X International Workshop on Differential Equations, Number Theory, Data Analysis Methods and Geometry University of Havana, February 19-23, 2007 From the Birch and Swinnerton Dyer Conjecture to the GL 2 Main Conjecture for elliptic curves by Otmar Venjakob
Arithmetic of elliptic curves E elliptic curve over Q : E : y 2 + A 1 xy + A 3 y = x 3 + A 2 x 2 + A 4 x + A 6 , A i ǫ Z . E ( K ) = ? for number fields, local fields, finite fields K l any prime, � reduction of E mod l, E # � E ( F l ) =: 1 − a l + l Hasse-Weil L -function of E : � (1 − a l l − s + ǫ ( l ) l 1 − 2 s ) − 1 , s ǫ C , ℜ ( s ) > 3 L ( E/ Q , s ) := 2 , l � 1 E has good reduction at l where ǫ ( l ) := 0 otherwise 2
Arithmetic of elliptic curves E elliptic curve over Q : E : y 2 + A 1 xy + A 3 y = x 3 + A 2 x 2 + A 4 x + A 6 , A i ǫ Z . E ( K ) = ? for number fields, local fields, finite fields K l any prime, � reduction of E mod l, E # � E ( F l ) =: 1 − a l + l Hasse-Weil L -function of E : � (1 − a l l − s + ǫ ( l ) l 1 − 2 s ) − 1 , s ǫ C , ℜ ( s ) > 3 L ( E/ Q , s ) := 2 , l � 1 E has good reduction at l where ǫ ( l ) := 0 otherwise 3
Arithmetic of elliptic curves E elliptic curve over Q : E : y 2 + A 1 xy + A 3 y = x 3 + A 2 x 2 + A 4 x + A 6 , A i ǫ Z . E ( K ) = ? for number fields, local fields, finite fields K l any prime, � reduction of E mod l, E # � E ( F l ) =: 1 − a l + l Hasse-Weil L -function of E : � (1 − a l l − s + ǫ ( l ) l 1 − 2 s ) − 1 , s ǫ C , ℜ ( s ) > 3 L ( E/ Q , s ) := 2 , l � 1 E has good reduction at l where ǫ ( l ) := 0 otherwise 4
Mordell-Weil Theorem E ( Q ) is a finitely generated abelian group Birch & Swinnerton-Dyer Conjecture If the Taylor expansion at s = 1 is L ( E/ Q , s ) = L ∗ ( E/ Q )( s − 1) r + . . . , then I . r = rk Z E ( Q ) (order of vanishing) � L ∗ ( E/ Q ) # X ( E/ Q ) II . = c l ǫ Q (# E ( Q ) tors ) 2 Ω + R E l (rationality, integrality) X ( E/ Q ) Tate-Shafarevich group R E = det( < P i , P j > ) i,j regulator of E ω N´ eron Differential Ω + = � γ + ω real period of E c l = [ E ( Q l ) : E ns ( Q l )] Tamagawa-number at l 5
Mordell-Weil Theorem E ( Q ) is a finitely generated abelian group Birch & Swinnerton-Dyer Conjecture If the Taylor expansion at s = 1 is L ( E/ Q , s ) = L ∗ ( E/ Q )( s − 1) r + . . . , then I . r = rk Z E ( Q ) (order of vanishing) � L ∗ ( E/ Q ) # X ( E/ Q ) II . = c l ǫ Q (# E ( Q ) tors ) 2 Ω + R E l (rationality, integrality) X ( E/ Q ) Tate-Shafarevich group R E = det( < P i , P j > ) i,j regulator of E ω N´ eron Differential Ω + = � γ + ω real period of E c l = [ E ( Q l ) : E ns ( Q l )] Tamagawa-number at l 6
Mordell-Weil Theorem E ( Q ) is a finitely generated abelian group Birch & Swinnerton-Dyer Conjecture If the Taylor expansion at s = 1 is L ( E/ Q , s ) = L ∗ ( E/ Q )( s − 1) r + . . . , then I . r = rk Z E ( Q ) (order of vanishing) � L ∗ ( E/ Q ) # X ( E/ Q ) II . = c l ǫ Q (# E ( Q ) tors ) 2 Ω + R E l (rationality, integrality) X ( E/ Q ) Tate-Shafarevich group R E = det( < P i , P j > ) i,j regulator of E ω N´ eron Differential Ω + = � γ + ω real period of E c l = [ E ( Q l ) : E ns ( Q l )] Tamagawa-number at l 7
The Selmer group of E Assumption: p ≥ 5 prime such that E has good ordinary reduction at p, i.e. # � E ( F p )[ p ] = p. For any finite extension K/ Q we have the ( p -primary) Selmer group Sel ( E/K ) � E ( K ) ⊗ Z Q p / Z p � Sel ( E/K ) � X ( E/K )( p ) � 0 0 Thus, assuming # X ( E/K ) < ∞ , it holds for the Pon- tryagin dual of the Selmer group Sel ( E/K ) ∨ := Hom( Sel ( E/K ) , Q p / Z p ) , that rk Z E ( K ) = rk Z p Sel ( E/K ) ∨ 8
The Selmer group of E Assumption: p ≥ 5 prime such that E has good ordinary reduction at p, i.e. # � E ( F p )[ p ] = p. For any finite extension K/ Q we have the ( p -primary) Selmer group Sel ( E/K ) � 0 � E ( K ) ⊗ Z Q p / Z p � Sel ( E/K ) � X ( E/K )( p ) 0 Thus, assuming # X ( E/K ) < ∞ , it holds for the Pon- tryagin dual of the Selmer group Sel ( E/K ) ∨ := Hom( Sel ( E/K ) , Q p / Z p ) , that rk Z E ( K ) = rk Z p Sel ( E/K ) ∨ 9
Towers of number fields K n := Q ( E [ p n ]) , 1 ≤ n ≤ ∞ , G n := G ( K n / Q ) G := G ∞ K ∞ G ⊆ GL 2 ( Z p ) closed subgroup i.e. a p -adic Lie group G ∞ K n G n Q X ( E/K n ) := Sel ( E/K n ) ∨ is a compact Z p [ G n ]-module Sel ( E/K n ) ∨ is a finitely gener- X := X ( E/K ∞ ) := lim ← − n ated Λ( G )-module, where Λ( G ) = lim Z p [ G n ] ← − n denotes the Iwasawa algebra of G, a noehterian possibly non-commutative ring. 10
Towers of number fields K n := Q ( E [ p n ]) , 1 ≤ n ≤ ∞ , G n := G ( K n / Q ) G := G ∞ K ∞ G ⊆ GL 2 ( Z p ) closed subgroup i.e. a p -adic Lie group G ∞ K n G n Q X ( E/K n ) := Sel ( E/K n ) ∨ is a compact Z p [ G n ]-module Sel ( E/K n ) ∨ is a finitely gener- X := X ( E/K ∞ ) := lim ← − n ated Λ( G )-module, where Λ( G ) = lim Z p [ G n ] ← − n denotes the Iwasawa algebra of G, a noehterian possibly non-commutative ring. 11
Twisted L -functions Irr( G n ) irreducible representations of G n , ρ : G → GL ( V ρ ) , realized over a number field ⊆ C or a local field ⊆ Q l ( ρ, V ρ ) ǫ Irr( G n ) , n < ∞ L ( E, ρ, s ) L -function of E × ρ � 1 L ( E, ρ, s ) := det(1 − Frob − 1 q T | ( H 1 l ( E ) ⊗ Q V ρ ) I q ) | T = q − s q H 1 l ( E ) := Hom( H 1 ( E ( C ) , Z ) , Q l ) 12
Twisted L -functions Irr( G n ) irreducible representations of G n , ρ : G → GL ( V ρ ) , realized over a number field ⊆ C or a local field ⊆ Q l ( ρ, V ρ ) ǫ Irr( G n ) , n < ∞ L ( E, ρ, s ) L -function of E × ρ : � 1 L ( E, ρ, s ) := det(1 − Frob − 1 q T | ( H 1 l ( E ) ⊗ Q V ρ ) I q ) | T = q − s q H 1 l ( E ) := Hom( H 1 ( E ( C ) , Z ) , Q l ) 13
From BSD to the Main Conjecture algebraic analytic L ( E/K n ) = � Irr( G n ) L ( E, ρ, s ) n ρ X ( E/K n ) ∼ as G n -module p -adic families X ( E/K ∞ ) ∼ ( L ( E, ρ, 1)) ρ ǫ Irr( G n ) ,n< ∞ p -adic L -functions F E := F X L E Characteristic analytic p -adic Element L -function Main Conjecture F E ≡ L E 14
What is new? Example (CM-case): E : y 2 = x 3 − x End( E ) ∼ = Z [ i ] � = Z , i.e. E admits complex multiplica- tion (CM), thus = Z p 2 × finite group G ∼ is abelian. Main conjecture is a Theorem of Rubin in many cases,i.e. the theory is rather well known! Example ( GL 2 -case): E : y 2 + y = x 3 − x 2 End( E ) ∼ = Z , i.e. E does not admit complex multipli- cation, thus G ⊆ o GL 2 ( Z p ) open subgroup is not abelian. It was not even known how to formulate a main con- jecture! New: existence of characteristic elements 15
What is new? Example (CM-case): E : y 2 = x 3 − x End( E ) ∼ = Z [ i ] � = Z , i.e. E admits complex multiplica- tion (CM), thus = Z p 2 × finite group G ∼ is abelian. Main conjecture is a Theorem of Rubin in many cases,i.e. the theory is rather well known! Example ( GL 2 -case): E : y 2 + y = x 3 − x 2 End( E ) ∼ = Z , i.e. E does not admit complex multipli- cation, thus G ⊆ o GL 2 ( Z p ) open subgroup is not abelian. It was not even known how to formulate a main con- jecture! New: existence of characteristic elements 16
Localization of Iwasawa algebras ( joint work with: Coates, Fukaya, Kato and Sujatha ) Assumption: H � G with Γ := G/H ∼ = Z p (is satisfied in our application because K ∞ contains the cyclotomic Z p -extension Q cyc of Q ) We define a certain multiplicatively closed subset T of Λ := Λ( G ) associated with H. Question Can one localize Λ with respect to T ? In general, this is a very difficult question for non- commutative rings! If yes, the localisation with respect to T should be re- lated - by construction - to the following subcategory of the category of Λ-torsion modules: M H ( G ) category of Λ-modules M such that modulo Z p -torsion M is finitely gen- erated over Λ( H ) ⊆ Λ( G ) . ⇐ ⇒ Λ T ⊗ Λ M = 0 17
Localization of Iwasawa algebras ( joint work with: Coates, Fukaya, Kato and Sujatha ) Assumption: H � G with Γ := G/H ∼ = Z p (is satisfied in our application because K ∞ contains the cyclotomic Z p -extension Q cyc of Q ) We define a certain multiplicatively closed subset T of Λ := Λ( G ) associated with H. Question Can one localize Λ with respect to T ? In general, this is a very difficult question for non- commutative rings! If yes, the localisation with respect to T should be re- lated - by construction - to the following subcategory of the category of Λ-torsion modules: M H ( G ) category of Λ-modules M such that modulo Z p -torsion M is finitely gen- erated over Λ( H ) ⊆ Λ( G ) . ⇐ ⇒ Λ T ⊗ Λ M = 0 18
Characteristic Elements Theorem. The localization Λ T of Λ with respect to T exists and there is a surjective map ∂ : K 1 (Λ T ) ։ K 0 ( M H ( G )) arising from K -theory, whose kernel is the image of K 1 (Λ) . Fact: K 1 (Λ T ) ∼ = (Λ T ) × / [(Λ T ) × , (Λ T ) × ] Definition. Any F M ǫ K 1 (Λ T ) with ∂ [ F M ] = [ M ] is called characteristic element of M ǫ M H ( G ) . Property Any f ǫ K 1 (Λ T ) can be interpreted as a map on the isomorphism classes of (continuous) represen- tations ρ : G → Gl n ( O K ) , [ K : Q p ] < ∞ : ρ �→ f ( ρ ) ǫ K ∪ {∞} . 19
Recommend
More recommend
