Mixed Tate motives and the unit equation Stefan Wewers, Ishai - PowerPoint PPT Presentation

Mixed Tate motives and the unit equation Stefan Wewers, Ishai Dan-Cohen Ulm/Essen December 4th, 2014

The unit equation Notation: ◮ X := P 1 − { 0 , 1 , ∞} ◮ S finite set of primes ◮ R := Z [ S − 1 ] Then X ( R ) = { ( u , v ) ∈ ( R × ) 2 | u + v = 1 } .

The unit equation Notation: ◮ X := P 1 − { 0 , 1 , ∞} ◮ S finite set of primes ◮ R := Z [ S − 1 ] Then X ( R ) = { ( u , v ) ∈ ( R × ) 2 | u + v = 1 } . Theorem (Mahler,Siegel) X ( R ) is finite.

The unit equation Notation: ◮ X := P 1 − { 0 , 1 , ∞} ◮ S finite set of primes ◮ R := Z [ S − 1 ] Then X ( R ) = { ( u , v ) ∈ ( R × ) 2 | u + v = 1 } . Theorem (Mahler,Siegel) X ( R ) is finite. Remark X ( R ) is computable, using e.g. Baker’s method and LLL.

The Chabauty-Kim method For p �∈ S define pro-unipotent algebraic groups over Q p : Π et , [ n ] Π dR , [ n ] Π et Π dR X = lim , = lim . ← − X X ← − X n n Q , � E.g. Π et X is the Malcev completion of π et 1 ( X ¯ 01) over Q p (with G Q -action).

The Chabauty-Kim method For p �∈ S define pro-unipotent algebraic groups over Q p : Π et , [ n ] Π dR , [ n ] Π et Π dR X = lim , = lim . ← − X X ← − X n n Q , � E.g. Π et X is the Malcev completion of π et 1 ( X ¯ 01) over Q p (with G Q -action). For each z ∈ X ( R ) we get Π et , [ n ] -torsor X Π et , [ n ] ( � 01 , z ) , X classified by a family of elements κ ( p ) n ( z ) ∈ Sel ( p ) f ( G S , Π et , [ n ] S , n := H 1 ) X in a Selmer variety .

� � � The main diagram Using p -adic Hodge theory and Coleman integration we obtain a commutative diagram X ( R ) � � X ( Z p ) κ ( p ) α n n h ( p ) Sel ( p ) � Π dR , [ n ] n . S , n X Here h ( p ) is an algebraic map, while α n is locally analytic. n

� � � The main diagram Using p -adic Hodge theory and Coleman integration we obtain a commutative diagram X ( R ) � � X ( Z p ) κ ( p ) α n n h ( p ) Sel ( p ) � Π dR , [ n ] n . S , n X Here h ( p ) is an algebraic map, while α n is locally analytic. n We set h ( p ) n ( Sel ( p ) X ( Z p ) n := α − 1 � � S , n ) ⊂ X ( Z p ) . n Then X ( Z p ) 1 ⊃ X ( Z p ) 2 ⊃ . . . ⊃ X ( R ) .

� � � Kim’s conjecture X ( R ) � � X ( Z p ) . κ ( p ) α n n h ( p ) Sel ( p ) � Π dR , [ n ] n . S , n X Theorem (M. Kim, 2005) For n ≫ 0 we have dim Sel ( p ) S , n < dim Π dR , [ n ] X and hence | X ( Z p ) n | < ∞ . Corollary X ( R ) is finite.

� � � Kim’s conjecture X ( R ) � � X ( Z p ) . κ ( p ) α n n h ( p ) Sel ( p ) � Π dR , [ n ] n . S , n X Theorem (M. Kim, 2005) For n ≫ 0 we have dim Sel ( p ) S , n < dim Π dR , [ n ] X and hence | X ( Z p ) n | < ∞ . Conjecture (M. Kim, 2012) For n ≫ 0 we have X ( R ) = X ( Z p ) n .

Goals Our goals: ◮ algorithm to compute the map h ( p ) : Sel ( p ) S , n → Π dR , [ n ] n X ◮ test Kim’s conjecture numerically

Goals Our goals: ◮ algorithm to compute the map h ( p ) : Sel ( p ) S , n → Π dR , [ n ] n X ◮ test Kim’s conjecture numerically Problems: ◮ Galois cohomology difficult to compute ◮ h ( p ) may involve transcendental numbers n

Goals Our goals: ◮ algorithm to compute the map h ( p ) : Sel ( p ) S , n → Π dR , [ n ] n X ◮ test Kim’s conjecture numerically Problems: ◮ Galois cohomology difficult to compute ◮ h ( p ) may involve transcendental numbers n Solution: use mixed Tate motives

Mixed Tate motives Let MTM S denote the category of mixed Tate motives over Q , unramified outside of S . This is a Tannakian category with a canonical fiber functor ω = gr ∗ w : MTM S → Vec ( Q ) .

Mixed Tate motives Let MTM S denote the category of mixed Tate motives over Q , unramified outside of S . This is a Tannakian category with a canonical fiber functor ω = gr ∗ w : MTM S → Vec ( Q ) . Set G mot := π 1 ( MTM S , ω ) = U mot ⋊ G m . S S We will identify MTM S with the category of finite dimensional Z -graded representations of U mot . S

The ring of unipotent periods We work with the graded Hopf algebra A [ n ] A S := O ( U mot ) = ⊕ n ≥ 0 A S , n = lim S . S − → n

The ring of unipotent periods We work with the graded Hopf algebra A [ n ] A S := O ( U mot ) = ⊕ n ≥ 0 A S , n = lim S . S − → n The known structure of Ext i MTM S ( Q ( n ) , Q ( m )) shows that there is a noncanonical isomorphism A S ∼ = Q � g ℓ , ℓ ∈ S ; f 3 , f 5 , . . . � , with deg( g ℓ ) = 1, deg( f n ) = n .

The motivic fundamental group Deligne and Goncharov have constructed Π mot , [ n ] ( X , � Π mot := π mot 01) = lim . X 1 ← − X n We view Π mot as a pro-unipotent algebraic group with X G mot -action. S

The motivic fundamental group Deligne and Goncharov have constructed Π mot , [ n ] ( X , � Π mot := π mot 01) = lim . X 1 ← − X n We view Π mot as a pro-unipotent algebraic group with X G mot -action. S It has the following ‘explicit’ description: Π mot = Spec Q � e 0 , e 1 � , X where e 0 = dz dz z , e 1 = 1 − z . Action of U mot determined by motivic multiple zeta values . S

The motivic Selmer variety Set , Π mot , [ n ] Sel S , n := H 1 ( G mot ) . S X This is an affine Q -variety, isomorphic to A N Q . For z ∈ X ( R ), the path torsor Π mot , [ n ] ( � 01 , z ) X defines a class κ n ( z ) ∈ Sel S , n . The resulting map κ n : X ( R ) → Sel S , n is the motivic Kummer map .

Motivic multiple polylogarithms For z ∈ X ( R ) there exists a canonical path ( � γ dR ∈ Π mot 01 , z ) z X (not fixed by G S !),

Motivic multiple polylogarithms For z ∈ X ( R ) there exists a canonical path ( � γ dR ∈ Π mot 01 , z ) z X (not fixed by G S !), defining a cocycle → Π mot , [ n ] , Π mot , [ n ] ∈ Z 1 ( G mot c n ( z ) : G mot ) S S X X representing κ n ( z ).

Motivic multiple polylogarithms For z ∈ X ( R ) there exists a canonical path ( � γ dR ∈ Π mot 01 , z ) z X (not fixed by G S !), defining a cocycle → Π mot , [ n ] , Π mot , [ n ] ∈ Z 1 ( G mot c n ( z ) : G mot ) S S X X representing κ n ( z ). The coefficients of the dual map w �→ Li mot Q � e 0 , e 1 � → A S , ( z ) , w are the motivic multi polylogarithms . Formally, � Li mot ( z ) = w . w γ dR z

The p -adic period map For p �∈ S there is a p-adic period map per p : A S → Q p (Chatzistamatiou-¨ Unver). One shows that Li ( p ) w ( z ) := per p ( Li mot ( z )) ∈ Q p , w where Li ( p ) : X ( Z p ) → Q p w are Furusho’s p -adic multiple polylogarithms (which appear in Kim’s diagram).

The map h n Lemma Every class in Sel S , n is represented by a unique cocycle c whose dual map c ∗ : Q � e 0 , e 1 � → A S respects the grading.

The map h n Lemma Every class in Sel S , n is represented by a unique cocycle c whose dual map c ∗ : Q � e 0 , e 1 � → A S respects the grading. We define → Π mot , [ n ] h n : Sel S , n × U mot S X by h n ([ c ] , σ ) := c ( σ ) .

The map h n Lemma Every class in Sel S , n is represented by a unique cocycle c whose dual map c ∗ : Q � e 0 , e 1 � → A S respects the grading. We define → Π mot , [ n ] h n : Sel S , n × U mot S X by h n ([ c ] , σ ) := c ( σ ) . Then h ( p ) = h n ( · , σ p ) : Sel ( p ) = Sel S , n ⊗ Q p → Π dR , [ n ] S , n ∼ , n X where σ p ∈ U mot ( Q p ) corresponds to per p . S

Exhaustion of A S by iterated integrals Assumption (Exhaustion) There exists ¯ S ⊃ S and a set I (resp. I n ) of motivic iterated integrals of the form I mot ( a 0 ; a 1 , . . . , a r ; a r +1 ) , S (resp. A [ n ] with a i ∈ X (¯ R ) ∪ { 0 , 1 , ∞} and such that A ¯ S ) is ¯ generated by I (resp. I n ).

Exhaustion of A S by iterated integrals Assumption (Exhaustion) There exists ¯ S ⊃ S and a set I (resp. I n ) of motivic iterated integrals of the form I mot ( a 0 ; a 1 , . . . , a r ; a r +1 ) , S (resp. A [ n ] with a i ∈ X (¯ R ) ∪ { 0 , 1 , ∞} and such that A ¯ S ) is ¯ generated by I (resp. I n ). Remark ◮ For ¯ S = ∅ (resp. ¯ S = { 2 } ), this is a theorem of Brown (resp. Deligne).

Exhaustion of A S by iterated integrals Assumption (Exhaustion) There exists ¯ S ⊃ S and a set I (resp. I n ) of motivic iterated integrals of the form I mot ( a 0 ; a 1 , . . . , a r ; a r +1 ) , S (resp. A [ n ] with a i ∈ X (¯ R ) ∪ { 0 , 1 , ∞} and such that A ¯ S ) is ¯ generated by I (resp. I n ). Remark ◮ For ¯ S = ∅ (resp. ¯ S = { 2 } ), this is a theorem of Brown (resp. Deligne). ◮ True for A [2] S if ¯ S contains all primes ≤ max S. ¯

Exhaustion of A S by iterated integrals Assumption (Exhaustion) There exists ¯ S ⊃ S and a set I (resp. I n ) of motivic iterated integrals of the form I mot ( a 0 ; a 1 , . . . , a r ; a r +1 ) , S (resp. A [ n ] with a i ∈ X (¯ R ) ∪ { 0 , 1 , ∞} and such that A ¯ S ) is ¯ generated by I (resp. I n ). Remark ◮ For ¯ S = ∅ (resp. ¯ S = { 2 } ), this is a theorem of Brown (resp. Deligne). ◮ True for A [2] S if ¯ S contains all primes ≤ max S. ¯ ◮ Goncharov conjectures this if ¯ S contains all primes.

Decomposition Theorem (D.-C.,W.,Brown) Assuming exhaustion, there is an ‘algorithm’ which determines 1. a subset B ⊂ I such that A S = Q [ B ] , 2. expression of all elements of I as polynomials in B.