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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

  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. Theorem (Greenberg, Perrin-Riou) Algebraic Functional Equ: Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 5 / 12

  9. 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

  10. 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

  11. 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

  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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. Serre: E non-CM elliptic curve over Q . Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 6 / 12

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. Theorem (J., Ochiai, 2020) Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 7 / 12

  47. 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

  48. 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

  49. 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

  50. 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

  51. 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

  52. 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

  53. 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

  54. 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

  55. 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

  56. 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

  57. 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

  58. 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

  59. 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

  60. 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

  61. Example J ∞ / Q false-Tate extension. Somnath Jha (IIT Kanpur) Algebraic functional equation for Selmer groups 23 November 2020 8 / 12

  62. 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

  63. 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

  64. 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

  65. 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

  66. 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

  67. 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

  68. 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

  69. 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

  70. 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

  71. 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

  72. 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

  73. 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

  74. 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

  75. 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

  76. 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

  77. 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

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