Typically bounding torsion on elliptic curves with rational j-invariant
aa r X i v : . [ m a t h . N T ] F e b TYPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITHRATIONAL j -INVARIANT TYLER GENAO
Abstract.
A family F of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups E ( F )[tors] of those elliptic curves E /F ∈ F can be made uniformly bounded after removing from F those whose numberfield degrees lie in a subset of Z + with arbitrarily small upper density. For everynumber field F , we prove unconditionally that the family E F of elliptic curves overnumber fields with F -rational j -invariants is typically bounded in torsion. For anyinteger d ∈ Z + , we also strengthen a result on typically bounding torsion for thefamily E d of elliptic curves over number fields with degree d j -invariants. Contents
1. Introduction 21.1. Bounds on torsion subgroups 21.2. Typically bounding torsion for other families 31.3. Strategy of proof 41.4. An additional result 51.5. Acknowledgments 61.6. Notation 62. Isogeny Characters 63. Orbits Under the Galois Representation 73.1. The semisimplification of a Borel subgroup 73.2. Orbits in the diagonal case 83.3. Uniform orbit divisibility 93.4. Action on ℓ -torsion 104. Part One of the Proof: Allowing Rationally Defined CM 114.1. Reducing to the split prime case 114.2. Noncommutative case 124.3. Commutative case 145. Part Two of the Proof: Removing GRH 14References 16 Mathematics Subject Classification. [email protected] . This material is based upon work supported by the National ScienceFoundation Graduate Research Fellowship under Grant No. 1842396. Partial support was also providedby the Research and Training Group grant DMS-1344994 funded by the National Science Foundation. Introduction
Bounds on torsion subgroups.
Let E /F denote an elliptic curve over a numberfield. The Mordell-Weil theorem states that the abelian group E ( F ) of F -rationalpoints on E is finitely generated. In particular, one has a decomposition E ( F ) ∼ = Z r ⊕ E ( F )[tors]where E ( F )[tors] is its torsion subgroup over F . In the present paper we prove resultson asymptotic upper bounds for the sizes of these torsion subgroups.In 1977, Mazur [Maz77] gives a complete classification of torsion subgroups E ( Q )[tors]of all elliptic curves E / Q . Following this, a combination of work by Kamienny, Kenkuand Momose [KM88] [Kam92a] [Kam92b] leads to a complete classification of ellipticcurve torsion subgroups E ( F )[tors] over all quadratic number fields F . One consequenceof their work is that there are finitely many possibilities for torsion subgroups overquadratic number fields, suggesting a “uniform bound” on torsion subgroups when thedegree of the number field is fixed.In 1996, Merel proves the “strong uniform boundedness conjecture” suggested previ-ously by work of Kamienny, Kenku and Momose. Theorem ([Mer96]) . For each integer d > there exists a constant B ( d ) ∈ Z + so thatfor all number fields F/ Q of degree d and all elliptic curves E /F one has E ( F )[tors] ≤ B ( d ) . In lieu of Merel’s result, we can ask for the sharpest upper bound B ( d ), i.e., thelargest size of any elliptic curve torsion subgroup over all degree d number fields. Letus define a function which records such values: for d ∈ Z + we set T ( d ) := max E /F , [ F : Q ]= d E ( F )[tors] . Only a few values of T ( d ) are currently known. Via a complete classification of torsionsubgroups over degree d ≤ T (1) = 16 [Maz77], T (2) = 24[KM88] [Kam92a] [Kam92b] and T (3) = 28 [DEvHMZB]. Lower bounds for T ( d ) areknown for 4 ≤ d ≤ T ( d )which are, unfortunately, greater than exponential in d [Par99].Let us also define the CM-analogue of T ( d ), T CM ( d ) := max CM E /F , [ F : Q ]= d E ( F )[tors] . This arithmetic function records the largest size over all CM elliptic curve torsionsubgroups over all degree d number fields. One clearly has T CM ( d ) ≤ T ( d ) < ∞ .Via a complete classification of CM torsion subgroups over number fields of a fixeddegree, many values of T CM ( d ) are known. For example, if d ≤
13 then T CM ( d ) isrecorded [CCRS14]. For all primes ℓ > T CM ( ℓ ) = 6 [BCS17]. For each n ∈ Z + one has for sufficiently large primes ℓ ≫ n T CM ( ℓ n ) = 6 [BCP17]. In fact, for anyodd d ∈ Z + T CM ( d ) is also known [BP17].Bourdon, Clark and Pollack [BCP17] also treat T CM ( d ) as an arithmetic function andstudy its behavior as d → ∞ . As noted above, T CM ( d ) = 6 for infinitely many d ∈ Z + YPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITH RATIONAL j -INVARIANT 3 [BCS17]. In fact, they show that T CM ( d ) ≥ d ∈ Z + , whence the “lower order”of T CM ( d ) is lim inf d →∞ T CM ( d ) = 6 . Similarly, Clark and Pollack have shown the “upper order” result (Theorem 1.1 [CP17])lim sup d →∞ T CM ( d ) d log log d = e γ π √ . Bourdon, Clark and Pollack [BCP17] also study the typical order of T CM ( d ), e.g., itsstatistical behavior away from certain sets of arbitrarily small upper density. Recallthat the upper (asymptotic) density of a subset S ⊆ Z + is¯ δ ( S ) := lim sup x →∞ S ∩ [1 , x ]) x . They prove the following typical order result on T CM ( d ). Theorem (Theorem 1.1.(i) [BCP17]) . For all ǫ > there exists B ǫ ∈ Z + so that ¯ δ ( { d ∈ Z + : T CM ( d ) ≥ B ǫ } ) ≤ ǫ. The above theorem says that for any ǫ > S ⊆ Z + of upper density ≤ ǫ , there exists a uniform bound B ǫ on all CM torsionsubgroups over all degree d S number fields; writing this out, E ( F )[tors] ≤ B ǫ for any CM elliptic curve E /F with [ F : Q ] S . We say that the family of CM ellipticcurves over number fields is typically bounded in torsion. Compare this result to the following, which shows that torsion is decidedly not typi-cally bounded on the family A (1) of all elliptic curves over all number fields. Theorem (Theorem 1.7 [CMP18]) . For all B ∈ Z + the set { d ∈ Z + : T ( d ) ≥ B } is cofinite in Z + , whence its upper density is equal to 1. Typically bounding torsion for other families.
Despite the failure of A (1)to be typically bounded in torsion, we saw that the proper subfamily A CM (1) of CMelliptic curves is. We will see there are also other familiar subfamilies of A (1) whichare typically bounded in torsion.Let F = { E i/F i } i ∈ I be a family of elliptic curves E i defined over number fields F i .From here on out, we assume that any such F is “closed under rational isomorphism”,e.g., if E /F ∈ F and E /F is an elliptic curve with E ∼ = F E , then E /F ∈ F . Wesay that F is typically bounded in torsion if the torsion subgroups E ( F )[tors] of ellipticcurves E /F ∈ F can be made uniformly bounded after removing those elements whosenumber field degrees all lie in a certain subset of Z + of arbitrarily small upper density.Formally stated, F is typically bounded in torsion if for all ǫ > B ǫ > { d ∈ Z + : ∃ E /F ∈ F so that [ F : Q ] = d and E ( F )[tors] ≥ B ǫ } TYLER GENAO has upper density at most ǫ .For a family F of elliptic curves, one can define an F -analogue of T ( d ), T F ( d ) := max E /F ∈F , [ F : Q ]= d E ( F )[tors] . By Merel [Mer96] one has T F ( d ) < ∞ . We see that T CM ( d ) = T A CM (1) ( d ). It is clearthat F is typically bounded in torsion iff for all ǫ > B ǫ > δ ( { d ∈ Z + : T F ( d ) ≥ B ǫ } ) ≤ ǫ. Compare this to Theorem 1.1.(i) [BCP17] above.One important family of elliptic curves is the “base extension” family of all ellipticcurves E /F which are defined over Q . For example, Lozano-Robledo [LR13] gives anexplicit bound on all primes which divide E ( F )[tors], and this bound is linear in thedegree [ F : Q ]. On the other hand, Gonz´alez-Jim´enez and Najman [GJN20] show that E ( F )[tors] = E ( Q )[tors] when 2 , , , ∤ [ F : Q ]. A (subtly) larger family is thatof elliptic curves E /F with Q -rational j -invariant. For such a family, Propp [Pro18]provides (conditional) bounds on prime divisors of E ( F )[tors] when [ F : Q ] is fixed.Clark and Pollack [CP18] also show that for each ǫ > C ( ǫ ) > E ( F )[tors] ≤ C ( ǫ )[ F : Q ] + ǫ .Our primary family of study is a natural generalization of the above. For a fixednumber field F , we consider the family of elliptic curves over number fields with F -rational j -invariants, E F := { E /L : L ⊇ F, [ L : Q ] < ∞ , j ( E ) ∈ F } . Clark, Milosevic and Pollack study E F [CMP18]. They show (Theorem 1.8 [CMP18])that torsion is typically bounded in E F , provided that F contains no Hilbert class fieldsof imaginary quadratic fields and that the Generalized Riemann Hypothesis (GRH) istrue.Our main result is a generalization of Theorem 1.8 [CMP18]: we prove it uncondi-tionally. Theorem 1.
For any number field F , torsion is typically bounded on the family E F ofelliptic curves E /L defined over number fields L ⊇ F and whose j -invariant j ( E ) ∈ F . Strategy of proof.
Our proof of Theorem 1 will be broken down into severalsteps, following [CMP18]. Let us introduce a condition P2 on a family F of ellipticcurves. P2):
There exists a constant c := c ( F ) ∈ Z + such that for all primes ℓ ∈ Z + and all E /F ∈ F , if E ( F ) has a point of order ℓ then ℓ − | c [ F : Q ] . For an integer d ∈ Z + let us consider the family E d of elliptic curves over number fieldswith degree d j -invariant, E d := { E /F : [ F : Q ] < ∞ , [ Q ( j ( E )) : Q ] = d } . YPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITH RATIONAL j -INVARIANT 5 By Theorems 3.2.b and 3.5 [CMP18] any subfamily of a finite union of E d will betypically bounded in torsion if and only if the subfamily satisfies P2. In particular,from the containment E F ⊆ n [ d =1 E d , to prove Theorem 1 it suffices to show that E F satisfies P2.Following the proof of Theorem 4.3 [CMP18], to show that P2 holds for E F it sufficesto construct a constant c := c ( F ) ∈ Z + such that for any prime ℓ ≫ F E /F with an F -rational ℓ -isogeny, one has for all nontrivial P ∈ E [ ℓ ]that(1) ℓ − | c [ F ( P ) : Q ] . Since E has an F -rational ℓ -isogeny, the image G of its mod − ℓ Galois representation isconjugate to a subgroup of upper triangular matrices in GL ( ℓ ). We will use this factto compare the orbits of the representation acting on the standard unit vectors, to theorbits of its semisimplification.First, a careful analysis of the description of our isogeny character from case 1 ofTheorem 3 (see the following section) will allow us to relate the semisimplification of G to the mod − ℓ Galois representation of some CM elliptic curve E ′ /F with F -rational CM.We will show that (1) holds when we replace E with E ′ , and then use an orbit-divisibilityargument to show that such a divisibility will also hold for E with an appropriateconstant c . Following this, we will construct a similar constant c for when we are incase 2 of Theorem 3. The cases where G is and is not commutative will be addressedseparately.1.4. An additional result.
In our proof of Theorem 1, the constant c := c ( F ) whichwe will construct to show that (1) holds will depend only on the degree of F . We willuse this to improve Theorem 1.10 [CMP18].We denote by SI( d ) the strong isogeny conjecture in degree d , which asserts that thereexists a prime ℓ := ℓ ( d ) ∈ Z + so that for all primes ℓ > ℓ there do not exist non-CMelliptic curves E /F with an F -rational ℓ -isogeny when [ F : Q ] = d . The strong isogenyconjecture SI( d ) would follow from a stronger conjecture of Serre which has been usedto prove other results, see e.g. Theorem 1.6 [BELOV].Let us denote by LV( d ) the hypothesis that the set S F of primes from Theorem 3(see the next section) depends only on the degree [ F : Q ]. One may check that LV( d )is a weaker hypothesis than SI( d ). Theorem 2.
Fix an integer d ∈ Z + . If LV( d ) holds, then the family E d := { E /F : [ F : Q ] < ∞ , [ Q ( j ( E )) : Q ] = d } is typically bounded in torsion. The proof of Theorem 2 is similar to the proof of Theorem 1.8 [CMP18], which boilsdown to the proof of Theorem 4.3 [CMP18]. By Theorem 3.5 [CMP18], to show that E d is typically bounded in torsion it suffices to show it satisfies P2. In turn, this reduces TYLER GENAO to showing that for any prime ℓ ≫ d c := c ( d ) ∈ Z + such that forany non-CM elliptic curve E / Q ( j ( E )) with a Q ( j ( E ))-rational ℓ -isogeny, the divisibility ℓ − | c [ Q ( j ( E ))( P ) : Q ]holds for all P ∈ E [ ℓ ] • . They cite SI( d ) to exclude this case. But if one assumesLV( d ) instead, then we will see that our proof of Theorem 1 also applies here, since wemay take c to be the appropriate constant c := c ( F ) produced through our proof, e.g., c := 864[ Q ( j ( E )) : Q ]. This constant will then depend only on the degree of j ( E ) over Q .1.5. Acknowledgments.
The author thanks Pete L. Clark for several insightful con-versations about typically bounding torsion, and a careful review of earlier drafts ofthis paper. The author also thanks Nicholas Triantafillou for his help with the proof ofLemma 4.1.6.
Notation.
Given an extension F/ Q , we let G F := Gal( F sep /F ) denote the abso-lute Galois group of F . Let us write the mod − ℓ cyclotomic character of F as χ ℓ : G F → F × ℓ . For an elliptic curve E /F , we will write its mod − N Galois representation as ρ E,N : G F → Aut( E [ N ]) . If { P, Q } is a Z /N Z -basis for E [ N ], then when working with explicit matrix represen-tations we may also write ρ E,N,P,Q : G F → GL ( N ) , where GL ( N ) := GL ( Z /N Z ) is the general linear group of degree 2 over Z /N Z .Suppose E /F has an F -rational cyclic N -isogeny C N , i.e., a one-dimensional G F -invariant submodule of E [ N ]. Then one has an isogeny character which describes theaction of G F on C N , r : G F → ( Z /N Z ) × . Given a group homomorphism f : G → H , we sometimes write im f for its image f ( G ).2. Isogeny Characters
A crucial component of the proof of Theorem 1 is an analysis of the description ofour isogeny character from a result of Larson and Vaintrob [LV14].
Theorem 3 (Theorem 1 [LV14]) . Let F be a number field. Then there is a finite setof primes S F ⊆ Z + such that for all ℓ S F , if E /F is an elliptic curve with an F -rational ℓ -isogeny, then for the corresponding isogeny character r : G F → F × ℓ one of thefollowing holds:1. There exists a CM elliptic curve E ′ defined over F such that its CM field K :=End( E ′ ) ⊗ Q is with K ⊆ F . Furthermore, there exists a character ψ ′ : G F → F ℓ × forwhich we have both similarity ρ E ′ ,ℓ ⊗ F ℓ F ℓ ( G F ) ∼ im (cid:20) ψ ′ ∗ χ ℓ ψ ′− (cid:21) YPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITH RATIONAL j -INVARIANT 7 and equality r = ψ ′ .
2. GRH fails for F ( √− ℓ ) , and r = χ ℓ where χ ℓ : G F → F × ℓ is the mod − ℓ cyclotomic character. As per the theory of complex multiplication, the Hilbert class field of an imaginaryquadratic field K will equal K ( j ( E ′ )) for any O K -CM elliptic curve E ′ . In particular,for F not to contain Hilbert class fields of imaginary quadratic fields is equivalent tothere being no CM elliptic curve E ′ /F with End( E ′ ) ⊗ Q ⊆ F . For such a field F , if onealso assumes that GRH is true then Theorem 3 tells us that for ℓ ≫ F F -rational ℓ -isogenies on any elliptic curve. It is this observation, coupled with the factthat if F satisfies P2 then one can exclude finitely many primes ℓ ∈ Z + and still haveP2 hold for F , which shows us that E F satisfies P2 – this is Theorem 4.3.b [CMP18].In particular, torsion is typically bounded on E F , conditionally.To prove Theorem 1 we will neither assume that GRH is true nor that F does notcontain the Hilbert class field of any imaginary quadratic field. Instead, we will leveragethe extra information from Theorem 3 on our possible isogeny characters to construct aconstant c for which (1) holds for all ℓ ≫ F E /F with an F -rational ℓ -isogeny.3. Orbits Under the Galois Representation
In our study of the mod − ℓ Galois representations of an elliptic curve E /F , it will helpto understand the orbits of the F ℓ [ G F ]-module E [ ℓ ] under various subgroups of GL ( ℓ ),and in a more general sense.3.1. The semisimplification of a Borel subgroup.
Fix a prime ℓ , and suppose asubgroup G ⊆ GL ( ℓ ) is contained in the subgroup of upper triangular matrices ofGL ( ℓ ). Let us consider its semisimplification G ss := (cid:26)(cid:20) a d (cid:21) ∈ GL ( ℓ ) : ∃ b ∈ F ℓ (cid:20) a b d (cid:21) ∈ G (cid:27) . The group GL ( ℓ ) acts on the F ℓ -vector space V spanned by the column vectors e := (cid:20) (cid:21) and e := (cid:20) (cid:21) , via multiplication on the left. Consequently, its subgroups G and G ss also act on V via left multiplication.Ultimately, we will compare divisibility of the sizes of such orbits from the action ofGal( F ( E [ ℓ ]) /F ) on E [ ℓ ] • := E [ ℓ ] r { O } . The following lemma is our first step towardssuch a comparison. Lemma 4.
Let ℓ ∈ Z + be prime. Let G be an upper triangular subgroup of GL ( ℓ ) .(1) G is not commutative then G ss ⊆ G .(2) If G is commutative then it is diagonalizable over F ℓ .Proof. it is clear that G consists exclusively of diagonal matrices iff G ss = G , and insuch a case we are done. Let us suppose that G = G ss . We will break the proof intothree cases; in the first two cases, we will show containment G ss ⊆ G . TYLER GENAO
Case 1: G contains a matrix of the form (cid:20) a b a (cid:21) with b = 0. We write such a matrixas γ ; its power γ ℓ − equals(2) (cid:20) λ (cid:21) ∈ G where λ := ( ℓ − ba ℓ − = 0. It follows that G contains the transvection (cid:20) (cid:21) = γ A ( ℓ − ∈ G, where A ∈ Z may be taken so that A ≡ λ − (mod ℓ ). We conclude that for each n ∈ Z we have (cid:20) n (cid:21) = (cid:20) (cid:21) n ∈ G. It follows that G ss ⊆ G . Indeed, for any element δ := (cid:20) a b d (cid:21) ∈ G we check that δ (cid:20) n (cid:21) = (cid:20) a an + b d (cid:21) , so taking n ∈ Z such that n ≡ − ba − (mod ℓ ) shows that (cid:20) a d (cid:21) ∈ G . Case 2: G is not commutative. Then choosing any γ , γ ∈ G with nontrivialcommutator, their commutator γ γ γ − γ − must be of the form (cid:20) λ (cid:21) for some λ ∈ F × ℓ .Then the steps following (2) show that G ss ⊆ G . Case 3:
Neither Case 1 nor Case 2 holds. Then G is commutative, and its non-diagonal matrices have distinct F ℓ -rational eigenvalues. In particular, each non-diagonalmatrix is diagonalizable over F ℓ , and since all matrices in G commute we deduce thatthere is a simultaneous diagonalization for these matrices over F ℓ . The conclusion isthat G is conjugate to a subgroup of diagonal matrices. (cid:3) Orbits in the diagonal case.
In our describing the orbits of the action ofGal( F ( E [ ℓ ]) /F ) on E [ ℓ ] • , we will repeatedly relate them to the orbits of E [ ℓ ] • un-der its semisimplification, the latter of which is contained in the split Cartan subgroupof GL ( ℓ ), i.e., is a subgroup of diagonal matrices. To this end, we first study orbitsunder such subgroups.Let G ′ ⊆ GL ( ℓ ) be a subgroup of diagonal matrices. Associated to this group arethe two characters which describe the diagonal entries, χ : G ′ → F × ℓ , (cid:20) a d (cid:21) a and χ : G ′ → F × ℓ , (cid:20) a d (cid:21) d. For a proof of this fact, see e.g. [Con].
YPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITH RATIONAL j -INVARIANT 9 Let us study the orbits of elements in V • := F × ℓ h e , e i r { } under G ′ . First, we claimthat there are I := [ F × ℓ : im χ ] distinct orbits of the form O G ′ ( ae ) where a ∈ F × ℓ . Tosee this, let us check that the map F × ℓ / im χ → { Orbits O G ′ ( ae ) : a ∈ F × ℓ } , a
7→ O G ′ ( ae )is a well-defined bijection:(1) Well-defined: if a = χ ( γ ) b then γ = (cid:20) ab − d (cid:21) for some d ∈ F × ℓ , and thus O G ′ ( be ) = O G ′ ( γbe ) = O G ′ ( ae ) . (2) Injective: suppose that O G ′ ( ae ) = O G ′ ( be ). Then for some γ ∈ G ′ we have ab − e = γe , whence we have χ ( γ ) = ab − , i.e., a · im χ ≡ b · im χ in F × ℓ / im χ .(3) Surjective: obvious.We conclude that e contributes I distinct orbits of the form O G ′ ( ae ) with a ∈ F × ℓ ,each of size χ . An identical argument shows that e contributes I := [ F × ℓ : im χ ]distinct orbits of the form O G ′ ( de ) where d ∈ F × ℓ , each of size χ . The remainingorbits are of the form O G ′ ( ae + de ) where ad = 0, and a similar argument using F ℓ -linear independence of e and e shows us that O G ′ ( ae + de ) = O G ′ ( e + e ).Since the orbits above partition V • := F ℓ h e , e i r { } , one counts sizes and findsthat I χ + I χ + X O G ′ ( e + e ) = ℓ − X denotes the number of orbits of the form O G ′ ( ae + de ) with ad = 0. Inparticular, since I χ = I χ = ℓ − X O G ′ ( e + e ) = ( ℓ − . To summarize: for our diagonal subgroup G ′ ⊆ GL ( ℓ ) one has two characters χ , χ which describe the diagonal entries of elements of G ′ . The number of orbits of the form O G ′ ( ae ) (resp. O G ′ ( de )) with a ∈ F × ℓ (resp. d ∈ F × ℓ ) equals the index [ F × ℓ : im χ ](resp. [ F × ℓ : im χ ]), and each such orbit is of size χ (resp. χ ). Orbits of theform O G ′ ( ae + de ) with a, d ∈ F × ℓ are in bijection with the orbit O G ′ ( e + e ), and thenumber X of such orbits satisfies (3).3.3. Uniform orbit divisibility.
Suppose that a group G acts on a set X on the left.For an element x ∈ X we write O G ( x ) to denote its orbit. Let H ⊆ G be a finite indexsubgroup; write its coset representatives as g − , . . . , g − G : H ] . Then one has O G ( x ) = [ G : H ] G i =1 O H ( g i x ) . It follows that for all x ∈ X one has(4) O G ( x ) = [ G : H ] X i =1 O H ( γ i x ) . Therefore, if an integer M ∈ Z + “uniformly divides” all H -orbits, i.e., if for an integer c ∈ Z + one has for each x ∈ X that M | c O H ( x ) , then it follows by (4) that M uniformly divides G -orbits, i.e., for each x ∈ X one has M | c O G ( x ) . One also has a “reverse direction” to uniform orbit divisibility: for M ∈ Z + , if for aconstant C ∈ Z + one has for all x ∈ X that M | C O G ( x ) , then as per the divisibility O G ( x ) | [ G : H ] O H ( x ) one has for all x ∈ X that M | C [ G : H ] O H ( x ) . The key is that if c (resp. C ) is not “trivial”, e.g., if c = M (resp. C = M ), then C := c (resp. c := C [ G : H ]) need not be trivial either.Our conclusion is that for a finite group G acting on a set X , a finite index subgroup H ⊆ G and an integer M ∈ Z + : there is a constant c ∈ Z + so that for all x ∈ XM | c O H ( x ) , iff there is a constant C ∈ Z + so that for all x ∈ XM | C O G ( x ) . Thus, M ∈ Z + uniformly divides H -orbits iff M uniformly divides G -orbits; and moreimportantly, in passing from H to G we may take the constant C := c , and in passingfrom G to H we may take c := C [ G : H ].3.4. Action on ℓ -torsion. Let F be a number field and E /F an elliptic curve. Through-out this paper we are considering for prime ℓ ∈ Z + the mod − ℓ Galois representationsof the absolute Galois group G F acting on the various F ℓ -modules E [ ℓ ]. At each level ℓ ,our action reduces to a faithful representation ρ E,ℓ : Gal( F ( E [ ℓ ]) /F ) ֒ → Aut( E [ ℓ ]). Wewill often work with an explicit basis { P, Q } of E [ ℓ ] in mind, in which case our Galoisrepresentation G := ρ E,ℓ,P,Q (Gal( F ( E [ ℓ ]) /F )) is identified with a subgroup of GL ( ℓ ).In studying orbits under subgroups G of GL ( ℓ ), we may restrict to the action on theset E [ ℓ ] • := E [ ℓ ] r { O } . For a mod − ℓ Galois representation G as above, Our G -orbitsof points R ∈ E [ ℓ ] • are O G ( R ) = { R σ : σ ∈ Gal( F ( E [ ℓ ]) /F ) } , and their stabilizers are the subgroups of automorphisms which fix R ,Stab( R ) = Gal( F ( E [ ℓ ]) /F ( R )) . By the orbit-stabilizer theorem, the size of an element’s orbit is the index of its stabilizer,whence the degree of an ℓ -torsion point R ∈ E [ ℓ ] satisfies[ F ( R ) : F ] = O G ( R ) . Consequently, torsion points in the same orbit share the same degree over F . YPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITH RATIONAL j -INVARIANT 11 Part One of the Proof: Allowing Rationally Defined CM
To recapitulate our goals: we will prove Theorem 1, which is the unconditional formof Theorem 1.8 [CMP18]. This means we will allow our number field F to containHilbert class fields of imaginary quadratic fields, and we will not assume that GRHis true. Following the proof of Theorem 4.3 [CMP18], we will show the following: forall non-CM elliptic curves E /F whose mod − ℓ Galois representation G := ρ E,ℓ ( G F ) iscontained in a Borel subgroup – i.e., G is conjugate to a subgroup of upper triangularmatrices – one has that all G -orbits are uniformly divisible by ℓ −
1. Specifically, thereexists a constant c ∈ Z + such that for all ℓ -torsion R ∈ E [ ℓ ] • Equation (1) holds, ℓ − | c · [ F ( R ) : F ] . Furthermore, this constant c can be chosen to be independent of the choice of E /F and ℓ . For this section, let us assume that GRH is true. This will land us in case 1 of The-orem 3, where our isogeny character agrees with a CM “mod − ℓ associated character”up to twelfth powers.4.1. Reducing to the split prime case.
As per Remark 3.1 [CMP18] we can excludeany finite amount of primes from our consideration. We will show why this lets usassume that ℓ ≫ F O -CM elliptic curve E ′ defined over F where K := O ⊗ Q ⊆ F , we have thatthe ring class field K ( O ) = K ( j ( E ′ )) ⊆ F . Since F may contain only finitely many suchring class fields, we need only to consider finitely many imaginary quadratic orders O and (up to isomorphism) finitely many O -CM E ′ /F . In particular, we may assume thatthe following primes ℓ are unramified in each O . It will follow that our CM associatedcharacter in Theorem 3 will give us a diagonalization of our CM Galois representationover F ℓ .Suppose E /F is an elliptic curve whose mod − ℓ Galois representation is contained in aBorel subgroup; equivalently, suppose E has an F -rational ℓ -isogeny. Let r : G F → F × ℓ denote its isogeny character. Assuming ℓ ≫ F ℓ S F where S F is as inTheorem 3, there is an elliptic curve E ′ /F with CM by an imaginary quadratic order O ;with CM field K := O ⊗ Q ⊆ F ; and there exists a character ψ ′ : G F → F ℓ × for whichboth(5) ( ρ E ′ ,ℓ ⊗ F ℓ )( G F ) ∼ im (cid:20) ψ ′ χ ℓ ψ ′− (cid:21) and(6) r = ψ ′ . We note that Equation (6) implies(7) gcd(12 , ψ ′ ( G F )) · r ( G F ) = gcd(12 , r ( G F )) · ψ ′ ( G F ) . This is an application of Heilbronn’s classic result on imaginary quadratic class numbers [Hei34]towards showing that the class number of an imaginary quadratic order O tends to infinity as itsdiscriminant ∆( O ) → −∞ . Suppose that ℓ is inert in O . In this case, up to conjugation the CM image ρ E ′ ,ℓ ( G F )lands in the non-split Cartan subgroup C ns ( ℓ ) := (cid:26)(cid:20) a bǫb a (cid:21) : ( a, b ) ∈ F ℓ , ( a, b ) = (0 , (cid:27) ;here, ǫ is the least positive generator of F × ℓ . For ℓ >
2, one has an isomorphism F ℓ ( √ ǫ ) × ∼ = C ns ( ℓ ) by sending each a + b √ ǫ to its matrix representation (cid:20) a bǫb a (cid:21) . Theeigenvalues of such matrices are a ± b √ ǫ ; and since the Galois group Gal( F ℓ ( √ ǫ ) / F ℓ )is generated by the Frobenius automorphism α α ℓ , these eigenvalues are Galoisconjugates. This implies that ψ ′ and χ ℓ ψ ′− are Galois conjugates. Therefore, Equation(5) shows us that ρ E ′ ,ℓ ( G F ) = ψ ′ ( G F ) . Since r ( G F ) | ( ℓ − ρ E ′ ,ℓ ( G F ) | ℓ − . On the other hand, O -CM theory with ℓ inert tells us that for any torsion point P ′ ∈ E ′ [ ℓ ] • one has ( ℓ − | O × · [ F : Q ] h O · [ F ( P ′ ) : F ]where h O is the class number of O and O × is its unit group, see Theorem 7.2 [BC20].In particular, from [ F ( P ′ ) : F ] | [ F ( E ′ [ ℓ ]) : F ] = ρ E ′ ,ℓ ( G F )we deduce that ( ℓ − ≤ O × · [ F : Q ] h O · ρ E ′ ,ℓ ( G F ) . Comparing this with (8), we conclude that ℓ < [ F : Q ] h O − ℓ splits in any order O we are considering. This im-plies that the CM image ρ E ′ ,ℓ ( G F ) lands in a split Cartan subgroup of GL ( ℓ ), and so ρ E ′ ,ℓ ( G F ) is diagonalizable over F ℓ . In particular, our character ψ ′ must be the isogenycharacter of an F -rational ℓ -isogeny h P ′ i of E ′ .In determining a constant c such that (1) holds, we will split our proof into two caseson G .4.2. Noncommutative case.
Suppose that G is not commutative. As G is containedin a Borel, we may choose an upper triangular basis { P, Q } of E [ ℓ ]; without loss ofgenerality, suppose that h P i is an F -rational ℓ -isogeny, so that G := ρ E,ℓ,P,Q ( G F ) = im (cid:20) r ∗ χ ℓ r − (cid:21) where ∗ : G F → F × ℓ is not necessarily a character.Since G is not commutative, by Lemma 4 we have the semisimplification G ss ⊆ G .Let us take ℓ ≫ F ℓ S F . Then Theorem 3 applies:we have an O -CM E ′ /F and h P ′ i as in the previous subsection. It follows that there is a YPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITH RATIONAL j -INVARIANT 13 complementary basis element Q ′ to P ′ so that the mod − ℓ Galois representation of E ′ is diagonal with respect to the basis { P ′ , Q ′ } . In particular, this means that the twocharacters ψ ′ , χ ℓ ψ ′− are both isogeny characters. Let us write G ′ := ρ E ′ ,ℓ,P ′ ,Q ′ ( G F ) = im (cid:20) ψ ′ χ ℓ ψ ′− (cid:21) . We have written our actions here as left multiplication on the two-dimensional F ℓ -vectorspace V = F ℓ h e , e i with e i the unit column vectors. So for each v = ae + de ∈ V ,under G one has that v corresponds to R v := aP + dQ , and under G ′ one has that v corresponds to R ′ v := aP ′ + dQ ′ .Since E ′ /F has CM, the image G ′ of its mod − ℓ Galois representation lands inside themod − ℓ Cartan subgroup C ℓ ( O ) := ( O /ℓ O ) × . Since ℓ splits in O , one has that C ℓ ( O )may be identified with the split Cartan subgroup C s ( ℓ ) of diagonal matrices in GL ( ℓ ).Furthermore, since both G ss ⊆ G and r = ψ ′ , we have the inclusions(9) C s ( ℓ ) ⊇ G ′ ⊇ G ′ ⊆ G ss ⊆ G. Suppose we have a constant C ∈ Z + so that uniform orbit divisibility for C s ( ℓ ) by ℓ − v ∈ V • ℓ − | C O C s ( ℓ ) ( v ) . Then by our remarks in subsection 3.3 on uniform orbit divisibility, in making thepassage C s ( ℓ ) G ′ G ′ G ss G we find that for all v ∈ V • ℓ − | C [ C s ( ℓ ) : G ′ ] · [ G ′ : G ′ ] · O G ( v ) , i.e., for all R ∈ E [ ℓ ] • ℓ − | C [ C s ( ℓ ) : G ′ ] · [ G ′ : G ′ ] · [ F ( R ) : F ] . Since [ G ′ : G ′ ] | (12) this simplifies to(10) ℓ − | C · [ C s ( ℓ ) : G ′ ] · [ F ( R ) : F ] . Therefore, we may conclude that (1) holds with a constant which depends only on F ,if we can show that uniform orbit divisibility by ℓ − C s ( ℓ ) holds with a constantwhich depends only on F , and that the index [ C s ( ℓ ) : G ′ ] is also bounded in terms of F .First, we claim that uniform orbit divisibility by ℓ − C s ( ℓ ) with constant C := 1 – i.e., all orbit sizes are divisible by ℓ −
1. As observed in subsection 3.2, thereare exactly three orbits of C s ( ℓ ) via its action on V • : they are O C s ( ℓ ) ( e ) , O C s ( ℓ ) ( e ) and O C s ( ℓ ) ( e + e ), and each are of size ℓ − , ℓ − ℓ − , respectively. This provesour claim.Next, we are to show that the index [ C s ( ℓ ) : G ′ ] is bounded only in terms of F . Ingeneral, for an O -CM elliptic curve E ′ /F with F -rational CM – i.e., O ⊗ Q ⊆ F – onehas for any N ∈ Z + that its mod- N Galois representation will land inside the mod- N Cartan subgroup C N ( O ) := ( O /N O ) × . In fact, the indices of all mod − N Galoisrepresentations of E ′ are uniformly bounded in terms which involve F . Theorem 5 (Corollary 1.5, [BC20]) . Let K be an imaginary quadratic field. Thenthe index of the mod − N Galois representation of any O -CM elliptic curve E ′ /F with O ⊗ Q = K ⊆ F satisfies [ C N ( O ) : ρ E ′ ,N ( G F )] | O × [ F : K ( j ( E ′ ))] . By Theorem 5, we have that the index of our CM mod − ℓ Galois representation G ′ satisfies [ C s ( ℓ ) : G ′ ] | F : Q ] . Combining this with (10) shows us that for all R ∈ E [ ℓ ] • one has ℓ − | F : Q ] · [ F ( R ) : F ] . We conclude that (1) holds for E with the constant c := 864[ F : Q ].4.3. Commutative case.
Next, we suppose that G is commutative. As shown inLemma 4 we may diagonalize G over F ℓ . Picking a diagonal basis { P, Q } , we get an F -rational ℓ -isogeny h P i , say with character r . Appealing to Theorem 3 as before, we getan O -CM elliptic curve E ′ /F with an F -rational ℓ -isogeny h P ′ i with isogeny character ψ ′ , so that r = ψ ′ . We may diagonalize the mod − ℓ Galois representation of E ′ with respect to P ′ ; saysuch a basis is { P ′ , Q ′ } . Then we observe that our two groups G := ρ E,ℓ,P,Q ( G F ) and G ′ := ρ E ′ ,ℓ,P ′ ,Q ′ ( G F ) are so that G = G ′ . Then our containments are C s ( ℓ ) ⊇ G ′ ⊇ G ′ ⊆ G. In particular, our previous work shows that the constant c := 864[ F : Q ] is so that (1)holds for all R ∈ E [ ℓ ] • . This concludes our proof of Theorem 1 when one assumes thatGRH is true. 5. Part Two of the Proof: Removing GRH
Let E /F be a non-CM elliptic curve with an F -rational ℓ -isogeny for ℓ ≫ F
0. For anappropriate constant c ∈ Z + we would like to show that (1) holds, without assumingGRH. In doing so, there is an additional case which may appear from Theorem 3 thatwill be dealt with in this section.Let h P i be an F -rational ℓ -isogeny of E . Let us write its isogeny character as r : G F → F × ℓ . Suppose we are in case 2 of Theorem 3, so that(11) r = χ ℓ where χ ℓ : G F → F × ℓ is the mod − ℓ cyclotomic character.First, let us suppose that G is not commutative. Then by Lemma 4 we have G ss ⊆ G .In particular, if we pick Q ∈ E [ ℓ ] so that { P, Q } is an F ℓ -basis for E [ ℓ ] then we have G := ρ E,ℓ,P,Q ( G F ) = im (cid:20) r ∗ χ ℓ r − (cid:21) , and thus G ss = im (cid:20) r χ ℓ r − (cid:21) . YPICALLY BOUNDING TORSION ON ELLIPTIC CURVES WITH RATIONAL j -INVARIANT 15 By Equation (11) we have ( χ ℓ r − ) = r , whence we deduce that( G ss ) = im (cid:20) r r (cid:21) is a subgroup of scalars. As per the inclusions ( G ss ) ⊆ G ss ⊆ G , to find a constantso that (1) holds it suffices to show that ℓ − G ss ) -orbits with aconstant c := c ( F ) ∈ Z + . In particular, for such an integer c we must show that for all v ∈ V • := F ℓ h e , e i • one has ℓ − | c O ( G ss ) ( v ) . By our discussion in subsection 3.2, the sizes of orbits of the form O ( G ss ) ( ae ) and O ( G ss ) ( de ) with ad = 0 are r ( G F ), and the rest of the orbits O ( G ss ) ( ae + de ) with ad = 0 share the same size as the orbit O ( G ss ) ( e + e ). Since the action under G ss is scalar, it is clear that O ( G ss ) ( e + e ) = r ( G F ). Therefore, to find a constant c ∈ Z + for which (1) holds it suffices to find such a constant for which(12) ℓ − | c r ( G F ) . Since r ( G F ) is cyclic, we observe that r ( G F ) = r ( G F )gcd(12 , r ( G F )) . On the other hand, since ℓ ≫ F − ℓ cyclotomic character χ ℓ : G F → F × ℓ is surjective. Since χ ℓ ( G F ) is also cyclic, the image of its sixth power is χ ℓ ( G F ) = ℓ − , ℓ − . Since χ ℓ = r , we compare these sizes and find that( ℓ − · gcd(12 , r ( G F )) = gcd(6 , ℓ − · r ( G F ) , whence we have ℓ − | r ( G F ) . Since we also have r ( G F ) = r ( G F ) · gcd(6 , r ( G F ))it follows that ℓ − | · r ( G F ) . So we may take again c := 864[ F : Q ] when G is noncommutative to have that (12)holds, and thus (1) holds for all R ∈ E [ ℓ ] • .Next we suppose that G is commutative. As per Lemma 4, we may diagonalize G sothat it takes the form G = im (cid:20) r χ ℓ r − (cid:21) where r : G F → F × ℓ is an isogeny character of E . Then by Equation (11) one has G = im (cid:20) r r (cid:21) , so arguing as before we see that c := 864[ F : Q ] works for (1). This concludes our proofof Theorem 1. References [BC20] A. Bourdon and P.L. Clark,
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