Algebraic functional equation for Selmer groups Fields Institute - PowerPoint PPT Presentation

n Q ( µ p n ) of Q s.t. Γ := Gal ( Q cyc / Q ) ∼ ∃ ! field extension Q cyc ⊂ ∪ = Z p . Q ⊂ Q n ⊂ Q cyc s.t. Γ n = Gal ( Q n / Q ) ∼ Z p [Γ n ] ∼ Z p n Z . Z p [[Γ]] := lim = Z p [[ T ]] . = ← − n S ( E / Q cyc ) := lim S ( E / Q n ) is a cofinitely generated Z p [[Γ]] module. − → n Assumption : E has good, ordinary reduction at p . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

n Q ( µ p n ) of Q s.t. Γ := Gal ( Q cyc / Q ) ∼ ∃ ! field extension Q cyc ⊂ ∪ = Z p . Q ⊂ Q n ⊂ Q cyc s.t. Γ n = Gal ( Q n / Q ) ∼ Z p [Γ n ] ∼ Z p n Z . Z p [[Γ]] := lim = Z p [[ T ]] . = ← − n S ( E / Q cyc ) := lim S ( E / Q n ) is a cofinitely generated Z p [[Γ]] module. − → n Assumption : E has good, ordinary reduction at p . Theorem (Mazur and Swinnerton-Dyer) ∃ ! g E ( T ) � = 0 ∈ Z p [[Γ]] ⊗ Z p Q p s. t. Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

n Q ( µ p n ) of Q s.t. Γ := Gal ( Q cyc / Q ) ∼ ∃ ! field extension Q cyc ⊂ ∪ = Z p . Q ⊂ Q n ⊂ Q cyc s.t. Γ n = Gal ( Q n / Q ) ∼ Z p [Γ n ] ∼ Z p n Z . Z p [[Γ]] := lim = Z p [[ T ]] . = ← − n S ( E / Q cyc ) := lim S ( E / Q n ) is a cofinitely generated Z p [[Γ]] module. − → n Assumption : E has good, ordinary reduction at p . Theorem (Mazur and Swinnerton-Dyer) ∃ ! g E ( T ) � = 0 ∈ Z p [[Γ]] ⊗ Z p Q p s. t. for any finite order character φ of Γ , g E ( φ ( T )) = L alg E ( 1 , φ ) . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

n Q ( µ p n ) of Q s.t. Γ := Gal ( Q cyc / Q ) ∼ ∃ ! field extension Q cyc ⊂ ∪ = Z p . Q ⊂ Q n ⊂ Q cyc s.t. Γ n = Gal ( Q n / Q ) ∼ Z p [Γ n ] ∼ Z p n Z . Z p [[Γ]] := lim = Z p [[ T ]] . = ← − n S ( E / Q cyc ) := lim S ( E / Q n ) is a cofinitely generated Z p [[Γ]] module. − → n Assumption : E has good, ordinary reduction at p . Theorem (Mazur and Swinnerton-Dyer) ∃ ! g E ( T ) � = 0 ∈ Z p [[Γ]] ⊗ Z p Q p s. t. for any finite order character φ of Γ , g E ( φ ( T )) = L alg 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

n Q ( µ p n ) of Q s.t. Γ := Gal ( Q cyc / Q ) ∼ ∃ ! field extension Q cyc ⊂ ∪ = Z p . Q ⊂ Q n ⊂ Q cyc s.t. Γ n = Gal ( Q n / Q ) ∼ Z p [Γ n ] ∼ Z p n Z . Z p [[Γ]] := lim = Z p [[ T ]] . = ← − n S ( E / Q cyc ) := lim S ( E / Q n ) is a cofinitely generated Z p [[Γ]] module. − → n Assumption : E has good, ordinary reduction at p . Theorem (Mazur and Swinnerton-Dyer) ∃ ! g E ( T ) � = 0 ∈ Z p [[Γ]] ⊗ Z p Q p s. t. for any finite order character φ of Γ , g E ( φ ( T )) = L alg E ( 1 , φ ) . cyclotomic Iwasawa Main Conjecture for E ; Kato, Skinner-Urban S ( E / Q ∞ ) ∨ : torsion Z p [[Γ]] module Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

n Q ( µ p n ) of Q s.t. Γ := Gal ( Q cyc / Q ) ∼ ∃ ! field extension Q cyc ⊂ ∪ = Z p . Q ⊂ Q n ⊂ Q cyc s.t. Γ n = Gal ( Q n / Q ) ∼ Z p [Γ n ] ∼ Z p n Z . Z p [[Γ]] := lim = Z p [[ T ]] . = ← − n S ( E / Q cyc ) := lim S ( E / Q n ) is a cofinitely generated Z p [[Γ]] module. − → n Assumption : E has good, ordinary reduction at p . Theorem (Mazur and Swinnerton-Dyer) ∃ ! g E ( T ) � = 0 ∈ Z p [[Γ]] ⊗ Z p Q p s. t. for any finite order character φ of Γ , g E ( φ ( T )) = L alg E ( 1 , φ ) . cyclotomic Iwasawa Main Conjecture for E ; Kato, Skinner-Urban S ( E / Q ∞ ) ∨ : torsion Z p [[Γ]] module and Char Z p [[Γ]] ( S ( E / Q ∞ ) ∨ ) = ( g E ( T )) . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 4 / 12

n Q ( µ p n ) of Q s.t. Γ := Gal ( Q cyc / Q ) ∼ ∃ ! field extension Q cyc ⊂ ∪ = Z p . Q ⊂ Q n ⊂ Q cyc s.t. Γ n = Gal ( Q n / Q ) ∼ Z p [Γ n ] ∼ Z p n Z . Z p [[Γ]] := lim = Z p [[ T ]] . = ← − n S ( E / Q cyc ) := lim S ( E / Q n ) is a cofinitely generated Z p [[Γ]] module. − → n Assumption : E has good, ordinary reduction at p . Theorem (Mazur and Swinnerton-Dyer) ∃ ! g E ( T ) � = 0 ∈ Z p [[Γ]] ⊗ Z p Q p s. t. for any finite order character φ of Γ , g E ( φ ( T )) = L alg E ( 1 , φ ) . cyclotomic Iwasawa Main Conjecture for E ; Kato, Skinner-Urban S ( E / Q ∞ ) ∨ : torsion Z p [[Γ]] module and Char Z p [[Γ]] ( S ( E / Q ∞ ) ∨ ) = ( g E ( T )) . 1 g E ( T ) = u E g E ( 1 + T − 1 ) , u E : a unit in Z p [[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: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. 1 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 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: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] � � is infinite ∀ n Example: T , ( 1 + T ) pn − 1 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[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: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: 2 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n r θ Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n are finite and uniformly bounded. r θ Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n are finite and uniformly bounded. r θ Pick θ s.t. S ( E / Q cyc ) ∨ ( θ ) Γ n is finite ∀ n . 3 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n are finite and uniformly bounded. r θ Pick θ s.t. S ( E / Q cyc ) ∨ ( θ ) Γ n is finite ∀ n . Generalized Cassles-Tate paring 3 by Flach: Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n are finite and uniformly bounded. r θ Pick θ s.t. S ( E / Q cyc ) ∨ ( θ ) Γ n is finite ∀ n . Generalized Cassles-Tate paring 3 by Flach: S ( E θ / Q n ) ∼ = S (( E θ ) ∗ ( 1 ) / Q n ) ∨ . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n are finite and uniformly bounded. r θ Pick θ s.t. S ( E / Q cyc ) ∨ ( θ ) Γ n is finite ∀ n . Generalized Cassles-Tate paring 3 by Flach: S ( E θ / Q n ) ∼ = S (( E θ ) ∗ ( 1 ) / Q n ) ∨ . A pseudoisomorphism S ( E / Q cyc ) ∨ − Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι , Z p [[Γ]]) . → Ext 1 4 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n are finite and uniformly bounded. r θ Pick θ s.t. S ( E / Q cyc ) ∨ ( θ ) Γ n is finite ∀ n . Generalized Cassles-Tate paring 3 by Flach: S ( E θ / Q n ) ∼ = S (( E θ ) ∗ ( 1 ) / Q n ) ∨ . A pseudoisomorphism S ( E / Q cyc ) ∨ − Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι , Z p [[Γ]]) . → Ext 1 4 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: Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ) = Char Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι ) . A twisting lemma: M : finitely gen. torsion Z p [[Γ]] -module. Then ∃ a 1 p s.t. M ( θ ) Γ pn := H 0 (Γ p n , M ( θ )) is finite ∀ n . continuous character θ : Γ → Z × Z p [[ T ]] Z p [[ T ]] � � is infinite ∀ n but � � is finite ∀ n . Example: T , ( 1 + T ) pn − 1 T − p , ( 1 + T ) pn − 1 Let Γ = < γ > . Any θ where θ ( γ ) µ p n − 1 � = a root of Char Z p [[Γ]] ( M ) for any n , works. Mazur’s control theorem: The kernel and cokernel of 2 n : S ( E θ / Q n ) → S ( E θ / Q cyc ) Γ n are finite and uniformly bounded. r θ Pick θ s.t. S ( E / Q cyc ) ∨ ( θ ) Γ n is finite ∀ n . Generalized Cassles-Tate paring 3 by Flach: S ( E θ / Q n ) ∼ = S (( E θ ) ∗ ( 1 ) / Q n ) ∨ . A pseudoisomorphism S ( E / Q cyc ) ∨ − Z p [[Γ]] ( S ( E / Q cyc ) ∨ ι , Z p [[Γ]]) . → Ext 1 4 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , J ∞ := ∪ 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − p s.t. for every open normal subgroup U of G, M ( θ ) U := H 0 ( 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − p s.t. for every open normal subgroup U of G, M ( θ ) U := H 0 ( 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − p s.t. for every open normal subgroup U of G, M ( θ ) U := H 0 ( U , M ( θ )) is finite. K : number field and K ∞ / K : an admissible p -adic Lie extension. K finite extension of Q 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − p s.t. for every open normal subgroup U of G, M ( θ ) U := H 0 ( U , M ( θ )) is finite. K : number field and K ∞ / K : an admissible p -adic Lie extension. K finite extension of Q p , 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − p s.t. for every open normal subgroup U of G, M ( θ ) U := H 0 ( U , M ( θ )) is finite. K : number field and K ∞ / K : an admissible p -adic Lie extension. K finite extension of Q p , ring of integers O . V ∼ = K ⊕ d − an "ordinary" p -adic Galois representation of G K . 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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − p s.t. for every open normal subgroup U of G, M ( θ ) U := H 0 ( U , M ( θ )) is finite. K : number field and K ∞ / K : an admissible p -adic Lie extension. K finite extension of Q p , ring of integers O . V ∼ = K ⊕ d − an "ordinary" p -adic Galois representation of G K . T ⊂ V , a G K -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 ( E p ∞ ) / Q ) open subgroup of GL 2 ( Z p ) for any prime p . H := Gal ( Q ( E p ∞ ) / Q cyc ) . Then G / H ∼ = Z p . False Tate-curve extension: m ∈ N , p -power free. Q ⊂ Q cyc ⊂ J ∞ , where n Q ( µ p ∞ , m 1 / p n ) , G := Gal ( J ∞ / Q ) ∼ = Z × J ∞ := ∪ p ⋊ Z p , H = Gal ( J ∞ / Q cyc ) . K 1 (Λ O ( G )) − → K 1 (Λ O ( G ) S ∗ ) − → K 0 ( M H ( 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 = Z p . M M fin. gen. Z p [[ G ]] -module, M ( p ) fin. gen. over Z p [[ H ]] . Then ∃ a continuous → Z × character θ : Γ − p s.t. for every open normal subgroup U of G, M ( θ ) U := H 0 ( U , M ( θ )) is finite. K : number field and K ∞ / K : an admissible p -adic Lie extension. K finite extension of Q p , ring of integers O . V ∼ = K ⊕ d − an "ordinary" p -adic Galois representation of G K . T ⊂ V , a G K -stable O -lattice, A := V / T . → S ( A θ / K ∞ ) U be the natural restriction map. Let r θ U , A : S ( A θ / K U ) − 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) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , 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) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , 1 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 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) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , 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) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . − U Ker ( res A ∗ ( 1 ) (2b) lim ) is a finitely generated Z p -module. ← U Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . − U Ker ( res A ∗ ( 1 ) (2b) lim ) is a finitely generated Z p -module. [ Z p ] = 0 in ← U K 0 ( M H ( G )) . Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . − U Ker ( res A ∗ ( 1 ) (2b) lim ) is a finitely generated Z p -module. [ Z p ] = 0 in ← U K 0 ( M H ( G )) . Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . � � S ( A / K ∞ ) ∨ ι , O [[ G ]] Some hypotheses so that Ext i = 0 for i ≥ 2 . 3 O [[ G ]] Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . − U Ker ( res A ∗ ( 1 ) (2b) lim ) is a finitely generated Z p -module. [ Z p ] = 0 in ← U K 0 ( M H ( G )) . Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . � � S ( A / K ∞ ) ∨ ι , O [[ G ]] Some hypotheses so that Ext i = 0 for i ≥ 2 . 3 O [[ G ]] Then Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . − U Ker ( res A ∗ ( 1 ) (2b) lim ) is a finitely generated Z p -module. [ Z p ] = 0 in ← U K 0 ( M H ( G )) . Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . � � S ( A / K ∞ ) ∨ ι , O [[ G ]] Some hypotheses so that Ext i = 0 for i ≥ 2 . 3 O [[ G ]] Then [ S ( A / K ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / K ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

Theorem (J., Ochiai, 2020) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . − U Ker ( res A ∗ ( 1 ) (2b) lim ) is a finitely generated Z p -module. [ Z p ] = 0 in ← U K 0 ( M H ( G )) . Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . � � S ( A / K ∞ ) ∨ ι , O [[ G ]] Some hypotheses so that Ext i = 0 for i ≥ 2 . 3 O [[ G ]] Then [ S ( A / K ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / K ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 E A 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) S ( B / K ∞ ) ∨ Assume for B ∈ { A , A ∗ ( 1 ) } , S ( B / K ∞ ) ∨ ( p ) is finitely gen. over O [[ H ]] . Moreover, For every continuous character θ : Γ cyc → Z × p , Ker ( r θ U , A ∗ ( 1 ) ) and 1 Coker ( r θ U , A ∗ ( 1 ) ) are finite groups for each U. Either (2a) or (2b) holds. 2 (2a) The order of Ker ( r U , A ∗ ( 1 ) ) is bounded independently of U. Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . − U Ker ( res A ∗ ( 1 ) (2b) lim ) is a finitely generated Z p -module. [ Z p ] = 0 in ← U K 0 ( M H ( G )) . Further, some technical condition on the growth of coker ( r U , A ∗ ( 1 ) ) . � � S ( A / K ∞ ) ∨ ι , O [[ G ]] Some hypotheses so that Ext i = 0 for i ≥ 2 . 3 O [[ G ]] Then [ S ( A / K ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / K ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 E A 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 ∈ S k (Γ 0 ( 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 ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , 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 ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ If S ( A / J ∞ ) ∨ ( p ) and 1 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 ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, 2 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . If N squarefree, f ∈ S K (Γ 0 ( N )) , j = k / 2. 3 Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . If N squarefree, f ∈ S K (Γ 0 ( N )) , j = k / 2. Then in K 0 ( M H ( G )) , 3 0 ] = [ � [ E A q ∈ P 1 ∪ P 2 Ind G G q T ( − 1 )] . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . If N squarefree, f ∈ S K (Γ 0 ( N )) , j = k / 2. Then in K 0 ( M H ( G )) , 3 0 ] = [ � [ E A q ∈ P 1 ∪ P 2 Ind G G q T ( − 1 )] . For any Artin representation η of G , 0 ) = � P q ( f ,η, q − k 2 ) η ( ξ E A 2 ) modulo p -adic units. q ∈ P 1 ∪ P 2 P q ( f ,η ∗ , q − k Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

Example J ∞ / Q false-Tate extension. f ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . If N squarefree, f ∈ S K (Γ 0 ( N )) , j = k / 2. Then in K 0 ( M H ( G )) , 3 0 ] = [ � [ E A q ∈ P 1 ∪ P 2 Ind G G q T ( − 1 )] . For any Artin representation η of G , 0 ) = � P q ( f ,η, q − k 2 ) η ( ξ E A 2 ) modulo p -adic units. q ∈ P 1 ∪ P 2 P q ( f ,η ∗ , q − k Here P 1 , P 2 ⊂ P 0 = { q prime in Q : q � = p , q | m & A G J ∞ , 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 ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . If N squarefree, f ∈ S K (Γ 0 ( N )) , j = k / 2. Then in K 0 ( M H ( G )) , 3 0 ] = [ � [ E A q ∈ P 1 ∪ P 2 Ind G G q T ( − 1 )] . For any Artin representation η of G , 0 ) = � P q ( f ,η, q − k 2 ) η ( ξ E A 2 ) modulo p -adic units. q ∈ P 1 ∪ P 2 P q ( f ,η ∗ , q − k Here P 1 , P 2 ⊂ P 0 = { q prime in Q : q � = p , q | m & A G J ∞ , w � = 0 ∀ w | q } . Example: 2 d + 1-th symmetric power of V p E 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 ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . If N squarefree, f ∈ S K (Γ 0 ( N )) , j = k / 2. Then in K 0 ( M H ( G )) , 3 0 ] = [ � [ E A q ∈ P 1 ∪ P 2 Ind G G q T ( − 1 )] . For any Artin representation η of G , 0 ) = � P q ( f ,η, q − k 2 ) η ( ξ E A 2 ) modulo p -adic units. q ∈ P 1 ∪ P 2 P q ( f ,η ∗ , q − k Here P 1 , P 2 ⊂ P 0 = { q prime in Q : q � = p , q | m & A G J ∞ , w � = 0 ∀ w | q } . Example: 2 d + 1-th symmetric power of V p E 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 ∈ S k (Γ 0 ( N ) , χ ) newform, k ≥ 2 , p ∤ N and v p ( a p ( f )) = 0. T ⊂ V f , p ( j ) lattice with 1 ≤ j ≤ k − 1 and A := T ⊗ Q p / Z p . Then S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] , then If S ( A / J ∞ ) ∨ ( p ) and 1 [ S ( A / J ∞ ) ∨ ] + [ E A ∗ ( 1 ) ] = [ S ( A ∗ ( 1 ) / J ∞ ) ∨ ι ] in K 0 ( M H ( G )) . 0 Whenever S ( A / Q ( µ p ∞ )) ∨ is a finitely generated Z p -module, then 2 S ( A / J ∞ ) ∨ S ( A ∗ ( 1 ) / J ∞ ) ∨ S ( A / J ∞ ) ∨ ( p ) and S ( A ∗ ( 1 ) / J ∞ ) ∨ ( p ) are finitely generated over O f [[ H ]] . If N squarefree, f ∈ S K (Γ 0 ( N )) , j = k / 2. Then in K 0 ( M H ( G )) , 3 0 ] = [ � [ E A q ∈ P 1 ∪ P 2 Ind G G q T ( − 1 )] . For any Artin representation η of G , 0 ) = � P q ( f ,η, q − k 2 ) η ( ξ E A 2 ) modulo p -adic units. q ∈ P 1 ∪ P 2 P q ( f ,η ∗ , q − k Here P 1 , P 2 ⊂ P 0 = { q prime in Q : q � = p , q | m & A G J ∞ , w � = 0 ∀ w | q } . Example: 2 d + 1-th symmetric power of V p E 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