Diophantine Geometry and Non-Abelian Reciprocity Laws Minhyong Kim - PowerPoint PPT Presentation

Diophantine Geometry and Non-Abelian Reciprocity Laws Minhyong Kim Madrid, December, 2014

Diophantine Geometry: Abelian Case The Hasse-Minkowski theorem says that ax 2 + by 2 = c has a solution in a number field F and only if it has a solution in F v for all v . There are straightforward algorithms for determining this. For example, we need only check for v = ∞ and v | 2 abc , and there, a solution exists if and only if ( a , b ) v ( b , c ) v ( c , a ) v ( c , − 1 ) v = 1 .

Diophantine Geometry: Main Local-to-Global Problem 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 of satisfactory sort for conic equations.

Diophantine Geometry: Main Local-to-Global Problem In that case, assume for simplicity that there is a rational point (and that the points at infinity are rational), so that X ≃ G m . Then 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 Here, G ab F = Gal ( F ab / F ) , and F ab is the maximal abelian algebraic extension of F .

Diophantine Geometry: Abelian Class Field Theory Artin’s reciprocity law : The map Rec ✲ G ab F ∗ ⊂ ✲ A × F F is zero. Key point is that the reciprocity law becomes a result of Diophantine geometry. That is, the reciprocity map gives a defining equation for G m ( 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 NA X ( F ) ⊂ ✲ X ( A F ) Rec some target with base-point ✲ in such way that Rec NA = base-point becomes an equation for X ( F ) .

Diophantine Geometry: Non-Abelian Reciprocity To rephrase: we would like to construct class field theory with coefficients in a general variety X generalizing CFT with coefficients in G m Will describe a version that works for smooth hyperbolic curves.

Diophantine Geometry: Non-Abelian Reciprocity (Joint with Jonathan Pridham) 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 : Adeles of F A S F : S -integral adeles of F . F = Gal ( F S / F ) , where F S is the maximal extension of F G S unramified outside S .

Diophantine Geometry: Non-Abelian Reciprocity X : a smooth curve over F with genus at least two; b ∈ X ( F ) (sometimes tangential). X , b ) ( 2 ) : ∆ = π 1 ( ¯ 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 ] .

Diophantine Geometry: Non-Abelian Reciprocity We then have a nilpotent class field theory with coefficients in X made up of a filtration X ( A F ) = X ( A F ) 1 ⊃ X ( A F ) 2 ⊃ X ( A F ) 3 ⊃ · · · and a sequence of maps ✲ G n ( X ) rec n : X ( A F ) n to a sequence G n ( X ) of profinite abelian groups in such a way that X ( A F ) n + 1 = rec − 1 n ( 0 ) .

Diophantine Geometry: Non-Abelian Reciprocity X ( A F ) 3 = rec − 1 X ( A F ) 2 = rec − 1 · · · ⊂ 2 ( 0 ) ⊂ 1 ( 0 ) ⊂ X ( A F ) 1 = X ( A F ) · · · rec 3 rec 2 rec 1 ❄ ❄ ❄ · · · G 3 ( X ) G 2 ( X ) G 1 ( X ) rec n is defined not on all of X ( A F ) , but only on the kernel (the inverse image of 0) of all the previous rec i .

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 When X = G m , then G n ( X ) = 0 for n ≥ 2 and G 1 = Hom [ H 1 ( G F , D (ˆ Z ( 1 ))) , Q / Z ] = Hom [ H 1 ( G F , Q / Z ) , Q / Z ] = G ab F .

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 1 ( ¯ X ; b , x )] . Because the homotopy classes of étale paths π 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

Diophantine Geometry: Non-Abelian Reciprocity The first reciprocity map is just defined using x ∈ X ( A F ) �→ d 1 ( j loc 1 ( x )) , where S M S M 1 )) ∨ loc ∗ � H 1 ( G v , ∆ M � H 1 ( G v , D (∆ M ✲ H 1 ( G S M F , D (∆ M 1 )) ∨ , d 1 : 1 ) ✲ is obtained from Tate duality and the dual of localization. One needs first to work with a pro- M quotient for a finite set of primes M and S M = S ∪ M . Here, S M � H 1 ( G v , ∆ M � H 1 ( G v , ∆ M � H 1 ( G v / I v , ∆ M 1 ) = 1 ) × n ) . v ∈ S M v / ∈ S M

Diophantine Geometry: Non-Abelian Reciprocity To define the higher reciprocity maps, we use the exact sequences ✲ H 1 c ( G S M F , T M ✲ H 1 z ( G S M F , ∆ M ✲ H 1 z ( G S M 0 n + 1 ) n + 1 ) F , ∆ n ) δ n + 1 ✲ H 2 c ( G S M F , T M n + 1 ) for non-abelian cohomology with support and Poitou-Tate duality c ( G S M F , T M n + 1 ) ≃ H 1 ( G S M F , D ( T M n + 1 )) ∨ . d n + 1 : H 2

Diophantine Geometry: Non-Abelian Reciprocity Essentially, n + 1 = d n + 1 ◦ δ n + 1 ◦ loc − 1 ◦ j n . rec M S M − 1 n ) loc j loc � H 1 ( G v , ∆ M ✲ H 1 n ( x ) ( G S M F , ∆ M n ✲ x ∈ X ( A F ) n + 1 n ) j loc δ n + 1 d n + 1 ✲ H 2 c ( G S M ✲ H 1 ( G S M F , T M F , D ( T M n + 1 )) ∨ . n + 1 ) We take a limit over M to get the reciprocity maps.

Diophantine Geometry: Non-Abelian Reciprocity Put X ( A F ) ∞ = ∩ ∞ n = 1 X ( A F ) n . Theorem (Non-abelian reciprocity) X ( F ) ⊂ X ( A F ) ∞ .

Diophantine Geometry: Non-Abelian Reciprocity Remark: When F = Q and p is a prime of good reduction, suppose there is a finite set T of places such that H 1 ( G S F , ∆ p ✲ � H 1 ( G v , ∆ p n ) n ) v ∈ T is injective. Then the reciprocity law implies finiteness of X ( F ) .

Diophantine Geometry: Non-Abelian Reciprocity ✲ X ( A F ) X ( F ) j g j loc n n ❄ ❄ loc H 1 ( G S M ✲ � F , ∆ M H 1 ( G v , ∆ M n ) n ) H 1 ( G S M F , ∆ M n + 1 ) ✲ j g 1 + n j g ❄ n H 1 ( G S M F , ∆ M X ( F ) n + 1 ) ✲

Diophantine Geometry: Non-Abelian Reciprocity If x ∈ X ( A F ) comes from a global point x g ∈ X ( F ) , then there will be a class j n ( x ) ( G S M j g n ( x g ) ∈ H 1 F , ∆ M n ) for every n corresponding to the global torsor π et , M ( ¯ X ; b , x g ) . 1 That is, j g n ( x g ) = loc − 1 ( j loc n ( x )) and δ n + 1 ( j g n ( x g )) = 0 for every n .

A non-abelian conjecture of Birch and Swinnerton-Dyer type Let ✲ X ( F v ) Pr v : X ( A F ) be the projection to the v -adic component of the adeles. Define X ( F v ) n := Pr v ( X ( A F ) n ) . Thus, X ( F v ) = X ( F v ) 1 ⊃ X ( F v ) 2 ⊃ X ( F v ) 3 ⊃ · · · ⊃ X ( F v ) ∞ ⊃ X ( F ) . Conjecture: Let X / Q be a projective smooth curve of genus at least 2. Then for any prime p of good reduction, we have X ( Q p ) ∞ = X ( Q ) .

A non-abelian conjecture of Birch and Swinnerton-Dyer type Can consider more generally integral points on affine hyperbolic X as well. Conjecture: Let X be an affine smooth curve with non-abelian fundamental group and S a finite set of primes. Then for any prime p / ∈ S of good reduction, we have X ( Z [ 1 / S ]) = X ( Z p ) ∞ . Should allow us to compute X ( Q ) ⊂ X ( Q p ) or X ( Z [ 1 / S ]) ⊂ X ( Z p ) .

A non-abelian conjecture of Birch and Swinnerton-Dyer type Whenever we have an element k n ∈ H 1 ( G T , Hom ( T M n , Q p ( 1 ))) , we get a function rec n k n ✲ H 1 ( G T , D ( T M n )) ∨ ✲ Q p X ( A Q ) n that kills X ( Q ) ⊂ X ( A Q ) n . Need an explicit reciprocity law that describes the image X ( Q p ) n .

A non-abelian conjecture of Birch and Swinnerton-Dyer type Computational approaches all rely on the theory of U ( X , b ) , the Q p -pro-unipotent fundamental group of ¯ X with Galois action, and the diagram ✲ X ( Q p ) X ( Q ) j DR n j g j p n n ✲ loc p ❄ ❄ ≃ D n f ( G T ✲ H 1 ✲ U DR H 1 / F 0 Q , U n ) f ( G p , U n ) n

A non-abelian conjecture of Birch and Swinnerton-Dyer type The key point is that the map j DR ✲ U DR / F 0 X ( Q p ) can be computed explicitly using iterated integrals, and X ( Q ) ⊂ X ( Q p ) n ⊂ [ j DR ] − 1 [ Im ( D ◦ loc p n )] . n

A non-abelian conjecture of Birch and Swinnerton-Dyer type Two more key facts: 1. As soon as D ◦ loc p n has non-dense image, X ( Q p ) n is finite. This follows from analytic properties of Coleman functions and the fact that j DR has dense image. That is, in this case, n Im ( j DR ) ∩ Im ( D ◦ loc p ) is finite. n X ( Q ) ✛ ✲ H 1 f ( G T Q , U n ) X ( Q p ) ✲ ✛ U DR / F 0 n

A non-abelian conjecture of Birch and Swinnerton-Dyer type 2. If A DR denotes the coordinate ring of U DR / F 0 , then the n n functions [ j DR n + 1 ] ∗ ( A DR n + 1 ) contains many elements algebraically independent from [ j DR ] ∗ ( A DR n ) . n U DR n + 1 / F 0 ✲ j DR n + 1 ❄ j DR n ✲ U DR / F 0 X ( Q p ) n