aa r X i v : . [ m a t h . N T ] N ov SPECIALIZATION OF MONODROMY GROUP AND ℓ -INDEPENDENCE CHUN YIN HUI
Abstract.
Let E be an abelian scheme over a geometrically connected variety X definedover k , a finitely generated field over Q . Let η be the generic point of X and x ∈ X a closedpoint. If g l and ( g l ) x are the Lie algebras of the l -adic Galois representations for abelianvarieties E η and E x , then ( g l ) x is embedded in g l by specialization. We prove that the set { x ∈ X closed point | ( g l ) x ( g l } is independent of l and confirm Conjecture 5.5 in [2]. § . Introduction Let E be an abelian scheme of relative dimension n over a geometrically connected variety X defined over k , a finitely generated field over Q . If K is the function field of X and η isthe generic point of X , then A := E η is an abelian variety of dimension n defined over K .The structure morphism X → Spec( k ) induces at the level of ´ etale fundamental groups ashort exact sequence of profinite groups:(0.1) 1 → π ( X k ) → π ( X ) → Γ k := Gal( k/k ) → . Any closed point x : Spec( k ( x )) → X induces a splitting x : Γ k ( x ) → π ( X k ( x ) ) of equation(0 .
1) for π ( X k ( x ) ).Let Γ K = Gal( K/K ) the absolute Galois group of K . For each prime number l , we havethe Galois representation ρ l : Γ K → GL( T l ( A )) where T l ( A ) is the l -adic Tate module of A . This representation is unramified over X and factors through ρ l : π ( X ) → GL( T l ( A ))(still denote the map by ρ l for simplicity). The image of ρ l is a compact l -adic Lie subgroupof GL( T l ( A )) ∼ = GL n ( Z l ). Any closed point x : Spec( k ( x )) → X induces an l -adic Galoisrepresentation by restricting ρ l to x (Γ k ( x ) ). This representation is isomorphic to the Galoisrepresentation of Γ k ( x ) on the l -adic Tate module of E x , the abelian variety specialized at x .For simplicity, write G l := ρ l ( π ( X )), g l := Lie( G l ), ( G l ) x := ρ l ( x (Γ k ( x ) )) and ( g l ) x :=Lie(( G l ) x ). We have ( g l ) x ⊂ g l . We set X cl the set of closed points of X and define theexceptional set X ρ E,l := { x ∈ X cl | ( g l ) x ( g l } . The main result (Theorem 1.4) of this note is that the exceptional set X ρ E,l is independentof l . Conjecture 5.5 in [Cadoret & Tamagawa 2] is then a direct application of our theorem. § . l -independence of X ρ E,l
Theorem 1.1. (Serre [5 § A be an abelian variety defined over a field K finitelygenerated over Q and let Γ K = Gal( K/K ). If ρ l : Γ K → GL( T l ( A )) is the l -adic representa-tion of Γ K , then the Lie algebra g l of ρ l (Γ K ) is algebraic and the rank of g l is independentof the prime l .Since V l := T l ( A ) ⊗ Z l Q l is a semisimple Γ K -module (Faltings and W¨ustholz [3 Chap. 6]),the action of the Zariski closure of ρ l (Γ K ) in GL V l is also semisimple on V l . Therefore it isa reductive algebraic group (Borel [1]). By Theorem 1.1, g l is algebraic. So the rank of g l isjust the dimension of maximal tori. We state two more theorems: Theorem 1.2. (Faltings and W¨ustholz [3 Chap. 6]) Let A be an abelian variety de-fined over a field k finitely generated over Q and let Γ k = Gal( k/k ). Then the mapEnd k ( A ) ⊗ Z Q l → End G k ( V l ( A )) is an isomorphism. Theorem 1.3. (Zarhin [6 § V be a finite dimensional vector space over a field ofcharacteristic 0. Let g ⊂ g ⊂ End( V ) be Lie algebras of reductive subgroups of GL V . Weassume that the centralizers of g and g in End( V ) are equal and that the ranks of g and g are equal. Then g = g .We are now able to prove our main theorem. Theorem 1.4.
The set X ρ E,l is independent of l . Proof.
Suppose x ∈ X cl \ X ρ l , then ( g l ) x = g l . It suffices to show g l ′ = ( g l ′ ) x :=Lie( ρ l ′ ( x (Γ k ( x ) ))) for any prime number l ′ . Since base change with finite field extensionof k ( x ) does not change the Lie algebras, End k ( E x ) is finitely generated, and we have theexponential map from Lie algebras to Lie groups, we may assume that End k ( E x ) = End k ( E x )and End Γ k ( V l ( E x )) = End ( g l ) x ( V l ( E x )). We do the same for the abelian variety E η /K . Wetherefore have dim Q l ′ (End g l ′ ( V p ( E η ))) = dim Q l ′ (End K ( E η ) ⊗ Z Q l ′ ) = dim Q l (End K ( E η ) ⊗ Z Q l ) = dim Q l (End g l ( V l ( E η ))) = dim Q l (End ( g l ) x ( V l ( E x ))) = dim Q l (End k ( E x ) ⊗ Z Q l ) = dim Q l ′ (End k ( E x ) ⊗ Z Q p ) = dim Q l ′ (End ( g l ′ ) x ( V l ′ ( E x ))) . Theorem 1.2 implies the first, third, fifth and seventh equality. The dimensions of End K ( E η ) ⊗ Z Q l and End k ( E x ) ⊗ Z Q l as vector spaces are independent of l imply the second and the sixthequality. g l = ( g l ) x implies the fourth equality.We have End g l ′ ( V l ′ ( E η )) = End ( g l ′ ) x ( V l ′ ( E x )) because the left one is contained in the rightone. In other words, the centralizer of ( g l ′ ) x is equal to the centralizer of g l ′ . We know that( g l ′ ) x ⊂ g l ′ are both reductive from the semisimplicity of Galois representaion (Faltings and PECIALIZATION OF MONODROMY GROUP AND ℓ -INDEPENDENCE 3 W¨ustholz [3 Chap. 6]). By Theorem 1.1 on l -independence of reductive ranks and g l = ( g l ) x ,we have: rank( g l ′ ) = rank( g l ) = rank( g l ) x = rank( g l ′ ) x . Therefore, by Theorem 1.3 we conclude that ( g l ′ ) x = g l ′ and thus prove the theorem. (cid:3) Corollary 1.5 (Conjecture 5.5 [2]).
Let k be a field finitely generated over Q , X asmooth, separated, geometrically connected curve over k with quotient field K . Let η bethe generic point of X and E an abelian scheme over X . Let ρ l : π ( X ) → GL( T l ( E η )) bethe l -adic representation. Then there exists a finite subset X E ⊂ X ( k ) such that for anyprime l , X ρ E,l = X E , where X ρ E,l is the set of all x ∈ X ( k ) such that ( G l ) x is not open in G l := ρ l ( π ( X )). Proof.
The uniform open image theorem for GSRP representations [2 Thm. 1.1] impliesthe finiteness of X ρ E,l . Theorem 1.4 implies l -independence. (cid:3) Corollary 1.6.
Let A be an abelian variety of dimension n ≥ K finitely generated over Q . Let Γ K = Gal( K/K ) denote the absolute Galois group of K . Foreach prime number l , we have the Galois representation ρ l : Γ K → GL( T l ( A )) where T l ( A )is the l -adic Tate module of A . If the Mumford-Tate conjecture for abelian varieties overnumber fields is true, then there is an algebraic subgroup H of GL n defined over Q suchthat ρ l (Γ K ) ◦ is open in H ( Q l ) for all l . Proof.
There exists an abelian scheme E over a variety X defned over a number field k such that the function field of X is K and E η = A where η is the generic point of X (see, e.g.,Milne [4 § § x ∈ X such that ( g l ) x = g l . Therefore, wehave ( g l ) x = g l for any prime l by Theorem 1.4. Since all Lie algebras are algebraic (Theorem1.1), if we take H as the Mumford-Tate group of E x , ρ l (Γ K ) ◦ is then open in H ( Q l ) for all l . (cid:3) Question.
Is the algebraic group H in Corollary 1.6 isomorphic to the Mumford-Tategroup of the abelian variety A ? Acknowledgement.
This work grew out of an attempt to prove Conjecture 5.5 in [2]suggested by my advisor, Professor Michael Larsen. I am grateful to him for the sugges-tion, guidance and encouragement. I would also like to thank Anna Cadoret for her usefulcomments on an earlier version of this note.
References
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Linear Algebraic Groups , Graduate Texts in Mathematics, 126 (2nd ed.), Springer-Verlag 1991.2. A. Cadoret and A. Tamagawa,
A uniform open image theorem for l -adic representations I , preprint.3. G. Faltings and G. W¨ustholz, Rational Points , Seminar Bonn/Wuppertal 1983-1984, Vieweg 1984.4. J. S. Milne, “Jacobian Varieties”. Arithmetic Geometry . New York: Springer-Verlag 1986.5. J-P. Serre,
Letter to K. A. Ribet , Jan. 1, 1981.6. Y. G. Zarhin,
Abelian varieties having a reduction of K type , Duke Math. J. 65 (1992), 511-527. CHUN YIN HUI