Non-abelian Cohomology and Diophantine Geometry Minhyong Kim - PowerPoint PPT Presentation

Non-abelian Cohomology and Diophantine Geometry Minhyong Kim Bures-sur-Yvette, April, 2018

Some Coleman functions In Q 2 , ∞ 2 n � n 2 =? n = 1

Some Coleman functions ∞ 2 n � n 2 = 0 n = 1 in the 2-adics. Actually, also true in Q p for all p . � z ∞ z n dt dt � n 2 = 1 − t =: ℓ 2 ( z ) t 0 n = 1

Some Coleman functions Right hand side can be defined via � z � z  1 b dt / t 0 ( dt / t )( dt / ( 1 − t ))  � z M z b = 0 1 b dt / ( 1 − t )   0 0 1 where M z b is the holonomy of the rank 3 unipotent connection on P 1 \ { 0 , 1 , ∞} given by the connection form   0 dt / t 0 − 0 0 dt / ( 1 − t )   0 0 0

Some Coleman functions Locally, we are solving the equations d ℓ 1 = dt / ( 1 − t ); d ℓ 2 = ℓ 1 dt / t . Given a unipotent connection ( V , ∇ ) and two points x , y ∈ P 1 \ { 0 , 1 , ∞} ( Q p ) (possibly tangental), there is a canonical isomorphism ≃ x [) ∇ = 0 ✲ V (]¯ y [) ∇ = 0 M y x ( V , ∇ ) : V (]¯ determined by the property that it’s compatible with Frobenius pull-backs. This is the holonomy matrix above.

Some Coleman functions More generally, the k -logarithm � z ℓ k ( z ) := ( dt / t )( dt / t ) · · · ( dt / t )( dt / ( 1 − t )) 0 is defined as the upper right hand corner of the holonomy matrix arising from the ( k + 1 ) × ( k + 1 ) connection form  0 dt / t 0 0 . . . 0 0  0 0 dt / t 0 . . . 0 0     0 0 0 0 0 dt / t . . .   − .   .   . . . . dt / t 0     0 0 0 0 . . . 0 dt / ( 1 − t )   0 0 0 0 . . . 0 0

Some Coleman functions Coleman derived the following functional equations: ℓ k ( z ) + ( − 1 ) k ℓ k ( z − 1 ) = − 1 k ! log k ( z ); D 2 ( z ) = − D 2 ( z − 1 ); D 2 ( z ) = − D 2 ( 1 − z ); where D 2 ( z ) = ℓ 2 ( z ) + ( 1 / 2 ) log ( z ) log ( 1 − z ) . (The upper right hand corner of the log of the holonomy matrix.)

Some Coleman functions From this, we get D 2 ( − 1 ) = − D 2 ( 1 / ( − 1 )) = 0 , and D 2 ( 2 ) = − D 2 ( 1 − 2 ) = 0 . But D 2 ( 2 ) = ℓ 2 ( 2 ) + ( 1 / 2 ) log ( 2 ) log ( − 1 ) = ℓ 2 ( 2 ) . Also, D 2 ( 1 / 2 ) = − D 2 ( 2 ) = 0 .

Some Coleman functions Note: {− 1 , 2 , 1 / 2 } are exactly the 2-integral points of P 1 \ { 0 , 1 , ∞} , and one can give a global proof of the vanishing. Use, in some sense, the arithmetic geometry of Spec ( Z ) \ { 2 , p , ∞} .

Some Coleman functions [Ishai Dan-Cohen and Stefan Wewers] Let D 4 ( z ) = ζ ( 3 ) ℓ 4 ( z ) + ( 8 / 7 )[ log 3 2 / 24 + ℓ 4 ( 1 / 2 ) / log 2 ] log ( z ) ℓ 3 ( z ) +[( 4 / 21 )( log 3 2 / 24 + ℓ 4 ( 1 / 2 ) / log 2 ) + ζ ( 3 ) / 24 ] log 3 ( z ) log ( 1 − z ) . = ζ ( 3 ) ℓ 4 ( z ) + A log ( z ) ℓ 3 ( z ) + B log 3 ( z ) log ( 1 − z ) .

Some Coleman functions Then [ P 1 \ { 0 , 1 , ∞} ]( Z [ 1 / 2 ]) ⊂ { D 2 ( z ) = 0 , D 4 ( z ) = 0 } and numerical computations for p ≤ 29 indicate equality. The inclusions above are examples of non-abelian explicit reciprocity laws . Remark: The extra equation is definitely necessary in general, since, √ for example, 5 ∈ Z 11 , and √ √ √ D 2 ( − 1 ± 5 ) = D 2 ( 3 ± 5 ) = D 2 ( 1 ± 5 ) = 0 . 2 2 2

Diophantine Geometry: Main Local-to-Global Problem Given number field F and X / F smooth variety (with an integral model), locate ′ � X ( F ) ⊂ X ( A F ) = X ( F v ) v The question is How do the global points sit inside the local points? In fact, there is a classical answer for X = G m , in which case X ( F ) = F ∗ , X ( F v ) = F ∗ v . Problem becomes that of locating F ∗ ⊂ A × F .

Diophantine Geometry: Abelian Class Field Theory We have the Artin reciprocity map � rec v : A × ✲ G ab rec = F , F v and the reciprocity law , which says that the composed map rec F ∗ ⊂ ✲ A × ✲ G ab F F is zero. That is, the reciprocity map gives a defining equation for G m ( F ) ⊂ G m ( A F ) .

Diophantine Geometry: Non-Abelian Reciprocity? We would like to generalize this to other equations by way of a non-abelian reciprocity law . Start with a rather general variety X for which we would like to understand X ( F ) via rec NA X ( F ) ⊂ ✲ X ( A F ) some target with base-point 0 ✲ in such way that rec NA = 0 . becomes an equation for X ( F ) .

Diophantine Geometry: Non-Abelian Reciprocity Notation: F : number field. G F = Gal ( ¯ F / F ) . G v = Gal ( ¯ F v / F v ) for a place v of F . S : finite set of places of F . A F : finite Adeles of F A S F : finite S -integral adeles of F . G S = Gal ( F S / F ) , where F S is the maximal extension of F unramified outside S . � S : product over non-Archimedean places in S . � S H 1 ( G v , A ) : product over non-Archimedean places in S and ‘unramified cohomology’ outside of S .

Diophantine Geometry: Non-Abelian Reciprocity X : a smooth variety over F . Fix base-point b ∈ X ( F ) (sometimes tangential). ∆ = π 1 ( ¯ X , b ) ( 2 ) , pro-finite prime-to-2, étale fundamental group of X = X × Spec ( F ) Spec ( ¯ ¯ F ) with base-point b . ∆ [ n ] , lower central series with ∆ [ 1 ] = ∆ . ∆ n = ∆ / ∆ [ n + 1 ] . T n = ∆ [ n ] / ∆ [ n + 1 ] . Denote by ∆ M , (∆ n ) M , T M pro- M quotients for various finite sets n of prime M .

Diophantine Geometry: Non-Abelian Reciprocity [Coh] For each n and M sufficiently large, T M is torsion-free. n This implies ′ n ) loc � H 1 ( G S F , T M H 1 ( G v , T M n ) ✲ is injective. Assuming [Coh], we get a non-abelian class field theory with coefficients in the nilpotent completion of X .

Diophantine Geometry: Non-Abelian Reciprocity This consists of a filtration X ( A F ) = X ( A F ) 1 ⊃ X ( A F ) 2 1 ⊃ X ( A F ) 2 ⊃ X ( A F ) 3 2 ⊃ X ( A F ) 3 ⊃ X ( A F ) 4 3 ⊃ · · · and a sequence of maps ✲ G n ( X ) rec n : X ( A F ) n rec n + 1 : X ( A F ) n + 1 ✲ G n + 1 ( X ) n n n to a sequence G n ( X ) , G n + 1 ( X ) of profinite abelian groups in such n a way that X ( A F ) n + 1 = rec − 1 n ( 0 ) n and X ( A F ) n + 1 = ( rec n + 1 ) − 1 ( 0 ) . n

Diophantine Geometry: Non-Abelian Reciprocity · · · rec − 1 1 ) − 1 ( 0 ) ⊂ rec − 1 2 ( 0 ) ⊂ ( rec 2 1 ( 0 ) ⊂ X ( A F ) || || || || · · · X ( A F ) 3 X ( A F ) 2 ⊂ X ( A F ) 2 ⊂ ⊂ X ( A F ) 1 2 1 rec 3 rec 2 rec 2 rec 1 2 1 ❄ ❄ ❄ ❄ G 3 G 2 · · · 2 ( X ) G 2 ( X ) 1 ( X ) G 1 ( X )

Diophantine Geometry: Non-Abelian Reciprocity The G n ( X ) are defined as G n ( X ) := Hom [ H 1 ( G F , D ( T n )) , Q / Z ] where D ( T n ) = lim Hom ( T n , µ m ) . − → m G n + 1 S ( T M s X 2 ( X ) := lim [ lim n + 1 )] n ← − − → M S where ′ � s X 2 S ( T M n + 1 ) = Ker [ H 2 ( G S F , T M H 2 ( G v , T M n + 1 ) n + 1 )] . ✲

Diophantine Geometry: Non-Abelian Reciprocity When X = G m , then G n ( X ) = 0 for n ≥ 2, G n + 1 ( X ) = 0 n for all n , and G 1 = Hom [ H 1 ( G F , D (ˆ Z ( 1 ) ( 2 ) )) , Q / Z ] = Hom [ H 1 ( G F , [ Q / Z ] ( 2 ) ) , Q / Z ] = [ G ( 2 ) ] ab F . In this case, rec 1 reduces to the prime-to-2 part of the usual reciprocity map.

Diophantine Geometry: Non-Abelian Reciprocity The reciprocity maps are defined using the local period maps j v : X ( F v ) ✲ H 1 ( G v , ∆); x �→ [ π ( 2 ) 1 ( ¯ X ; b , x )] . Because the homotopy classes of étale paths π ( 2 ) 1 ( ¯ X ; b , x ) form a torsor for ∆ with compatible action of G v , we get a corresponding class in non-abelian cohomology of G v with coefficients in ∆ .

Diophantine Geometry: Non-Abelian Reciprocity These assemble to a map j loc : X ( A F ) ✲ � H 1 ( G v , ∆) , which comes in levels ✲ � j loc H 1 ( G v , ∆ n ) . : X ( A F ) n Also have pro- M versions ′ � j loc H 1 ( G v , ∆ M : X ( A F ) n ) ✲ n and integral versions S j loc : X ( A S � H 1 ( G v , ∆ M F ) n ) . ✲ n

Diophantine Geometry: Non-Abelian Reciprocity To indicate the definition of the reciprocity maps, will just define pro- M versions on X ( A S F ) and assume that n ) loc S H 1 ( G S F , T M ✲ � H 1 ( G v , T M n ) S are injective. In general, one needs first to work with a pro- M quotient for a finite set of primes M and S ⊃ M . Then take a limit over S and M .

Diophantine Geometry: Non-Abelian Reciprocity The first reciprocity map is just defined using x ∈ X ( A F ) �→ d 1 ( j loc 1 ( x )) , where ∗ 1 )) ∨ loc � ✲ � ✲ H 1 ( G S 1 )) ∨ , H 1 ( G v , ∆ M H 1 ( G v , D (∆ M F , D (∆ M D 1 : 1 ) S S is obtained from Tate duality and the dual of localization.

Diophantine Geometry: Non-Abelian Reciprocity To define the higher reciprocity maps, we use the exact sequences p n + 1 ✲ H 1 ( G S F , T M ✲ H 1 ( G S F , ∆ M n ✲ H 1 ( G S 0 n + 1 ) n + 1 ) F , ∆ n ) δ n + 1 ✲ H 2 ( G S F , T M n + 1 ) for non-abelian cohomology and Poitou-Tate duality stating that D n + 1 ✲ � H 1 ( G S F , T M H 1 ( G v , T M ✲ H 1 ( G S , D ( T M n + 1 )) ∨ n + 1 ) n ) S is exact.

Diophantine Geometry: Non-Abelian Reciprocity We proceed as follows: rec 2 1 ( x ) = δ 2 ◦ loc − 1 ( j 1 ( x )) ∈ X 2 S ( T M 2 ) and rec 2 ( x ) = D 2 ( loc (( p 2 1 ) − 1 ( loc − 1 ( j 1 ( x )))) − j 2 ( x )) ∈ H 1 ( G S , D ( T M 2 )) ∨ .