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Algebraic functional equation for Selmer groups

Fields Institute Number Theory Seminar Somnath Jha

IIT Kanpur

23 November 2020

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 0 / 12

E(Q) := {(X, Y) ∈ Q×Q|Y 2 = X 3+aX +b, a, b ∈ Q, 4a3+27b2 = 0}+[0, 1, 0]

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 1 / 12

E(Q) := {(X, Y) ∈ Q×Q|Y 2 = X 3+aX +b, a, b ∈ Q, 4a3+27b2 = 0}+[0, 1, 0]

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 1 / 12

E(Q) := {(X, Y) ∈ Q×Q|Y 2 = X 3+aX +b, a, b ∈ Q, 4a3+27b2 = 0}+[0, 1, 0] ∃ an abelian group law on E(Q).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 1 / 12

E(Q) := {(X, Y) ∈ Q×Q|Y 2 = X 3+aX +b, a, b ∈ Q, 4a3+27b2 = 0}+[0, 1, 0] ∃ an abelian group law on E(Q).

Theorem (Mordell-Weil)

E(Q) : a finitely generated abelian group.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 1 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2, where Lp(t) = 1 − (p + 1 − #˜ E(Fp))t + pt2 for all but finitely may p.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2, where Lp(t) = 1 − (p + 1 − #˜ E(Fp))t + pt2 for all but finitely may p.

Theorem (functional equation for LE)

˜ LE(s) := Ns/2

E

(2π)−sΓ(s)LE(s)

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2, where Lp(t) = 1 − (p + 1 − #˜ E(Fp))t + pt2 for all but finitely may p.

Theorem (functional equation for LE)

˜ LE(s) := Ns/2

E

(2π)−sΓ(s)LE(s) - analyt. cont. on C, ˜ LE(s) = ±˜ LE(2 − s).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2, where Lp(t) = 1 − (p + 1 − #˜ E(Fp))t + pt2 for all but finitely may p.

Theorem (functional equation for LE)

˜ LE(s) := Ns/2

E

(2π)−sΓ(s)LE(s) - analyt. cont. on C, ˜ LE(s) = ±˜ LE(2 − s).

Theorem (Wiles et al.)

E/Q elliptic curve of conductor NE. Then ∃ newform f = fE with fE(q) =

n≥1

anqn of weight 2 level NE, s.t. LE(s) = LfE(s) =

n≥1 an ns .

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2, where Lp(t) = 1 − (p + 1 − #˜ E(Fp))t + pt2 for all but finitely may p.

Theorem (functional equation for LE)

˜ LE(s) := Ns/2

E

(2π)−sΓ(s)LE(s) - analyt. cont. on C, ˜ LE(s) = ±˜ LE(2 − s).

Theorem (Wiles et al.)

E/Q elliptic curve of conductor NE. Then ∃ newform f = fE with fE(q) =

n≥1

anqn of weight 2 level NE, s.t. LE(s) = LfE(s) =

n≥1 an ns .

Hecke: (2π)−sΓ(s)Lf(s) = ∞ f(iy)ys dy

y =

1

0 +

∞

1 .

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2, where Lp(t) = 1 − (p + 1 − #˜ E(Fp))t + pt2 for all but finitely may p.

Theorem (functional equation for LE)

˜ LE(s) := Ns/2

E

(2π)−sΓ(s)LE(s) - analyt. cont. on C, ˜ LE(s) = ±˜ LE(2 − s).

Theorem (Wiles et al.)

E/Q elliptic curve of conductor NE. Then ∃ newform f = fE with fE(q) =

n≥1

anqn of weight 2 level NE, s.t. LE(s) = LfE(s) =

n≥1 an ns .

Hecke: (2π)−sΓ(s)Lf(s) = ∞ f(iy)ys dy

y =

1

0 +

∞

1 .

f( −1

τ ) = τ 2f(τ), τ ∈ H.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition (The L-function of E over Q)

LE(s) =

• p prime

1 Lp(p−s) = n∈N an ns

for Re(s) ≥ 3/2, where Lp(t) = 1 − (p + 1 − #˜ E(Fp))t + pt2 for all but finitely may p.

Theorem (functional equation for LE)

˜ LE(s) := Ns/2

E

(2π)−sΓ(s)LE(s) - analyt. cont. on C, ˜ LE(s) = ±˜ LE(2 − s).

Theorem (Wiles et al.)

E/Q elliptic curve of conductor NE. Then ∃ newform f = fE with fE(q) =

n≥1

anqn of weight 2 level NE, s.t. LE(s) = LfE(s) =

n≥1 an ns .

Hecke: (2π)−sΓ(s)Lf(s) = ∞ f(iy)ys dy

y =

1

0 +

∞

1 .

f( −1

τ ) = τ 2f(τ), τ ∈ H.

The Birch & Swinnerton-Dyer conjecture predicts a deep relation between E(Q) and LE(s).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 2 / 12

Definition ( p∞-Selmer groups)

Sp(E/Q) = Ker

• H1(GQ, E(¯

Q)[p]) − →

• ℓ prime

H1(GQℓ, E(¯ Qℓ))

• Somnath Jha (IIT Kanpur)

Algebraic functional equation for Selmer groups 23 November 2020 3 / 12

Definition ( p∞-Selmer groups)

Sp(E/Q) = Ker

• H1(GQ, E(¯

Q)[p]) − →

• ℓ prime

H1(GQℓ, E(¯ Qℓ))

• S(E/Q) = Sp∞(E/Q) := lim

− →

n

Spn(E/Q).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 3 / 12

Definition ( p∞-Selmer groups)

Sp(E/Q) = Ker

• H1(GQ, E(¯

Q)[p]) − →

• ℓ prime

H1(GQℓ, E(¯ Qℓ))

• S(E/Q) = Sp∞(E/Q) := lim

− →

n

Spn(E/Q). 0 →

E(Q) pE(Q) −

→ Sp(E/Q) − → (E/Q)[p] → 0.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 3 / 12

Definition ( p∞-Selmer groups)

Sp(E/Q) = Ker

• H1(GQ, E(¯

Q)[p]) − →

• ℓ prime

H1(GQℓ, E(¯ Qℓ))

• S(E/Q) = Sp∞(E/Q) := lim

− →

n

Spn(E/Q). 0 →

E(Q) pE(Q) −

→ Sp(E/Q) − → (E/Q)[p] → 0. 0 − → E(Q) ⊗ Qp/Zp − → S(E/Q) − → (E/Q)(p) − → 0.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 3 / 12

Definition ( p∞-Selmer groups)

Sp(E/Q) = Ker

• H1(GQ, E(¯

Q)[p]) − →

• ℓ prime

H1(GQℓ, E(¯ Qℓ))

• S(E/Q) = Sp∞(E/Q) := lim

− →

n

Spn(E/Q). 0 →

E(Q) pE(Q) −

→ Sp(E/Q) − → (E/Q)[p] → 0. 0 − → E(Q) ⊗ Qp/Zp − → S(E/Q) − → (E/Q)(p) − → 0. Series of works of Bhargava et al. and Mazur-Rubin on Sp(E/K), Sp∞(E/K).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 3 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z pnZ.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn)

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module. Assumption : E has good, ordinary reduction at p.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module. Assumption : E has good, ordinary reduction at p.

Theorem (Mazur and Swinnerton-Dyer)

∃! gE(T) = 0 ∈ Zp[[Γ]]⊗ZpQp s. t.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module. Assumption : E has good, ordinary reduction at p.

Theorem (Mazur and Swinnerton-Dyer)

∃! gE(T) = 0 ∈ Zp[[Γ]]⊗ZpQp s. t. for any finite order character φ of Γ, gE(φ(T)) = Lalg

E (1, φ).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module. Assumption : E has good, ordinary reduction at p.

Theorem (Mazur and Swinnerton-Dyer)

∃! gE(T) = 0 ∈ Zp[[Γ]]⊗ZpQp s. t. for any finite order character φ of Γ, gE(φ(T)) = Lalg

E (1, φ).

cyclotomic Iwasawa Main Conjecture for E; Kato, Skinner-Urban

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module. Assumption : E has good, ordinary reduction at p.

Theorem (Mazur and Swinnerton-Dyer)

∃! gE(T) = 0 ∈ Zp[[Γ]]⊗ZpQp s. t. for any finite order character φ of Γ, gE(φ(T)) = Lalg

E (1, φ).

cyclotomic Iwasawa Main Conjecture for E; Kato, Skinner-Urban

S(E/Q∞)∨ : torsion Zp[[Γ]] module

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module. Assumption : E has good, ordinary reduction at p.

Theorem (Mazur and Swinnerton-Dyer)

∃! gE(T) = 0 ∈ Zp[[Γ]]⊗ZpQp s. t. for any finite order character φ of Γ, gE(φ(T)) = Lalg

E (1, φ).

cyclotomic Iwasawa Main Conjecture for E; Kato, Skinner-Urban

S(E/Q∞)∨ : torsion Zp[[Γ]] module and CharZp[[Γ]](S(E/Q∞)∨) = (gE(T)).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

∃! field extension Qcyc ⊂ ∪

n Q(µpn) of Q s.t. Γ := Gal(Qcyc/Q) ∼

= Zp. Q ⊂ Qn ⊂ Qcyc s.t. Γn = Gal(Qn/Q) ∼ =

Z

• pnZ. Zp[[Γ]] := lim

← −

n

Zp[Γn] ∼ = Zp[[T]]. S(E/Qcyc) := lim − →

n

S(E/Qn) is a cofinitely generated Zp[[Γ]] module. Assumption : E has good, ordinary reduction at p.

Theorem (Mazur and Swinnerton-Dyer)

∃! gE(T) = 0 ∈ Zp[[Γ]]⊗ZpQp s. t. for any finite order character φ of Γ, gE(φ(T)) = Lalg

E (1, φ).

cyclotomic Iwasawa Main Conjecture for E; Kato, Skinner-Urban

S(E/Q∞)∨ : torsion Zp[[Γ]] module and CharZp[[Γ]](S(E/Q∞)∨) = (gE(T)). gE(T) = uEgE(

1 1+T − 1), uE : a unit in Zp[[Γ]].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ:

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem:

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn are finite and uniformly bounded.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn are finite and uniformly bounded.

3

Pick θ s.t. S(E/Qcyc)∨(θ)Γn is finite ∀ n.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn are finite and uniformly bounded.

3

Pick θ s.t. S(E/Qcyc)∨(θ)Γn is finite ∀ n. Generalized Cassles-Tate paring by Flach:

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn are finite and uniformly bounded.

3

Pick θ s.t. S(E/Qcyc)∨(θ)Γn is finite ∀ n. Generalized Cassles-Tate paring by Flach: S(Eθ/Qn) ∼ = S((Eθ)∗(1)/Qn)∨.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn are finite and uniformly bounded.

3

Pick θ s.t. S(E/Qcyc)∨(θ)Γn is finite ∀ n. Generalized Cassles-Tate paring by Flach: S(Eθ/Qn) ∼ = S((Eθ)∗(1)/Qn)∨.

4

A pseudoisomorphism S(E/Qcyc)∨ − → Ext1

Zp[[Γ]](S(E/Qcyc)∨ι, Zp[[Γ]]).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn are finite and uniformly bounded.

3

Pick θ s.t. S(E/Qcyc)∨(θ)Γn is finite ∀ n. Generalized Cassles-Tate paring by Flach: S(Eθ/Qn) ∼ = S((Eθ)∗(1)/Qn)∨.

4

A pseudoisomorphism S(E/Qcyc)∨ − → Ext1

Zp[[Γ]](S(E/Qcyc)∨ι, Zp[[Γ]]).

Generalization to p-adic Lie extensions/other motives for Algebraic Functional Equation?

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou)

Algebraic Functional Equ: CharZp[[Γ]](S(E/Qcyc)∨) = CharZp[[Γ]](S(E/Qcyc)∨ι).

1

A twisting lemma: M: finitely gen. torsion Zp[[Γ]]-module. Then ∃ a continuous character θ : Γ → Z×

p s.t. M(θ)Γpn := H0(Γpn, M(θ)) is finite ∀n.

Example:

Zp[[T]]

• T,(1+T)pn −1

is infinite ∀n but

Zp[[T]]

• T−p,(1+T)pn −1

is finite ∀n. Let Γ =< γ >. Any θ where θ(γ)µpn − 1 = a root of CharZp[[Γ]](M) for any n, works.

2

Mazur’s control theorem: The kernel and cokernel of r θ

n : S(Eθ/Qn) → S(Eθ/Qcyc)Γn are finite and uniformly bounded.

3

Pick θ s.t. S(E/Qcyc)∨(θ)Γn is finite ∀ n. Generalized Cassles-Tate paring by Flach: S(Eθ/Qn) ∼ = S((Eθ)∗(1)/Qn)∨.

4

A pseudoisomorphism S(E/Qcyc)∨ − → Ext1

Zp[[Γ]](S(E/Qcyc)∨ι, Zp[[Γ]]).

Generalization to p-adic Lie extensions/other motives for Algebraic Functional Equation? First, twisting lemma ?

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Serre: E non-CM elliptic curve over Q.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn),

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

M(θ)U := H0(U, M(θ)) is finite.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

M(θ)U := H0(U, M(θ)) is finite. K : number field and K∞/K : an admissible p-adic Lie extension.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

M(θ)U := H0(U, M(θ)) is finite. K : number field and K∞/K : an admissible p-adic Lie extension. K finite extension of Qp,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

M(θ)U := H0(U, M(θ)) is finite. K : number field and K∞/K : an admissible p-adic Lie extension. K finite extension of Qp, ring of integers O.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

M(θ)U := H0(U, M(θ)) is finite. K : number field and K∞/K : an admissible p-adic Lie extension. K finite extension of Qp, ring of integers O. V ∼ = K⊕d− an "ordinary" p-adic Galois representation of GK.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

M(θ)U := H0(U, M(θ)) is finite. K : number field and K∞/K : an admissible p-adic Lie extension. K finite extension of Qp, ring of integers O. V ∼ = K⊕d− an "ordinary" p-adic Galois representation of GK. T ⊂ V, a GK-stable O-lattice, A := V/T.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Serre: E non-CM elliptic curve over Q. G := Gal(Q(Ep∞)/Q) open subgroup

• f GL2(Zp) for any prime p. H := Gal(Q(Ep∞)/Qcyc). Then G/H ∼

= Zp. False Tate-curve extension: m ∈ N, p-power free. Q ⊂ Qcyc ⊂ J∞, where J∞ := ∪

nQ(µp∞, m1/pn), G := Gal(J∞/Q) ∼

= Z×

p ⋊ Zp, H = Gal(J∞/Qcyc).

K1(ΛO(G)) − → K1(ΛO(G)S∗)− →K0(MH(G)) − → 0. (C-F-K-S-V)

Theorem (J., Ochiai, Zábrádi, 2016)

p odd. G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin. gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a continuous

character θ : Γ− →Z×

p s.t. for every open normal subgroup U of G,

M(θ)U := H0(U, M(θ)) is finite. K : number field and K∞/K : an admissible p-adic Lie extension. K finite extension of Qp, ring of integers O. V ∼ = K⊕d− an "ordinary" p-adic Galois representation of GK. T ⊂ V, a GK-stable O-lattice, A := V/T. Let r θ

U,A : S(Aθ/KU) −

→ S(Aθ/K∞)U be the natural restriction map.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

Theorem (J., Ochiai, 2020)

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p ,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1))

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)). (2b) lim ← −U Ker(resA∗(1)

U

) is a finitely generated Zp-module.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)). (2b) lim ← −U Ker(resA∗(1)

U

) is a finitely generated Zp-module. [Zp] = 0 in K0(MH(G)). Further, some technical condition on the growth of coker(rU,A∗(1)).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)). (2b) lim ← −U Ker(resA∗(1)

U

) is a finitely generated Zp-module. [Zp] = 0 in K0(MH(G)). Further, some technical condition on the growth of coker(rU,A∗(1)).

3

Some hypotheses so that Exti

O[[G]]

• S(A/K∞)∨ι, O[[G]]
• = 0 for i ≥ 2.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)). (2b) lim ← −U Ker(resA∗(1)

U

) is a finitely generated Zp-module. [Zp] = 0 in K0(MH(G)). Further, some technical condition on the growth of coker(rU,A∗(1)).

3

Some hypotheses so that Exti

O[[G]]

• S(A/K∞)∨ι, O[[G]]
• = 0 for i ≥ 2.

Then

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)). (2b) lim ← −U Ker(resA∗(1)

U

) is a finitely generated Zp-module. [Zp] = 0 in K0(MH(G)). Further, some technical condition on the growth of coker(rU,A∗(1)).

3

Some hypotheses so that Exti

O[[G]]

• S(A/K∞)∨ι, O[[G]]
• = 0 for i ≥ 2.

Then [S(A/K∞)∨] + [EA∗(1) ] = [S(A∗(1)/K∞)∨ι] in K0(MH(G)).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)). (2b) lim ← −U Ker(resA∗(1)

U

) is a finitely generated Zp-module. [Zp] = 0 in K0(MH(G)). Further, some technical condition on the growth of coker(rU,A∗(1)).

3

Some hypotheses so that Exti

O[[G]]

• S(A/K∞)∨ι, O[[G]]
• = 0 for i ≥ 2.

Then [S(A/K∞)∨] + [EA∗(1) ] = [S(A∗(1)/K∞)∨ι] in K0(MH(G)). EA

0 : ‘error term’ related to the Euler factor of L(V, s) at finitely many primes.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020)

Assume for B ∈ {A, A∗(1)},

S(B/K∞)∨ S(B/K∞)∨(p) is finitely gen. over O[[H]]. Moreover,

1

For every continuous character θ : Γcyc → Z×

p , Ker(r θ U,A∗(1)) and

Coker(r θ

U,A∗(1)) are finite groups for each U.

2

Either (2a) or (2b) holds. (2a) The order of Ker(rU,A∗(1)) is bounded independently of U. Further, some technical condition on the growth of coker(rU,A∗(1)). (2b) lim ← −U Ker(resA∗(1)

U

) is a finitely generated Zp-module. [Zp] = 0 in K0(MH(G)). Further, some technical condition on the growth of coker(rU,A∗(1)).

3

Some hypotheses so that Exti

O[[G]]

• S(A/K∞)∨ι, O[[G]]
• = 0 for i ≥ 2.

Then [S(A/K∞)∨] + [EA∗(1) ] = [S(A∗(1)/K∞)∨ι] in K0(MH(G)). EA

0 : ‘error term’ related to the Euler factor of L(V, s) at finitely many primes.

We also show the compatibility of the algebraic and the conjectural analytic functional equation.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Example

J∞/Q false-Tate extension.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p)

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

3

If N squarefree, f ∈ SK(Γ0(N)), j = k/2.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

3

If N squarefree, f ∈ SK(Γ0(N)), j = k/2. Then in K0(MH(G)), [EA

0 ] = [ q∈P1∪P2IndG GqT(−1)].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

3

If N squarefree, f ∈ SK(Γ0(N)), j = k/2. Then in K0(MH(G)), [EA

0 ] = [ q∈P1∪P2IndG GqT(−1)]. For any Artin representation η of G,

η(ξEA

0 ) =

q∈P1∪P2 Pq(f,η,q− k

2 )

Pq(f,η∗,q− k

2 ) modulo p-adic units.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

3

If N squarefree, f ∈ SK(Γ0(N)), j = k/2. Then in K0(MH(G)), [EA

0 ] = [ q∈P1∪P2IndG GqT(−1)]. For any Artin representation η of G,

η(ξEA

0 ) =

q∈P1∪P2 Pq(f,η,q− k

2 )

Pq(f,η∗,q− k

2 ) modulo p-adic units.

Here P1, P2 ⊂ P0 = {q prime in Q : q = p, q | m & AGJ∞,w = 0 ∀ w | q}.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

3

If N squarefree, f ∈ SK(Γ0(N)), j = k/2. Then in K0(MH(G)), [EA

0 ] = [ q∈P1∪P2IndG GqT(−1)]. For any Artin representation η of G,

η(ξEA

0 ) =

q∈P1∪P2 Pq(f,η,q− k

2 )

Pq(f,η∗,q− k

2 ) modulo p-adic units.

Here P1, P2 ⊂ P0 = {q prime in Q : q = p, q | m & AGJ∞,w = 0 ∀ w | q}. Example: 2d + 1-th symmetric power of VpE over J∞; E/Q elliptic curve.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

3

If N squarefree, f ∈ SK(Γ0(N)), j = k/2. Then in K0(MH(G)), [EA

0 ] = [ q∈P1∪P2IndG GqT(−1)]. For any Artin representation η of G,

η(ξEA

0 ) =

q∈P1∪P2 Pq(f,η,q− k

2 )

Pq(f,η∗,q− k

2 ) modulo p-adic units.

Here P1, P2 ⊂ P0 = {q prime in Q : q = p, q | m & AGJ∞,w = 0 ∀ w | q}. Example: 2d + 1-th symmetric power of VpE over J∞; E/Q elliptic curve. Earlier works of Zábrádi for elliptic curves.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example

J∞/Q false-Tate extension. f ∈ Sk(Γ0(N), χ) newform, k ≥ 2, p ∤ N and vp(ap(f)) = 0. T ⊂ Vf,p(j) lattice with 1 ≤ j ≤ k −1 and A := T ⊗Qp/Zp. Then

1

If

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]], then

[S(A/J∞)∨] + [EA∗(1) ] = [S(A∗(1)/J∞)∨ι] in K0(MH(G)).

2

Whenever S(A/Q(µp∞))∨ is a finitely generated Zp-module, then

S(A/J∞)∨ S(A/J∞)∨(p) and S(A∗(1)/J∞)∨ S(A∗(1)/J∞)∨(p) are finitely generated over Of[[H]].

3

If N squarefree, f ∈ SK(Γ0(N)), j = k/2. Then in K0(MH(G)), [EA

0 ] = [ q∈P1∪P2IndG GqT(−1)]. For any Artin representation η of G,

η(ξEA

0 ) =

q∈P1∪P2 Pq(f,η,q− k

2 )

Pq(f,η∗,q− k

2 ) modulo p-adic units.

Here P1, P2 ⊂ P0 = {q prime in Q : q = p, q | m & AGJ∞,w = 0 ∀ w | q}. Example: 2d + 1-th symmetric power of VpE over J∞; E/Q elliptic curve. Earlier works of Zábrádi for elliptic curves. different proofs.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

We prove a control theorem for r θ

U,A:

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite. Also assume for every finite prime v ∤ p of K,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite. Also assume for every finite prime v ∤ p of K, and

for any open subgroup Wv ⊂ GKv , V Wv = 0.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite. Also assume for every finite prime v ∤ p of K, and

for any open subgroup Wv ⊂ GKv , V Wv = 0. Further, (a) Assume AGK∞ is finite.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite. Also assume for every finite prime v ∤ p of K, and

for any open subgroup Wv ⊂ GKv , V Wv = 0. Further, (a) Assume AGK∞ is finite. Then Coker(r θ

U,A) is finite for every U.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite. Also assume for every finite prime v ∤ p of K, and

for any open subgroup Wv ⊂ GKv , V Wv = 0. Further, (a) Assume AGK∞ is finite. Then Coker(r θ

U,A) is finite for every U.

(b) Assume the Lie algebra attached to G is reductive.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite. Also assume for every finite prime v ∤ p of K, and

for any open subgroup Wv ⊂ GKv , V Wv = 0. Further, (a) Assume AGK∞ is finite. Then Coker(r θ

U,A) is finite for every U.

(b) Assume the Lie algebra attached to G is reductive. Then the group Coker(r θ

U,A) is finite for every U.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

We prove a control theorem for r θ

U,A:

Theorem (J., Ochiai, 2020)

(1) (a) Assume AGK∞ is finite. Then Ker(r θ

U,A) is a finite group whose order

is bounded independent of U. (b) The Lie algebra attached to G is reductive. Then Ker(r θ

U,A) is finite

for every U. And, lim ← −U Ker(r θ

U,A) is a finitely generated Zp-module.

(2) Assume for every prime v | p of K and for every prime w | v of K∞, (A/F +

v A)GK∞,w is finite. Also assume for every finite prime v ∤ p of K, and

for any open subgroup Wv ⊂ GKv , V Wv = 0. Further, (a) Assume AGK∞ is finite. Then Coker(r θ

U,A) is finite for every U.

(b) Assume the Lie algebra attached to G is reductive. Then the group Coker(r θ

U,A) is finite for every U.

Further, more precise estimate on the growth of Coker(r θ

U,A) can be

given.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 9 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G,

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G, M(θ)U := H0(U, M(θ)) is finite.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G, M(θ)U := H0(U, M(θ)) is finite.

If U ⊂ V then M(θ)U ։ M(θ)V. Note:

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G, M(θ)U := H0(U, M(θ)) is finite.

If U ⊂ V then M(θ)U ։ M(θ)V. Note: M(θ)U is finite ⇔ M(θ)U ⊗Zp Qp = 0. Can extend scalar up to ¯ Qp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G, M(θ)U := H0(U, M(θ)) is finite.

If U ⊂ V then M(θ)U ։ M(θ)V. Note: M(θ)U is finite ⇔ M(θ)U ⊗Zp Qp = 0. Can extend scalar up to ¯ Qp. Also M(θ)U ⊗Zp Qp ∼ = M(θ) ⊗Zp[[G]] Zp[ G

U ] ⊗Zp Qp ∼

= (M(θ) ⊗Zp Qp)U.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G, M(θ)U := H0(U, M(θ)) is finite.

If U ⊂ V then M(θ)U ։ M(θ)V. Note: M(θ)U is finite ⇔ M(θ)U ⊗Zp Qp = 0. Can extend scalar up to ¯ Qp. Also M(θ)U ⊗Zp Qp ∼ = M(θ) ⊗Zp[[G]] Zp[ G

U ] ⊗Zp Qp ∼

= (M(θ) ⊗Zp Qp)U. Assume M(θ)U ⊗Zp Qp ∼ =

Qp[ G

U ]

Qp[ G

U ](

γU−θ(γ)−1aU),

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G, M(θ)U := H0(U, M(θ)) is finite.

If U ⊂ V then M(θ)U ։ M(θ)V. Note: M(θ)U is finite ⇔ M(θ)U ⊗Zp Qp = 0. Can extend scalar up to ¯ Qp. Also M(θ)U ⊗Zp Qp ∼ = M(θ) ⊗Zp[[G]] Zp[ G

U ] ⊗Zp Qp ∼

= (M(θ) ⊗Zp Qp)U. Assume M(θ)U ⊗Zp Qp ∼ =

Qp[ G

U ]

Qp[ G

U ](

γU−θ(γ)−1aU), where

γ ∈ G − → γU ∈ G

U via

the natural projection and aU ∈ Qp[ G

U ] is the image of an element a via

the composite map Zp[[H]] ⊗Zp Qp − → Zp[[G]] ⊗Zp Qp − → Zp[ G

U ] ⊗Zp Qp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Theorem (J., Ochiai, Zábrádi, 2016)

G: compact p-adic Lie group, H: closed normal subgp, G/H ∼ = Γ = Zp. M fin.

• gen. Zp[[G]]-module,

M M(p) fin. gen. over Zp[[H]]. Then ∃ a cont. character θ :

Γ− →Z×

p s.t. ∀ open normal subgroup U of G, M(θ)U := H0(U, M(θ)) is finite.

If U ⊂ V then M(θ)U ։ M(θ)V. Note: M(θ)U is finite ⇔ M(θ)U ⊗Zp Qp = 0. Can extend scalar up to ¯ Qp. Also M(θ)U ⊗Zp Qp ∼ = M(θ) ⊗Zp[[G]] Zp[ G

U ] ⊗Zp Qp ∼

= (M(θ) ⊗Zp Qp)U. Assume M(θ)U ⊗Zp Qp ∼ =

Qp[ G

U ]

Qp[ G

U ](

γU−θ(γ)−1aU), where

γ ∈ G − → γU ∈ G

U via

the natural projection and aU ∈ Qp[ G

U ] is the image of an element a via

the composite map Zp[[H]] ⊗Zp Qp − → Zp[[G]] ⊗Zp Qp − → Zp[ G

U ] ⊗Zp Qp.

Qp[ G

U ] is isomorphic to products of matrix algebras nU i=1 Mri(Qp).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 10 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp). EVU,i := the set of roots of the polynomial PU,i(T) := det(γU,i − aU,iT).

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp). EVU,i := the set of roots of the polynomial PU,i(T) := det(γU,i − aU,iT). As γU,i ∈ GLri(Qp), PU,i(T) = 0 and #EVU,i < ∞.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp). EVU,i := the set of roots of the polynomial PU,i(T) := det(γU,i − aU,iT). As γU,i ∈ GLri(Qp), PU,i(T) = 0 and #EVU,i < ∞. Hence EVU := ∪

i EVU,i

is again a finite set.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp). EVU,i := the set of roots of the polynomial PU,i(T) := det(γU,i − aU,iT). As γU,i ∈ GLri(Qp), PU,i(T) = 0 and #EVU,i < ∞. Hence EVU := ∪

i EVU,i

is again a finite set. If θ(γ)−1 ∈ EVU ∩ Z×

p , then M(θ)U ⊗Zp ¯

Qp = 0.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp). EVU,i := the set of roots of the polynomial PU,i(T) := det(γU,i − aU,iT). As γU,i ∈ GLri(Qp), PU,i(T) = 0 and #EVU,i < ∞. Hence EVU := ∪

i EVU,i

is again a finite set. If θ(γ)−1 ∈ EVU ∩ Z×

p , then M(θ)U ⊗Zp ¯

Qp = 0. G is profinite and has a countable base {Un}n∈N at the identity.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp). EVU,i := the set of roots of the polynomial PU,i(T) := det(γU,i − aU,iT). As γU,i ∈ GLri(Qp), PU,i(T) = 0 and #EVU,i < ∞. Hence EVU := ∪

i EVU,i

is again a finite set. If θ(γ)−1 ∈ EVU ∩ Z×

p , then M(θ)U ⊗Zp ¯

Qp = 0. G is profinite and has a countable base {Un}n∈N at the identity. Then EVM := ∪

nEVUn ∩ Z× p is countable; choose θ(γ)−1 ∈ Z× p \ EVM.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

Apply ⊗QpQp to get M(θ)U ⊗Zp Qp ∼ = nU

i=1 Mri (Qp) Mri (Qp)(γU,i−θ(γ)−1aU,i). Note

• γU ∈ AutQp[ G

U ](Qp[ G

U ]) ⊂ AutQp(Qp[ G U ]). Extending scalar to ¯

Qp,

• γU ∈ Aut¯

Qp(¯

Qp[ G

U ]) ∼

= AutQp(nU

i=1 Mri(Qp)) ∼

= nU

i=1 GLri(Qp). The

projection to the i-th component is γU,i. Extending scalar to ¯ Qp, aU ∈ ¯ Qp[ G

U ] ∼

= nU

i=1 Mri(Qp) and aU,i is the projection to the i-th

component Mri(Qp). EVU,i := the set of roots of the polynomial PU,i(T) := det(γU,i − aU,iT). As γU,i ∈ GLri(Qp), PU,i(T) = 0 and #EVU,i < ∞. Hence EVU := ∪

i EVU,i

is again a finite set. If θ(γ)−1 ∈ EVU ∩ Z×

p , then M(θ)U ⊗Zp ¯

Qp = 0. G is profinite and has a countable base {Un}n∈N at the identity. Then EVM := ∪

nEVUn ∩ Z× p is countable; choose θ(γ)−1 ∈ Z× p \ EVM.

#M(p)U < ∞; can assume M is fin. gen. over Zp[[H]].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 11 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard)

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard) Assume G has no element of order p.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard) Assume G has no element of order p. Let G, H be as above; G has no p-torsion. M fin. gen. Zp[[G]] module, which is fin. gen. over Zp[[H]].

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard) Assume G has no element of order p. Let G, H be as above; G has no p-torsion. M fin. gen. Zp[[G]] module, which is fin. gen. over Zp[[H]]. Then, ∃ an open subgroup G0 ⊂ G containing H and a Zp[[G0]]-module N which is a free Zp[[H]]-module of finite rank, and a surjective Zp[[G0]]-linear homomorphism N ։ M.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard) Assume G has no element of order p. Let G, H be as above; G has no p-torsion. M fin. gen. Zp[[G]] module, which is fin. gen. over Zp[[H]]. Then, ∃ an open subgroup G0 ⊂ G containing H and a Zp[[G0]]-module N which is a free Zp[[H]]-module of finite rank, and a surjective Zp[[G0]]-linear homomorphism N ։ M. Reduces to the case (i) G a p-adic Lie group without any element of

• rder p, (ii) G/H ∼

= Γ (iii) M is a fin. gen. Zp[[G]] module and ∃ a Zp[[H]] ⊗Zp Qp linear isomorphism M ⊗Zp Qp ∼ = (Zp[[H]] ⊗Zp Qp)⊕d.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard) Assume G has no element of order p. Let G, H be as above; G has no p-torsion. M fin. gen. Zp[[G]] module, which is fin. gen. over Zp[[H]]. Then, ∃ an open subgroup G0 ⊂ G containing H and a Zp[[G0]]-module N which is a free Zp[[H]]-module of finite rank, and a surjective Zp[[G0]]-linear homomorphism N ։ M. Reduces to the case (i) G a p-adic Lie group without any element of

• rder p, (ii) G/H ∼

= Γ (iii) M is a fin. gen. Zp[[G]] module and ∃ a Zp[[H]] ⊗Zp Qp linear isomorphism M ⊗Zp Qp ∼ = (Zp[[H]] ⊗Zp Qp)⊕d. ∃ a matrix A ∈ Md(Zp[[H]] ⊗Zp Qp) and a Zp[[G]] ⊗Zp Qp isomorphism M ⊗Zp Qp ∼ =

(Zp[[G]]⊗Zp Qp)⊕d (Zp[[G]]⊗Zp Qp)⊕d( γ1d×d−A). Here < γ >= Γ,

γ ∈ G is a fixed lift

• f γ and elements in (Zp[[G]] ⊗Zp Qp)⊕d are row vectors.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard) Assume G has no element of order p. Let G, H be as above; G has no p-torsion. M fin. gen. Zp[[G]] module, which is fin. gen. over Zp[[H]]. Then, ∃ an open subgroup G0 ⊂ G containing H and a Zp[[G0]]-module N which is a free Zp[[H]]-module of finite rank, and a surjective Zp[[G0]]-linear homomorphism N ։ M. Reduces to the case (i) G a p-adic Lie group without any element of

• rder p, (ii) G/H ∼

= Γ (iii) M is a fin. gen. Zp[[G]] module and ∃ a Zp[[H]] ⊗Zp Qp linear isomorphism M ⊗Zp Qp ∼ = (Zp[[H]] ⊗Zp Qp)⊕d. ∃ a matrix A ∈ Md(Zp[[H]] ⊗Zp Qp) and a Zp[[G]] ⊗Zp Qp isomorphism M ⊗Zp Qp ∼ =

(Zp[[G]]⊗Zp Qp)⊕d (Zp[[G]]⊗Zp Qp)⊕d( γ1d×d−A). Here < γ >= Γ,

γ ∈ G is a fixed lift

• f γ and elements in (Zp[[G]] ⊗Zp Qp)⊕d are row vectors.

For d = 1, M ⊗Zp Qp ∼ =

Zp[[G]]⊗Zp Qp (Zp[[G]]⊗Zp Qp)( γ−a), a ∈ Zp[[H]] ⊗Zp Qp.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

A compact p-adic Lie group G has an open normal subgroup G′ without any element of order p. (Lazard) Assume G has no element of order p. Let G, H be as above; G has no p-torsion. M fin. gen. Zp[[G]] module, which is fin. gen. over Zp[[H]]. Then, ∃ an open subgroup G0 ⊂ G containing H and a Zp[[G0]]-module N which is a free Zp[[H]]-module of finite rank, and a surjective Zp[[G0]]-linear homomorphism N ։ M. Reduces to the case (i) G a p-adic Lie group without any element of

• rder p, (ii) G/H ∼

= Γ (iii) M is a fin. gen. Zp[[G]] module and ∃ a Zp[[H]] ⊗Zp Qp linear isomorphism M ⊗Zp Qp ∼ = (Zp[[H]] ⊗Zp Qp)⊕d. ∃ a matrix A ∈ Md(Zp[[H]] ⊗Zp Qp) and a Zp[[G]] ⊗Zp Qp isomorphism M ⊗Zp Qp ∼ =

(Zp[[G]]⊗Zp Qp)⊕d (Zp[[G]]⊗Zp Qp)⊕d( γ1d×d−A). Here < γ >= Γ,

γ ∈ G is a fixed lift

• f γ and elements in (Zp[[G]] ⊗Zp Qp)⊕d are row vectors.

For d = 1, M ⊗Zp Qp ∼ =

Zp[[G]]⊗Zp Qp (Zp[[G]]⊗Zp Qp)( γ−a), a ∈ Zp[[H]] ⊗Zp Qp.

Thank You

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12

References:

• S. Jha, T. Ochiai, G. Zábrádi, On twists of modules over

non-commutative Iwasawa algebras, Algebra Number Theory, 10(3) (2016) 685-694.

• S. Jha, T. Ochiai, Control theorem and functional equation of Selmer

groups over p-adic Lie extensions, Selecta Mathematica (N.S.), 26(5) (2020), Article no. 80, 58 pages.

Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 12 / 12