Non-existence of certain CM abelian varieties with prime power torsion
aa r X i v : . [ m a t h . N T ] D ec Non-existence of certain CM abelian varieties with primepower torsion
Yoshiyasu Ozeki ∗ Abstract
In this paper, we study a conjecture of Rasmussen and Tamagawa, on the finiteness ofthe set of isomorphism classes of abelian varieties with constrained prime power torsion. Ourresult is related with abelian varieties which have complex multiplication over their fields ofdefinition.
Let K be a finite extension of Q , ¯ K an algebraic closure of K and G K = Gal( ¯ K/K ) the absoluteGalois group of K . For a prime number ℓ , we denote by K ( µ ℓ ) the field generated by the ℓ -throots of unity. We denote by A ( K, g, ℓ ) the set of K -isomorphism classes of g -dimensional abelianvarieties A over K which satisfy the following:(RT ℓ ) K ( A [ ℓ ]) is an ℓ -extension of K ( µ ℓ ).(RT red ) The abelian variety A has good reduction away from ℓ over K .It follows from the condition (RT red ) and Faltings’ result on the Shafarevich Conjecture that A ( K, g, ℓ ) is a finite set. Rasmussen and Tamagawa suggested that such finiteness should hold ifwe take the union of these sets for ℓ varying over all primes. Conjecture 1.1 ([RT], Conjecture 1) . The set A ( K, g ) := { ([ A ] , ℓ ) | [ A ] ∈ A ( K, g, ℓ ) , ℓ :prime number } is finite, that is, the set A ( K, g, ℓ ) is empty for any prime number ℓ large enough. This conjecture is proved only in a few case. For example, Conjecture 1.1 in the case where K is the rational number field or certain quadratic field, with g = 1 is proved by Rasmussen andTamagwa in [RT]. Their proof is based on results on K -rational points of modular curves of [Ma]and [Mo]. Arguments for the moduli of algebraic points on Shimura curves ([Ar], [AM]) also giveresults on Conjecture 1.1 for QM-abelian surfaces and certain quadratic field K .In this paper, we prove Conjecture 1.1 for abelian varieties in A ( K, g, ℓ ) which satisfy thecondition that representations associated with their ℓ -adic Tate modules are abelian. In fact, weprove more general result as follows: Denote by A ′ ( K, g, ℓ ) ab the set of K -isomorphism classes of g -dimensional abelian varieties A over K which satisfy the following:(RT ℓ ) ′ For some finite extension L of K which is unramified at all places of K above ℓ , L ( A [ ℓ ])is an ℓ -extension of L ( µ ℓ ).(RT ab ) The representation ρ A,ℓ : G K → GL ( T ℓ ( A )) associated with the ℓ -adic Tate module T ℓ ( A ) of A has an abelian image.Our main result in this paper is Theorem 1.2.
The set A ′ ( K, g, ℓ ) ab is empty for any prime number ℓ large enough. ∗ Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan.e-mail: [email protected]
Partly supported by the Grant-in-Aid for Research Activity Start-up, The Ministry of Education, Culture, Sports,Science and Technology, Japan. A over K has complex multiplication over K (in the sense of [ST], Section 4), then it is well-known that ρ A,ℓ is abelian (cf. loc . cit ., Section 4, Corollary 2). Thus we obtain Corollary 1.3.
The set of K -isomorphism classes of abelian varieties in A ( K, g, ℓ ) which havecomplex multiplication over K is empty for any prime ℓ large enough. We want to replace “an abelian image” in the statement of (RT ab ) with “a potential abelianimage”. Assume that we obtain Theorem 1.2 with this replacement. Under this assumption, wesee that Conjecture 1.1 holds for CM abelian varieties, that is, if we denote by A ( K, g, ℓ ) CM theset of K -isomorphism classes of abelian varieties in A ( K, g, ℓ ) which have complex multiplicationover ¯ K , then the set A ( K, g ) CM := { ([ A ] , ℓ ) | [ A ] ∈ A ( K, g, ℓ ) CM , ℓ : prime number } is finite.The paper proceeds as follows. Section 2 is devoted to a study of compatible systems. In Section3, we recall some facts about Conjecture 1.1. Finally we prove our main theorem in Section 4. In this section, we use same notation as given in Introduction. To find conditions that compatiblesystems should be of a simple form is important for the proof of Theorem 1.2.
Let E be a finite extension of Q . For a finite place λ of E , we denote by ℓ λ the prime numberbelow λ , E λ the completion of E at λ and F λ the residue field of λ . We denote by E λ (resp. K v ) the completion of E at a finite place λ of E (resp. the completion of K at a finite place v of K ). Let S be a finite set of finite places of K and T a finite set of finite places of E . Put S ℓ = S ∪ { places of K above ℓ } . A representation ρ : G K → GL n ( E λ ) is said to be E -rational withramification set S if ρ is unramified outside S ℓ and the characteristic polynomial det( T − ρ (Fr v ))of Fr v has coefficients in E for each finite place v / ∈ S ℓ of K , where Fr v is an arithmetic Frobeniusof v .Now we give definitions of compatible systems of λ -adic (resp. mod λ ) representations, whichmainly follows from that in [Kh1] and [Kh2]. An E -rational strictly compatible system ( ρ λ ) λ of n -dimensional λ -adic representations of G K with defect set T and ramification set S , consists of,for each finite place λ of E not in T , a continuous representation ρ λ : G K → GL n ( E λ ) that is(i) ρ λ is unramified outside S ℓ λ ;(ii) for any finite place v / ∈ S of K , there exists a monic polynomial f v ( T ) ∈ E [ T ] such that forall places λ / ∈ T of E which is coprime to the residue characteristic of v , the characteristicpolynomial det( T − ρ λ (Fr v )) of Fr v is equal to f v ( T ).An E -rational strictly compatible system (¯ ρ λ ) λ of n -dimensional mod λ representations of G K withdefect set T and ramification set S , consists of, for each finite place λ of E not in T , a continuousrepresentation ¯ ρ λ : G K → GL n ( F λ ) that is(i) ¯ ρ λ is unramified outside S ℓ λ ;(ii) for any finite place v / ∈ S of K , there exists a monic polynomial f v ( T ) ∈ E [ T ] such that forall places λ / ∈ T of E which is coprime to the residue characteristic of v , f v ( T ) is integral at λ and the characteristic polynomial det( T − ρ λ (Fr v )) of Fr v is the reduction of f v ( T ) mod λ .2e will often suppress the sets S and T from the notations. Example 2.1.
Let X be a proper smooth variety over K . Let V ℓ := H r ´et ( X ¯ K , Q ℓ ) ∨ be the dual ofthe ℓ -adic ´etale cohomology group H r ´et ( X ¯ K , Q ℓ ) of X . Then the system ( V ℓ ) ℓ is a strict compatiblesystem whose defect set is all prime numbers and ramification set is the set of finite places v of K such that X has bad reduction at v . This fact follows from the Weil Conjecture which is provedby Deligne (cf. [De1], [De2]).It is conjectured that every E -rational strictly compatible system arises motivically. Conjecture 2.2 ([Kh1], Conjecture 1) . Any E -rational strictly compatible system of λ -adic (resp.mod λ ) representations arises motivically. In fact, this conjecture is true if representations are abelian.
Theorem 2.3 ([Kh2], Theorem 2 and Corollary 1) . An E -rational strictly compatible system ofabelian semisimple λ -adic (resp. mod λ ) representations of G K arises from n Hecke characters.
For a finite place v of K , we denote by G v a decomposition group of v and I v the inertiasubgroup of G v . An inertial level L of K is a collection ( L v ) v of open normal subgroups L v of I v for each finite place v of K such that L v = I v for almost all v . An inertial level L of a geometric λ -adic representation ρ λ of G K , where we use the notion of geometric in the sense of [FM], is thecollection ( L v ( ρ λ )) v of open normal subgroups L v ( ρ λ ) of I v for each finite place v of K , where L v ( ρ λ ) is the largest open subgroup of I v such that the restriction of ρ λ to L v ( ρ λ ) is semi-stable.By definition, we have L v ( ρ λ ) = I v for almost all v . A compatible system ( ρ λ ) λ of geometric λ -adic representations of G K has bounded inertial level if there exists an inertial level L = ( L v ) v such that L v ⊂ L v ( ρ λ ) for all λ and v . Let w , w , . . . , w n be integers. A λ -adic representation ρ λ is E -rational with Frobenius weights w , w , . . . , w n outside S if ρ λ is E -rational with ramificationset S and for all finite places v / ∈ S ℓ of K , the complex roots of the characteristic polynomialdet( T − ρ (Fr v )) of Fr v , for a chosen embedding of E into C , have their complex absolute values q w / v , q w / v , . . . , q w n / v where q v is the cardinality of the residue field of v . A strict compatiblesystem ( ρ λ ) λ is said to be E -rational strict compatible system with Frobenius weights w , w , . . . , w n if each ρ λ is E -rational with Frobenius weights w , w , . . . , w n outside a ramification set of ( ρ λ ) λ .We call w , w , . . . , w n the Frobenius weights of ρ λ (resp. ( ρ λ ) λ ), and ρ λ (resp. ( ρ λ ) λ ) is said to be pure if w = w = · · · = w n . A compatible system ( ρ λ ) λ of geometric λ -adic representations of G K has bounded Hodge-Tate weights if there exist integers a and b with a ≤ b such that, for any λ andfinite place v of K above ℓ λ , all the Hodge-Tate weights of ρ | G v viewed as a Q ℓ -representation arein [ a, b ]. Finally, a compatible system (¯ ρ λ ) λ of mod λ representations of G K is of bounded Artinconductor if there exists an ideal N of K such that, for any λ , the Artin conductor of ¯ ρ λ divides N . Proposition 2.4. (1) An E -rational strictly compatible system ( ρ λ ) λ of abelian semisimple λ -adicrepresentations of G K has bounded inertial level and bounded Hodge-Tate weights. (2) An E -rational strictly compatible system ( ρ λ ) λ of abelian semisimple mod λ representations of G K is of bounded Artin conductor.Proof. By Theorem 2.3, such ( ρ λ ) λ arises from Hecke characters. Hence the Proposition followsfrom standard properties of a representation arising from Hecke characters. Choose an algebraic closure ¯ F λ of F λ . Put χ λ : G K χ ℓλ −→ Z × ℓ λ ֒ → E × λ and ¯ χ λ : G K ¯ χ ℓλ −→ F × ℓ λ ֒ → F × λ ,where χ ℓ λ and ¯ χ ℓ λ are the ℓ λ -adic cyclotomic character and the mod ℓ λ cyclotomic character,respectively. For a representation ¯ ρ λ : G K → GL n ( F λ ) with abelian semisimplification, Schur’slemma shows that (¯ ρ λ ) ss ⊗ ¯ F λ conjugates to the direct sum of n characters, where the subscript“ss” means the semisimplification, and we call these n characters characters associated with ¯ ρ λ .3or a λ -adic representation ρ λ , we denote by ¯ ρ λ a residual representation of ρ λ (for a chosenlattice). Note that the isomorphism class of (¯ ρ λ ) ss is independent of the choice of a lattice by theBrauer-Nesbitt theorem. Theorem 2.5.
Let ( ρ λ ) λ be an E -rational strictly compatible system of n -dimensional geometricsemisimple λ -adic representations of G K . Suppose that there exists an infinite set Λ of finite placesof E which satisfies the following: (1) For any λ ∈ Λ , there exists a place v of K above ℓ λ such that ( a ) ρ λ is semi-stable at v . ( b ) there exist integers w ≤ w which are independent of the choice of λ ∈ Λ such that theHodge-Tate weights of ρ λ | G v are in [ w , w ] . (2) For any λ ∈ Λ , (¯ ρ λ ) ss is abelian and any character associated with ¯ ρ λ has the form ε ¯ χ aλ ,where a is an integer and ε : G K → ¯ F × λ is a character unramified at all places of K above ℓ λ . (3) The Artin conductor of (¯ ρ λ ) ss is bounded independently of the choice of λ ∈ Λ .Then there exist integers m , m , . . . , m n and a finite extension L of K such that, for any λ , therepresentation ρ λ is isomorphic to χ m λ ⊕ χ m λ ⊕ · · · ⊕ χ m n λ on G L . Remark 2.6.
If Conjecture 2.2 holds, then we can remove the conditions (1) and (3) of Theorem2.5 since these conditions are automatically satisfied.
Proof of Theorem 2.5.
By replacing Λ with its infinite subset, we may suppose that ℓ λ does notdivide the discriminant of K and ℓ λ > [ E : Q ] · n for any λ ∈ Λ. Furthermore, we may assume that,for any λ ∈ Λ and a finite place v of K above ℓ λ as in the condition (1), the Hodge-Tate weightsof ρ λ | G v viewed as a Q ℓ λ -representation are positive and bounded independently of the choice of λ ∈ Λ. By the condition (3), there exists an ideal n of O K such that, for any λ ∈ Λ, the Artinconductor outside ℓ λ of (¯ ρ λ ) ss divides n . If we denote by ψ a character associated with (¯ ρ λ ) ss for λ ∈ Λ and decompose ψ = ε ¯ χ aλ where ε is as the condition (2), then the Artin conductor outside ℓ λ of ε also divides n . Hence, replacing the field K with the strict ray class field of K associatedwith n , we may replace the condition (2) with the following condition (2) ′ :(2) ′ For any λ ∈ Λ, (¯ ρ λ ) ss is abelian and any character associated with ¯ ρ λ has the form ¯ χ aλ .Now take any λ ∈ Λ. Let ¯ χ a λ, λ , ¯ χ a λ, λ , . . . , ¯ χ a λ,n λ be all the characters associated with ¯ ρ λ . By thecondition (2) ′ and ℓ λ > [ E : Q ] · n , the representation (¯ ρ λ ) ss conjugates to the direct some of n char-acters (over F λ ) of the form ¯ χ aλ , which has values in F × ℓ λ . Hence if we regard the F λ -representation¯ ρ λ as an F ℓ λ -representation, its semisimplification is of a diagonal form whose diagonal componentsare the copies of ¯ χ a λ, ℓ λ , ¯ χ a λ, ℓ λ , . . . , ¯ χ a λ,n ℓ λ (here we note that ℓ λ > [ F λ : F ℓ λ ] · n ). Furthermore, it isa direct summand of the semisimplification of a residual representation of ρ λ viewed as a Q ℓ λ -representation. Therefore, by Caruso’s result on an upper bound for tame inertia weights ([Ca])and the condition (1), there exists a constant C >
0, which is independent of the choice of λ ∈ Λ,and an integer 0 ≤ b λ,i ≤ C such that( ♯ ) b λ,i ≡ a λ,i mod ℓ λ − i (recall that ℓ λ does not divide the discriminant of K ). Now we claim that the set { b λ, , b λ, , . . . , b λ,n } is independent of the choice of λ ∈ Λ large enough. Denote by S the rami-fication set of ( ρ λ ) λ . Take a v / ∈ S and decompose det( T − ρ λ (Fr v )) = Q nj =1 ( T − α v ,j ). Byconditions (2) ′ and ( ♯ ), we have the congruence Q nj =1 ( T − α v ,j ) ≡ Q nj =1 ( T − q b λ,j v ) in ¯ F λ [ T ]. If Here we use the following fact: Let F be a field of characteristic ℓ >
0. Let ρ and ρ ′ be n -dimensional semisimple F -representation of a group G . Assume that ℓ > n . If det( T − ρ ( g )) = det( T − ρ ′ ( g )) for any g ∈ G , then ρ isisomorphic to ρ ′ . λ is large enough (note that Λ is an infinite set), then we obtain that this congruence is in factan equality in E [ T ]: Q nj =1 ( T − α v ,j ) = Q nj =1 ( T − q b λ,j v ) . Therefore, the set { b λ, , b λ, , . . . , b λ,n } is independent of the choice of λ ∈ Λ with ℓ λ large enough. This proves the claim. We denote { b λ, , b λ, , . . . , b λ,n } by { m , m , . . . , m n } for such a λ ∈ Λ. By the compatibility of ( ρ λ ) λ , weobtain the equation det( T − ρ λ (Fr v )) = Q nj =1 ( T − q m j v ) for any λ and v / ∈ S ℓ λ . Therefore, therepresentation ρ λ is isomorphic to χ m λ ⊕ χ m λ ⊕ · · · ⊕ χ m n λ . By the compatibility of ( ρ λ ) λ , thisfinishes the proof. Corollary 2.7.
Let (¯ ρ λ ) λ be an E -rational strictly compatible system of abelian semisimple mod λ representations of G K . Suppose that, for infinitely many finite places λ of E , any characterassociated with ¯ ρ λ has the form ε ¯ χ aλ , where ε : G K → ¯ F × λ is a character unramified at all places of K above ℓ λ . Then there exist a finite extension L of K and integers m , m , . . . , m n such that, forany λ , the representation ¯ ρ λ is isomorphic to ¯ χ m λ ⊕ ¯ χ m λ ⊕ · · · ⊕ ¯ χ m n λ on G L .Proof. By Theorem 2.3, we know that there exist a finite extension E ′ of E and an E ′ -rationalabelian semisimple compatible system ( ρ λ ′ ) λ ′ of λ ′ -adic representations of G K which arises fromHecke characters such that ( ρ λ ′ ) λ ′ is a lift of (¯ ρ λ ) λ , that is, ¯ ρ λ ′ is isomorphic to ¯ ρ λ ⊗ F λ ′ for any λ and any finite place λ ′ of E ′ above λ . By standard properties of compatible systems of Galoisrepresentations arising from Hecke characters, we see that ( ρ λ ′ ) λ ′ satisfies all the assumptions (1),(2) and (3) in Theorem 2.5. Consequently we obtain the desired result. Corollary 2.8.
Let ( ρ λ ) λ be an E -rational strictly compatible system of n -dimensional semisimple λ -adic representations of G K . Suppose that (i) (¯ ρ λ ) ss is abelian for almost all λ ; (ii) for infinitely many λ , any character associated with (¯ ρ λ ) ss has the form ε ¯ χ aλ , where ε : G K → F × λ is a character unramified at all places of K above ℓ λ .Then there exist integers m , m , . . . , m n and a finite extension L of K such that, for any λ , therepresentation ρ λ is isomorphic to χ m λ ⊕ χ m λ ⊕ · · · ⊕ χ m n λ on G L .Proof. The result follows immediately by applying Corollary 2.7 to the compatible system ((¯ ρ λ ) ss ) λ .Let λ and λ ′ be finite places of E of different residual characteristics. Let ρ λ be an E -rational n -dimensional semisimple λ -adic representations of G K with ramification set S . Suppose that thereexists an semisimple λ ′ -adic representation ρ λ ′ of G K such thatdet( T − ρ λ (Fr v )) = det( T − ρ λ ′ (Fr v ))for any v / ∈ S ℓ λ ∪ S ℓ λ ′ . In the spirit of Fontaine-Mazur’s “Main Conjecture”, we hope that ρ λ ′ iscrystalline for any finite place v ′ of K above ℓ λ ′ when the residual characteristic of λ ′ is prime tothat of any place in S . However to prove this hope seems not to be easy. If ρ λ is abelian, the hopeis true by Theorem 2.3. If we consider representations which is pure, we can improve the statement(1) of Theorem 2.5 as below. (If the hope is true, it is not difficult to prove the proposition belowwithout the assumption of pureness by the similar method of the proof of Theorem 2.5.) Proposition 2.9.
Let ( ρ λ ) λ be an E -rational strictly compatible system of n -dimensional geometricsemisimple λ -adic representations of G K . Suppose that ( ρ λ ) λ is pure. Suppose that there exists aninfinite set Λ of finite places of K which satisfies the following: (1) For any λ ∈ Λ , there exists a place v of K above ℓ λ such that In fact ρ λ and ρ λ ′ shall come from an algebraic variety X and their ramification set S shall be “bad primes”of X . a ) there exists a constant C > which is independent of the choice of λ ∈ Λ such that [ I v : L v ( ρ λ )] < C . Here L v ( ρ λ ) is the inertial level of ρ λ at v ( see Section 2.1 ) . ( b ) there exist integers w ≤ w which are independent of the choice of λ ∈ Λ such that theHodge-Tate weights of ρ λ | G v are in [ w , w ] . (2) For any λ ∈ Λ , (¯ ρ λ ) ss is abelian and any character associated with ¯ ρ λ has the form ε ¯ χ aλ ,where ε : G K → ¯ F × λ is a character unramified at all places of K above ℓ λ . (3) For any λ ∈ Λ , the Artin conductor of (¯ ρ λ ) ss is bounded independently of the choice of λ ∈ Λ .Then there exist an integer m and a finite extension L of K such that, for any λ , the representation ρ λ is isomorphic to ( χ mλ ) ⊕ n on G L .Proof. Most parts of the first paragraph of this proof will proceed by the similar method as theproof of Theorem 2.5 and hence we will often omit precise arguments. First we may assume that,for any λ ∈ Λ,(2) ′ any character associated with ¯ ρ λ has the form ¯ χ aλ and furthermore, ρ λ | G v has Hodge-Tate weights in [0 , r ] for any λ and v as in the condition (1).Here r is a positive integer which is independent of the choice of λ ∈ Λ. Suppose λ is a finite placein Λ. Let ¯ χ a λ, λ , ¯ χ a λ, λ , . . . , ¯ χ a λ,n λ be all the characters associated with ¯ ρ λ . Taking a finite place v as in the condition (1), there exists a finite extension L w of K v such that ρ λ | G Lw is semi-stableand [ L w : K v ] ≤ C . If we denote by e w the absolute ramification index of L w , then it follows e w ≤ C [ K : Q ], and Caruso’s result on an upper bound for tame inertia weights ([Ca]) impliesthat there exists an integer 0 ≤ b ′ λ,i ≤ e w r which satisfies b ′ λ,i ≡ e w a λ,i mod ℓ λ − . Consequently,we see that there exist integers e >
D >
0, which are independent of the choice of λ ∈ Λand b λ,i ≡ ea λ,i mod ℓ λ − b λ,i ∈ [0 , D ]. Take any v / ∈ S ℓ λ and decomposedet( T − ρ λ (Fr v )) = Q nj =1 ( T − α v,j ). Then, by the similar arguments as the proof of Theorem 2.5,we can show that Q nj =1 ( T − α ev,j ) = Q nj =1 ( T − q b λ,j v ) if we take λ ∈ Λ with ℓ λ large enough. Since( ρ λ ) λ is pure, we have n Y j =1 ( T − α ev,j ) = n Y j =1 ( T − q bv )for some integer b . It follows from the compatibility of ( ρ λ ) λ that the above equation holds forany λ (which may not be in Λ) and v / ∈ S ℓ λ .In the argument below, we use the method of the proof of Proposition 1.2 of [KL]. Fix λ anddenote it by λ . Take a finite extension K ′ of K such that there exists a continuous character χ /eλ : G K ′ → E × λ which has values in the integer ring of E λ and ( χ /eλ ) e = χ λ . Replace this K ′ with K . Then we know that, for any v / ∈ S ℓ λ , all the roots of det( T − ρ ′ λ (Fr v )) are roots of unity,where ρ ′ λ is the twist of ρ λ by ( χ /eλ ) − b . Since there are only finitely many such roots of unity,there are only finitely many possibilities for the characteristic polynomial of Fr v . Hence the functionwhich takes g ∈ G K to det( T − ρ ′ λ ( g )) ∈ E [ T ] is continuous and takes only finitely many values byChebotarev’s density theorem. It follows that the set { g ∈ G K | det( T − ρ ′ λ ( g )) = ( T − n } is anopen subset of G K , which contains the identity map of ¯ K . Hence there exists a finite extension L of K such that G L ⊂ { g ∈ G K | det( T − ρ ′ λ ( g )) = ( T − n } . Then we see that ρ λ is isomorphicto (( χ /eλ ) b ) ⊕ n on G L . Since ρ λ is geometric, we know that b/e =: m is an integer and we finishthe proof by the compatibility of ( ρ λ ) λ . We continue to use same notation as in the previous section. Let g ≥ efinition 3.1. We denote by A ( K, g, ℓ ) the set of K -isomorphism classes of g -dimensional abelianvarieties A over K which satisfy the following:(RT ℓ ) K ( A [ ℓ ]) is an ℓ -extension of K ( µ ℓ ).(RT red ) The abelian variety A has good reduction away from ℓ over K .By (RT red ), the set A ( K, g, ℓ ) is a finite set (Theorem 5 of [Fa] and 1. Theorem of [Za]). Rasmussenand Tamagawa conjectured in [RT] that for any ℓ large enough, this set is in fact empty (seeConjecture 1.1 in Introduction). The following results on the Rasmussen-Tamagawa Conjectureare known:(i) ([RT], Theorem 2) If K = Q and g = 1, then the conjecture holds.(ii) ([RT], Theorem 4) If K is a quadratic number field other than the imaginary quadratic fieldsof class number one and g = 1, then the conjecture holds.(iii) ([Oz], Corollary 4.5) Let A ( K, g, ℓ ) st be the set of K -isomorphism classes of abelian varieties in A ( K, g, ℓ ) with semi-stable reduction everywhere. Then there exists an integer C = C ([ K : Q ] , g ),depending only on [ K : Q ] and g , such that A ( K, g, ℓ ) st is empty for any ℓ > C with ℓ ∤ d K . Here d K is the discriminant of K .(iv) ([Ar], Corollary 6.4 and [AM]) Let K be a quadratic number field other than the imaginaryquadratic fields of class number one. Let A ( K, , ℓ ) QM be the set of K -isomorphism classes ofQM-abelian surfaces in A ( K, , ℓ ). Then A ( K, , ℓ ) QM is empty for any ℓ large enough.For an abelian variety A , denote by ρ A,ℓ : G K → GL ( T ℓ ( A )) ≃ GL g ( Z p ) the representationdetermined by the action of G K on the ℓ -adic Tate module T ℓ ( A ) of A . Consider the followingproperties:(RT ℓ ) ′ For some finite extension L of K which is unramified at all places of K above ℓ , L ( A [ ℓ ])is an ℓ -extension of L ( µ ℓ ).(RT ab ) The representation ρ A,ℓ has an abelian image.It is clear that (RT ℓ ) implies (RT ℓ ) ′ . Definition 3.2.
We define sets A ( K, g, ℓ ) ab and A ′ ( K, g, ℓ ) ab of isomorphism classes of g -dimensionalabelian varieties A over K as follows:(1) [ A ] ∈ A ( K, g, ℓ ) ab if and only if A satisfies (RT ℓ ), (RT red ) and (RT ab ).(2) [ A ] ∈ A ′ ( K, g, ℓ ) ab if and only if A satisfies (RT ℓ ) ′ and (RT ab ).Clearly, we have A ( K, g, ℓ ) ⊃ A ( K, g, ℓ ) ab ⊂ A ′ ( K, g, ℓ ) ab . Note that abelian varieties in A ′ ( K, g, ℓ ) ab are not forced the reduction hypothesis (RT red ). Hence A ′ ( K, g, ℓ ) ab may be infinite (but the au-thor does not know an example such that A ′ ( K, g, ℓ ) ab is infinite). In this section, we use same notation as in the previous section. First we study the structure of A [ ℓ ] for an abelian variety A in A ′ ( K, g, ℓ ) ab . Let A be any g -dimensional abelian variety over K .We denote by ¯ ρ A,ℓ : G K → GL ( A [ ℓ ]) ≃ GL g ( F p ) the representation determined by the action of G K on A [ ℓ ]. Consider the following properties:(RT mod ) (¯ ρ A,ℓ ) ss conjugates to the direct sum of n characters which are of the form ¯ χ aℓ .(RT mod ) ′ (¯ ρ A,ℓ ) ss is abelian and characters associated with ¯ ρ A,ℓ are of the form ε ¯ χ aℓ , where ε : G K → ¯ F × ℓ is a continuous character which is unramified at all places above ℓ .The condition (RT ℓ ) is equivalent to the condition (RT mod ) by the Lemma below. Hence the K -isomorphism class [ A ] of g -dimensional abelian variety A over K is in A ( K, g, ℓ ) if and only if A satisfies (RT mod ) and (RT red ). Lemma 4.1.
Let A be a g -dimensional abelian variety over K . The abelian variety A satisfies (RT ℓ ) if and only if A satisfies (RT mod ) . (2) Suppose that the abelian variety A satisfies (RT ab ) . Then A satisfies (RT ℓ ) ′ if and only if A satisfies (RT mod ) ′ .Proof. The assertion (1) follows from the arguments of the proof of Lemma 3 in [RT] and thus weomit the proof. Suppose that an abelian variety A satisfies the condition (RT ab ) and denote by ψ , . . . , ψ g characters associated with ¯ ρ A,ℓ . If A satisfies (RT mod ) ′ , then we have ψ i = ε i ¯ χ a i ℓ forsome integer a i where ε i : G K → ¯ F × ℓ is a continuous character which is unramified at all places of K above ℓ . Let L be the composition field of all fields ¯ K ker ε i for all i . Then L is unramified atall places of K above ℓ . Since each ψ i | G L ( µℓ ) is trivial, we obtain (RT ℓ ) ′ . Conversely, suppose that(RT ℓ ) ′ holds and take a field L as in the statement of (RT ℓ ) ′ . By (1), we know that each ψ i | G L isequal to ¯ χ a i ℓ for some integer a i . Hence ε i := ψ i · ¯ χ − a i ℓ : G K → ¯ F × ℓ is unramified at all places above ℓ and this implies (RT mod ) ′ .We recall the following two propositions. Proposition 4.2 (Faltings) . Fix an integer w . The set of isomorphism classes of semisimple n -dimensional ℓ -adic representations G K → GL n ( Q ℓ ) which are Q -integral with Frobenius weights ≤ w outside S , is finite.Proof. The Proposition follows from the proof of Theorem 5 in [Fa]. See also [La], Chapter VIII,Section 5, Theorem 11.
Proposition 4.3 (Raynaud’s criterion of semi-stable reduction, [Gr], Proposition 4.7) . Suppose A is an abelian variety over a field F with a discrete valuation v , n is a positive integer not divisibleby the residue characteristic, and the points of A [ n ] are defined over an extension of F which isunramified over v . In particular, if A is an arbitrary abelian variety over a number field K , then A has semi-stable reduction everywhere over K ( A [12]) = K ( A [3] , A [4]) . For an integer g >
0, put D g := ♯GL g ( Z / Z ) · ♯GL g ( Z / Z ) . If ρ : G K → GL g ( Q ℓ ) is an abelian representation, then, for any integer k , we denote by ρ k therepresentation G K → GL g ( Q ℓ ) which is defined by ρ k ( s ) := ( ρ ( s )) k for any s ∈ G K . With thisnotation, we obtain the following lemma which plays an important role in the proof of Theorem1.2 to construct a good compatible system. Lemma 4.4.
Let g > be an integer and ℓ a prime number. Let A ℓ be the set of isomor-phism classes of representations ρ : G K → GL g ( Q ℓ ) which are isomorphic to ρ D g A,ℓ for some g -dimensional abelian variety A over K such that K ( A [ ℓ ∞ ]) is an abelian extension of K . Then A ℓ is finite.Proof. If A is an abelian variety over K such that K ( A [ ℓ ∞ ]) is an abelian extension of K , then A has potential good reduction everywhere. Putting L := K ( A [12]), such an abelian variety A hasgood reduction everywhere over L by Proposition 4.3. Since [ L : K ] divides D g , the representation ρ D g A,ℓ is unramified outside ℓ for any g -dimensional abelian variety A over K such that K ( A [ ℓ ∞ ])is an abelian extension of K . Take any finite place v of K not above ℓ . Let v L be a finite placeof L above v and denote by f the extension degree of F v L over F v , where F v L and F v are residuefields of v L and v , respectively. Noting that L is a Galois extension of K and A has good reductioneverywhere over L , we see that D g /f is an integer and obtain the equationdet( T − ρ D g A,ℓ (Fr v )) = det( T − ( ρ A,ℓ (Fr v L )) D g /f ) . Since A has good reduction everywhere over L , the polynomial det( T − ρ A,ℓ (Fr v L )) has rationalinteger coefficients and hence so is det( T − ( ρ A,ℓ (Fr v L )) D g /f ). Consequently, the representation8 D g A,ℓ is Q -integral with Frobenius weight D g / K above ℓ .Therefore, by Proposition 4.2, it is enough to prove that the representation ρ D g A,ℓ is semisimple.Note that it has already known that ρ A,ℓ is semisimple (Theorem 3 of [Fa]). Since ρ A,ℓ is abelianand geometric in the sense of [FM], the representation ρ A,ℓ is locally algebraic in the sense of[Se] (see also Proposition of Section 6 in [FM]). Therefore, by (MT 1) of [Ri], there exists amodulus of definition m and an algebraic homomorphism φ : S m → GL g over Q such that the ℓ -representation induced by φ is isomorphic to ρ A,ℓ Here, the definition of the commutativealgebraic group S m over Q is given in Chapter II of [Se]. Note that any ℓ -adic representationcoming from an algebraic morphism S m → GL g is automatically semisimple. Since ρ D g A,ℓ comesfrom the composition S m D g → S m φ → GL g where S m D g → S m is the multiplication by D g map, weobtain the fact that ρ D g A,ℓ is semisimple. Proof of Theorem 1.2.
First we note that, if an abelian variety A over K satisfies (RT ab ), then ρ A,ℓ ′ is abelian for any prime number ℓ ′ (cf. [Se], Chapter III, Section 2.3, Corollary 1). Fix aprime number ℓ and denote by A ℓ the set as in Lemma 4.4. Assume that there exist infinitelymany prime numbers ℓ such that A ′ ( K, g, ℓ ) ab is not empty. For every such ℓ , we obtain the ℓ -adicrepresentation ρ D g A,ℓ which is in the set A ℓ , where A is an abelian variety whose isomorphism classis in the set A ′ ( K, g, ℓ ) ab . By Lemma 4.4, we see that there exists a representation ρ ℓ in A ℓ suchthat for infinitely many ℓ and [ A ] ∈ A ′ ( K, g, ℓ ) ab , ρ D g A,ℓ is isomorphic to ρ ℓ . In particular, weknow the fact that the representation ρ ℓ extends to a Q -integral strict compatible system ( ρ ℓ ) ℓ of2 g -dimensional abelian semisimple ℓ -adic representations of G K . Furthermore, for infinitely manyprime numbers ℓ , the characters associated with a residual representation ¯ ρ ℓ of ρ ℓ are of the form ε ¯ χ aℓ by Lemma 4.1, where ε : G K → ¯ F × ℓ is a continuous character which is unramified at all places of K above ℓ . Applying Theorem 2.5 or Corollary 2.8, we see that there exist integers m , . . . , m g anda finite extension L of K such that ρ ℓ is isomorphic to χ m ℓ ⊕ χ m ℓ ⊕· · ·⊕ χ m g ℓ on G L . In particular,for some prime number ℓ and [ A ] ∈ A ′ ( K, g, ℓ ) ab , ρ D g A,ℓ is isomorphic to χ m ℓ ⊕ χ m ℓ ⊕ · · · ⊕ χ m g ℓ on G L . Therefore, looking at the eigenvalues of images of a Frobenius element (at some place) of ρ D g A,ℓ and χ m ℓ ⊕ χ m ℓ ⊕ · · · ⊕ χ m g ℓ , we know that D g / m = m = · · · = m g . Since ρ D g A,ℓ hasHodge-Tate weights 0 and D g at a place of L above ℓ , this is a contradiction. References [Ar] Keisuke Arai,
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