-spectrum by Construction Let , defjne Here, acts on the wedge product diagonally. Remark means the homotopy

• eigen-spectrum.

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Background Twisted J-spectra Relations with twisted Eisenstein series

The χ-twisted J-spectrum

Motivated by the Dirichlet equivariance of twisted Eisenstein series, we construct the χ-twisted J-spectrum by Construction Let Z[χ] = Z[ζn], defjne J(N)hχ ∶=(J(N) ∧ M(Z[1/n,χ−1]))h(Z/N)×. Here, (Z/N)× acts on the wedge product diagonally. Remark means the homotopy

• eigen-spectrum.

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Background Twisted J-spectra Relations with twisted Eisenstein series

The χ-twisted J-spectrum

Motivated by the Dirichlet equivariance of twisted Eisenstein series, we construct the χ-twisted J-spectrum by Construction Let Z[χ] = Z[ζn], defjne J(N)hχ ∶=(J(N) ∧ M(Z[1/n,χ−1]))h(Z/N)×. Here, (Z/N)× acts on the wedge product diagonally. Remark (−)hχ means the homotopy χ-eigen-spectrum.

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Background Twisted J-spectra Relations with twisted Eisenstein series

p-adic decompositions

Proposition Let be a Dirichlet character of conductor . The

• completion of the
• twisted
• spectrum decomposes as:

where is the Teichmüller character. Remark corresponds to

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Background Twisted J-spectra Relations with twisted Eisenstein series

p-adic decompositions

Proposition Let χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p > 2. The

• completion of the
• twisted
• spectrum decomposes as:

where is the Teichmüller character. Remark corresponds to

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Background Twisted J-spectra Relations with twisted Eisenstein series

p-adic decompositions

Proposition Let χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p > 2. The p-completion of the χ-twisted J-spectrum decomposes as: (J(p)hχ)

∧ p ≃

⋁

0≤a≤p−2 ker ωa=ker χ

(S0

K(1)(p)) hωa

, where ω ∶ (Z/p)× → Z×

p is the Teichmüller character.

Remark corresponds to

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

p-adic decompositions

Proposition Let χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p > 2. The p-completion of the χ-twisted J-spectrum decomposes as: (J(p)hχ)

∧ p ≃

⋁

0≤a≤p−2 ker ωa=ker χ

(S0

K(1)(p)) hωa

, where ω ∶ (Z/p)× → Z×

p is the Teichmüller character.

Remark (S0

K(1)(p)) hωa

∈ Picalg

K(1) ≃ End(Z× p) corresponds to

Z×

p

(Z/p)× Z×

p. ω−a

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

The homotopy eigen spectral sequence

Proposition The

• page of the HFPSS to compute

can be identifjed with where acts on by . For

• adic Dirichlet

characters, when , we further have and where acts on by .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

The homotopy eigen spectral sequence

Proposition The E2-page of the HFPSS to compute π∗ (J(N)hχ) can be identifjed with Es,t

2

≃ Exts

Z[(Z/N)×] (Z[1/n,χ],πt(J(N)))

⇒ πt−s (J(N)hχ), where (Z/N)× acts on Z[1/n,χ] by χ. For

• adic Dirichlet

characters, when , we further have and where acts on by .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

The homotopy eigen spectral sequence

Proposition The E2-page of the HFPSS to compute π∗ (J(N)hχ) can be identifjed with Es,t

2

≃ Exts

Z[(Z/N)×] (Z[1/n,χ],πt(J(N)))

⇒ πt−s (J(N)hχ), where (Z/N)× acts on Z[1/n,χ] by χ. For p-adic Dirichlet characters, when N = p, we further have Zp[χ] = Zp and Es,t

2

≃ Exts

ZpZ×

p (Zp,πt (K∧

p ))

⇒ πt−s ((S0

K(1)(p)) hχ

), where Z×

p acts on Zp by Z× p ↠ (Z/p)× χ

• → Z×

p.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Relations with twisted Eisenstein series

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

More numeric coincidences

Let p > 2 and χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p. Bk,χ ∈ Q[χ] is an algebraic number. Using SageMath, one can check that when , if else The HFPSS computation shows if else When is trivial, ifg and we recover the classical numeric coincidence. We will explain this coincidence via congruences of twisted Eisenstein series .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

More numeric coincidences

Let p > 2 and χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p. Bk,χ ∈ Q[χ] is an algebraic number. Using SageMath, one can check that when (−1)k = χ(−1), vp (Norm( 2k Bk,χ−1 )) = { vp(k) + 1, if kerωk = kerχ; 0, else. The HFPSS computation shows if else When is trivial, ifg and we recover the classical numeric coincidence. We will explain this coincidence via congruences of twisted Eisenstein series .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

More numeric coincidences

Let p > 2 and χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p. Bk,χ ∈ Q[χ] is an algebraic number. Using SageMath, one can check that when (−1)k = χ(−1), vp (Norm( 2k Bk,χ−1 )) = { vp(k) + 1, if kerωk = kerχ; 0, else. The HFPSS computation shows π2k−1 ((J(p)hχ)

∧ p) = { Z/pvp(k)+1,

if kerωk = kerχ; 0, else. When is trivial, ifg and we recover the classical numeric coincidence. We will explain this coincidence via congruences of twisted Eisenstein series .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

More numeric coincidences

Let p > 2 and χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p. Bk,χ ∈ Q[χ] is an algebraic number. Using SageMath, one can check that when (−1)k = χ(−1), vp (Norm( 2k Bk,χ−1 )) = { vp(k) + 1, if kerωk = kerχ; 0, else. The HFPSS computation shows π2k−1 ((J(p)hχ)

∧ p) = { Z/pvp(k)+1,

if kerωk = kerχ; 0, else. When χ is trivial, kerωk = kerχ = (Z/p)× ifg (p − 1) ∣ k and we recover the classical numeric coincidence. We will explain this coincidence via congruences of twisted Eisenstein series .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

More numeric coincidences

Let p > 2 and χ ∶ (Z/p)× → C× be a Dirichlet character of conductor p. Bk,χ ∈ Q[χ] is an algebraic number. Using SageMath, one can check that when (−1)k = χ(−1), vp (Norm( 2k Bk,χ−1 )) = { vp(k) + 1, if kerωk = kerχ; 0, else. The HFPSS computation shows π2k−1 ((J(p)hχ)

∧ p) = { Z/pvp(k)+1,

if kerωk = kerχ; 0, else. When χ is trivial, kerωk = kerχ = (Z/p)× ifg (p − 1) ∣ k and we recover the classical numeric coincidence. We will explain this coincidence via congruences of twisted Eisenstein series Ek,χ.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Strategy

Fix a Dirichlet character with .

1 Consider the stack

and the

• level

structures: Study the congruence of .

2 Reformulate a Riemann-Hilbert correspondence to show

is a trivial

• rep mod

3 For

, use chromatic resolution to show

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Strategy

Fix a Dirichlet character χ ∶ (Z/N)× → C×

p with vp(N) = v.

1 Consider the stack Mord

ell and the (µN)∧ p ≃ µpv-level

structures: Mord

ell (pv)

B(1 + pvZp) Mord

ell

BZ×

p

⌟ Study the congruence of .

2 Reformulate a Riemann-Hilbert correspondence to show

is a trivial

• rep mod

3 For

, use chromatic resolution to show

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Strategy

Fix a Dirichlet character χ ∶ (Z/N)× → C×

p with vp(N) = v.

1 Consider the stack Mord

ell and the (µN)∧ p ≃ µpv-level

structures: Mord

ell (pv)

B(1 + pvZp) Mord

ell

BZ×

p

⌟ Study the congruence of Ek,χ ∈ HomZp[(Z/N)×] (Zp[χ],H0(Mord

ell (pv),ω⊗k)).

2 Reformulate a Riemann-Hilbert correspondence to show

is a trivial

• rep mod

3 For

, use chromatic resolution to show

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Strategy

Fix a Dirichlet character χ ∶ (Z/N)× → C×

p with vp(N) = v.

1 Consider the stack Mord

ell and the (µN)∧ p ≃ µpv-level

structures: Mord

ell (pv)

B(1 + pvZp) Mord

ell

BZ×

p

⌟ Study the congruence of Ek,χ ∈ HomZp[(Z/N)×] (Zp[χ],H0(Mord

ell (pv),ω⊗k)).

2 Reformulate a Riemann-Hilbert correspondence to show

Ek,χ ≡ 1 mod I ⊴ Zp[χ] ⇐ ⇒ Z⊗k

p [χ−1] is a trivial Z× p-rep mod I.

3 For

, use chromatic resolution to show

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Strategy

Fix a Dirichlet character χ ∶ (Z/N)× → C×

p with vp(N) = v.

1 Consider the stack Mord

ell and the (µN)∧ p ≃ µpv-level

structures: Mord

ell (pv)

B(1 + pvZp) Mord

ell

BZ×

p

⌟ Study the congruence of Ek,χ ∈ HomZp[(Z/N)×] (Zp[χ],H0(Mord

ell (pv),ω⊗k)).

2 Reformulate a Riemann-Hilbert correspondence to show

Ek,χ ≡ 1 mod I ⊴ Zp[χ] ⇐ ⇒ Z⊗k

p [χ−1] is a trivial Z× p-rep mod I.

3 For M = Z⊗k

p [χ−1], use chromatic resolution to show

colim

m

((M/pm)Z×

p) ≃ (colim

m

(M/pm))

Z×

p

≃ H1

c (Z× p;M).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruence and group cohomology

Proposition Let be a

• representation in fjnite free
• modules with no

non-zero fjxed points, then . Proof. Apply to the short exact sequence: we get an isomorphism . The claim now follows from the isomorphism

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruence and group cohomology

Proposition Let M be a Z×

p-representation in fjnite free Zp-modules with no

non-zero fjxed points, then H1

c (Z× p;M) ≃ colimm ((M/pm)Z×

p).

Proof. Apply to the short exact sequence: we get an isomorphism . The claim now follows from the isomorphism

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruence and group cohomology

Proposition Let M be a Z×

p-representation in fjnite free Zp-modules with no

non-zero fjxed points, then H1

c (Z× p;M) ≃ colimm ((M/pm)Z×

p).

Proof. Apply H∗

c (Z× p;−) to the short exact sequence:

M p−1M M/p∞ 0, we get an isomorphism (M/p∞)Z×

p ≃ H1

c (Z× p;M).

The claim now follows from the isomorphism

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruence and group cohomology

Proposition Let M be a Z×

p-representation in fjnite free Zp-modules with no

non-zero fjxed points, then H1

c (Z× p;M) ≃ colimm ((M/pm)Z×

p).

Proof. Apply H∗

c (Z× p;−) to the short exact sequence:

M p−1M M/p∞ 0, we get an isomorphism (M/p∞)Z×

p ≃ H1

c (Z× p;M). The claim now

follows from the isomorphism colim

m

((M/pm)Z×

p)

∼

• → (colim

m

M/pm)

Z×

p

≃ (M/p∞)Z×

p.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

A Riemann-Hilbert correspondence

Let be a perfect fjeld of characteristic . Let be a fmat

• algebra such that

is an integrally closed domain over and that admits an endomorphism that lifts the Frobenius

• n

(the

• th power map). Then we have:

Theorem The following categories are equivalent: Projective

• modules
• f

rank with Continuous actions on

• divisible formal groups over
• f dimension and height
• torsors
• ver

I V III II IV

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

A Riemann-Hilbert correspondence

Let κ be a perfect fjeld of characteristic p. Let A be a fmat W(κ)-algebra such that A/p is an integrally closed domain over κ and that A admits an endomorphism ϕ ∶ A → A that lifts the Frobenius ϕ0 on A/p (the p-th power map). Then we have: Theorem The following categories are equivalent: Projective

• modules
• f

rank with Continuous actions on

• divisible formal groups over
• f dimension and height
• torsors
• ver

I V III II IV

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

A Riemann-Hilbert correspondence

Let κ be a perfect fjeld of characteristic p. Let A be a fmat W(κ)-algebra such that A/p is an integrally closed domain over κ and that A admits an endomorphism ϕ ∶ A → A that lifts the Frobenius ϕ0 on A/p (the p-th power map). Then we have: Theorem The following categories are equivalent: { Projective A-modules M of rank r with F ∶ ϕ∗M

∼

• → M }

{ Continuous π´

et 1 (A)

actions on Z⊕r

p

} { p-divisible formal groups over Spf A

• f dimension and height r

} { GLr(Zp)-torsors

• ver Spf A

}

I V III II IV

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruences in the RH correspondence

Proposition Let be a one-dimensional formal group of height

• ver

. Denote the Dieudonné module associated to by and the Galois descent data by . Then the followings are equivalent:

1

.

2 There is a generator

such that .

3

is trivial mod , i.e. the image of is contained in . In particular, when , the followings are equivalent:

1

.

2 There is a generator

such that .

3

is the trivial representation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruences in the RH correspondence

Proposition Let ̂ G be a one-dimensional formal group of height 1 over A. Denote the Dieudonné module associated to ̂ G by (M,F ∶ ϕ∗M

∼

• → M) and the Galois descent data by

ρ ∈ H1(π´

et 1 (A);Z× p).

Then the followings are equivalent:

1

.

2 There is a generator

such that .

3

is trivial mod , i.e. the image of is contained in . In particular, when , the followings are equivalent:

1

.

2 There is a generator

such that .

3

is the trivial representation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruences in the RH correspondence

Proposition Let ̂ G be a one-dimensional formal group of height 1 over A. Denote the Dieudonné module associated to ̂ G by (M,F ∶ ϕ∗M

∼

• → M) and the Galois descent data by

ρ ∈ H1(π´

et 1 (A);Z× p). Then the followings are equivalent:

1

̂ G[pm] ≃ µpm.

2 There is a generator γ ∈ M such that Fγ ≡ γ mod pm. 3 ρ is trivial mod pm, i.e. the image of ρ ∶ π´

et 1 (A) → Z× p is

contained in 1 + pmZp ⊆ Z×

p.

In particular, when , the followings are equivalent:

1

.

2 There is a generator

such that .

3

is the trivial representation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Congruences in the RH correspondence

Proposition Let ̂ G be a one-dimensional formal group of height 1 over A. Denote the Dieudonné module associated to ̂ G by (M,F ∶ ϕ∗M

∼

• → M) and the Galois descent data by

ρ ∈ H1(π´

et 1 (A);Z× p). Then the followings are equivalent:

1

̂ G[pm] ≃ µpm.

2 There is a generator γ ∈ M such that Fγ ≡ γ mod pm. 3 ρ is trivial mod pm, i.e. the image of ρ ∶ π´

et 1 (A) → Z× p is

contained in 1 + pmZp ⊆ Z×

p.

In particular, when m = ∞, the followings are equivalent:

1

̂ G ≃ ̂ Gm.

2 There is a generator γ ∈ M such that Fγ = γ. 3 ρ is the trivial representation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet equivariance and Galois descent

Construction As the invertible sheaf

• ver

is the pullback of the invertible sheaf

• ver

, there is a canonical isomorphism where . Then a

• adic Dirichlet

character

• f conductor

induces a Galois descent data: Denote the resulting sheaf over by . Lemma .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet equivariance and Galois descent

Construction As the invertible sheaf ω⊗k over Mord

ell (pv) is the pullback of the

invertible sheaf ω⊗k over Mord

ell , there is a canonical isomorphism

fσ ∶ ω⊗k

∼

• → σ∗ω⊗k,

σ ∈ AutMord

ell (Mord

ell (pv)) ≃ (Z/pv)× ,

where (fσ) = 1 ∈ H1((Z/pv)× ;Z×

p).

Then a

• adic Dirichlet

character

• f conductor

induces a Galois descent data: Denote the resulting sheaf over by . Lemma .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet equivariance and Galois descent

Construction As the invertible sheaf ω⊗k over Mord

ell (pv) is the pullback of the

invertible sheaf ω⊗k over Mord

ell , there is a canonical isomorphism

fσ ∶ ω⊗k

∼

• → σ∗ω⊗k,

σ ∈ AutMord

ell (Mord

ell (pv)) ≃ (Z/pv)× ,

where (fσ) = 1 ∈ H1((Z/pv)× ;Z×

p). Then a p-adic Dirichlet

character χ of conductor pv induces a Galois descent data: 1 ⊗ χ−1(σ) ∶ ω⊗k ⊗Zp Zp[χ−1]

∼

• → (σ ⊗ 1)∗(ω⊗k ⊗Zp Zp[χ−1]).

Denote the resulting sheaf over Mord

ell by Fk,χ.

Lemma .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Dirichlet equivariance and Galois descent

Construction As the invertible sheaf ω⊗k over Mord

ell (pv) is the pullback of the

invertible sheaf ω⊗k over Mord

ell , there is a canonical isomorphism

fσ ∶ ω⊗k

∼

• → σ∗ω⊗k,

σ ∈ AutMord

ell (Mord

ell (pv)) ≃ (Z/pv)× ,

where (fσ) = 1 ∈ H1((Z/pv)× ;Z×

p). Then a p-adic Dirichlet

character χ of conductor pv induces a Galois descent data: 1 ⊗ χ−1(σ) ∶ ω⊗k ⊗Zp Zp[χ−1]

∼

• → (σ ⊗ 1)∗(ω⊗k ⊗Zp Zp[χ−1]).

Denote the resulting sheaf over Mord

ell by Fk,χ.

Lemma HomZp[(Z/pv)×] (Zp[χ],H0(Mord

ell (pv),ω⊗k)) ≃ H0(Mord ell ,Fk,χ).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Igusa’s theorem

Using descent, we can endow with an isomorphism .Let be the

• representation

corresponding to . Notice factors as where corresponds to

• ver

. Further, we have Theorem (Igusa) is surjective. Proposition ifg the

• representation

is trivial mod .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Igusa’s theorem

Using descent, we can endow Fk,χ with an isomorphism F k,χ ∶ Fk,χ

∼

• → ϕ∗Fk,χ.

Let be the

• representation

corresponding to . Notice factors as where corresponds to

• ver

. Further, we have Theorem (Igusa) is surjective. Proposition ifg the

• representation

is trivial mod .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Igusa’s theorem

Using descent, we can endow Fk,χ with an isomorphism F k,χ ∶ Fk,χ

∼

• → ϕ∗Fk,χ.Let ρk,χ be the π´

et 1 (Mord ell )-representation

corresponding to (Fk,χ,F k,χ). Notice ρk,χ factors as ρk,χ ∶ π´

et 1 (Mord ell )

Z×

p

(Zp[χ])× AutZp(Zp[χ]),

ρ (−)k⋅̃ χ−1

where ρ corresponds to (ω,F) over Mord

ell .

Further, we have Theorem (Igusa) is surjective. Proposition ifg the

• representation

is trivial mod .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Igusa’s theorem

Using descent, we can endow Fk,χ with an isomorphism F k,χ ∶ Fk,χ

∼

• → ϕ∗Fk,χ.Let ρk,χ be the π´

et 1 (Mord ell )-representation

corresponding to (Fk,χ,F k,χ). Notice ρk,χ factors as ρk,χ ∶ π´

et 1 (Mord ell )

Z×

p

(Zp[χ])× AutZp(Zp[χ]),

ρ (−)k⋅̃ χ−1

where ρ corresponds to (ω,F) over Mord

ell . Further, we have

Theorem (Igusa) ρ ∶ π´

et 1 (Mord ell ) → Z× p is surjective.

Proposition ifg the

• representation

is trivial mod .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Igusa’s theorem

Using descent, we can endow Fk,χ with an isomorphism F k,χ ∶ Fk,χ

∼

• → ϕ∗Fk,χ.Let ρk,χ be the π´

et 1 (Mord ell )-representation

corresponding to (Fk,χ,F k,χ). Notice ρk,χ factors as ρk,χ ∶ π´

et 1 (Mord ell )

Z×

p

(Zp[χ])× AutZp(Zp[χ]),

ρ (−)k⋅̃ χ−1

where ρ corresponds to (ω,F) over Mord

ell . Further, we have

Theorem (Igusa) ρ ∶ π´

et 1 (Mord ell ) → Z× p is surjective.

Proposition Ek,χ ≡ 1 mod I ⊴ Zp[χ] ifg the Z×

p-representation Z⊗k p

⊗ Zp[χ−1] is trivial mod I.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let be a

• adic Dirichlet character. The

followings are equivalent:

1

.

2 Over

, there is such that:

The generates the module

• ver

; For any , ; .

3 The induced

• action on

is trivial mod .

4 The

• representation

is trivial mod .

5 There is a surjection

. Moreover, is the maximal congruence ifg

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let χ ∶ (Z/N)× → C×

p be a p-adic Dirichlet character. The

followings are equivalent:

1

.

2 Over

, there is such that:

The generates the module

• ver

; For any , ; .

3 The induced

• action on

is trivial mod .

4 The

• representation

is trivial mod .

5 There is a surjection

. Moreover, is the maximal congruence ifg

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let χ ∶ (Z/N)× → C×

p be a p-adic Dirichlet character. The

followings are equivalent:

1 Ek,χ ≡ 1 mod I ⊴ Zp[χ]. 2 Over

, there is such that:

The generates the module

• ver

; For any , ; .

3 The induced

• action on

is trivial mod .

4 The

• representation

is trivial mod .

5 There is a surjection

. Moreover, is the maximal congruence ifg

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let χ ∶ (Z/N)× → C×

p be a p-adic Dirichlet character. The

followings are equivalent:

1 Ek,χ ≡ 1 mod I ⊴ Zp[χ]. 2 Over Mord

ell (pv), there is γ ∈ ω⊗k ⊗Zp Zp[χ] such that:

The γ generates the module ω⊗k ⊗Zp Zp[χ]/I over Mord

ell (pv) ×Spf Zp Spf Zp[χ];

For any g ∈ (Z/pv)×, g ⋅ γ = χ(g)γ ; (F ⊗ 1)(γ) ≡ γ mod I.

3 The induced

• action on

is trivial mod .

4 The

• representation

is trivial mod .

5 There is a surjection

. Moreover, is the maximal congruence ifg

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let χ ∶ (Z/N)× → C×

p be a p-adic Dirichlet character. The

followings are equivalent:

1 Ek,χ ≡ 1 mod I ⊴ Zp[χ]. 2 Over Mord

ell (pv), there is γ ∈ ω⊗k ⊗Zp Zp[χ] such that:

The γ generates the module ω⊗k ⊗Zp Zp[χ]/I over Mord

ell (pv) ×Spf Zp Spf Zp[χ];

For any g ∈ (Z/pv)×, g ⋅ γ = χ(g)γ ; (F ⊗ 1)(γ) ≡ γ mod I.

3 The induced π´

et 1 (Mord ell )-action on Zp[χ] is trivial mod I.

4 The

• representation

is trivial mod .

5 There is a surjection

. Moreover, is the maximal congruence ifg

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let χ ∶ (Z/N)× → C×

p be a p-adic Dirichlet character. The

followings are equivalent:

1 Ek,χ ≡ 1 mod I ⊴ Zp[χ]. 2 Over Mord

ell (pv), there is γ ∈ ω⊗k ⊗Zp Zp[χ] such that:

The γ generates the module ω⊗k ⊗Zp Zp[χ]/I over Mord

ell (pv) ×Spf Zp Spf Zp[χ];

For any g ∈ (Z/pv)×, g ⋅ γ = χ(g)γ ; (F ⊗ 1)(γ) ≡ γ mod I.

3 The induced π´

et 1 (Mord ell )-action on Zp[χ] is trivial mod I.

4 The Z×

p-representation Z⊗k p [χ−1] is trivial mod I.

5 There is a surjection

. Moreover, is the maximal congruence ifg

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let χ ∶ (Z/N)× → C×

p be a p-adic Dirichlet character. The

followings are equivalent:

1 Ek,χ ≡ 1 mod I ⊴ Zp[χ]. 2 Over Mord

ell (pv), there is γ ∈ ω⊗k ⊗Zp Zp[χ] such that:

The γ generates the module ω⊗k ⊗Zp Zp[χ]/I over Mord

ell (pv) ×Spf Zp Spf Zp[χ];

For any g ∈ (Z/pv)×, g ⋅ γ = χ(g)γ ; (F ⊗ 1)(γ) ≡ γ mod I.

3 The induced π´

et 1 (Mord ell )-action on Zp[χ] is trivial mod I.

4 The Z×

p-representation Z⊗k p [χ−1] is trivial mod I.

5 There is a surjection H1 (Z×

p;Z⊗k p [χ−1]) ↠ Zp[χ]/I.

Moreover, is the maximal congruence ifg

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Recap

Theorem Let χ ∶ (Z/N)× → C×

p be a p-adic Dirichlet character. The

followings are equivalent:

1 Ek,χ ≡ 1 mod I ⊴ Zp[χ]. 2 Over Mord

ell (pv), there is γ ∈ ω⊗k ⊗Zp Zp[χ] such that:

The γ generates the module ω⊗k ⊗Zp Zp[χ]/I over Mord

ell (pv) ×Spf Zp Spf Zp[χ];

For any g ∈ (Z/pv)×, g ⋅ γ = χ(g)γ ; (F ⊗ 1)(γ) ≡ γ mod I.

3 The induced π´

et 1 (Mord ell )-action on Zp[χ] is trivial mod I.

4 The Z×

p-representation Z⊗k p [χ−1] is trivial mod I.

5 There is a surjection H1 (Z×

p;Z⊗k p [χ−1]) ↠ Zp[χ]/I.

Moreover, Ek,χ ≡ 1 mod I ⊴ Zp[χ] is the maximal congruence ifg H1 (Z×

p;Z⊗k p [χ−1]) ≃ Zp[χ]/I.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background Twisted J-spectra Relations with twisted Eisenstein series

Thanks for your attention!