Serre's uniformity problem in the split Cartan case
aa r X i v : . [ m a t h . N T ] M a r Serre’s Uniformity Problem in the Split Cartan Case
Yuri Bilu, Pierre Parent (Universit´e de Bordeaux I)October 30, 2018
Abstract
We prove that there exists an integer p such that X split ( p )( Q ) is made of cusps and CM-points for any prime p > p . Equivalently, for any non-CM elliptic curve E over Q and anyprime p > p the image of Gal( ¯ Q / Q ) by the representation induced by the Galois action onthe p -division points of E is not contained in the normalizer of a split Cartan subgroup. Thisgives a partial answer to an old question of Serre.AMS 2000 Mathematics Subject Classification 11G18 (primary), 11G05, 11G16 (secondary). Let N be a positive integer and G a subgroup of GL ( Z /N Z ) such that det G = ( Z /N Z ) × . Thenthe corresponding modular curve X G , defined as a complex curve as ¯ H / Γ, where ¯ H is the extendedPoincar´e upper half-plane and Γ is the pullback of G ∩ SL ( Z /N Z ) to SL ( Z ), is actually definedover Q , that is, it has a geometrically integral Q -model. As usual, we denote by Y G the finite partof X G (that is, X G deprived of the cusps). The curve X G has a natural (modular) model over Z that we still denote by X G . The cusps define a closed subscheme of X G over Z , and we define therelative curve Y G over Z as X G deprived of the cusps. The set of integral points Y G ( Z ) consists ofthose P ∈ Y G ( Q ) for which j ( P ) ∈ Z , where j is, as usual, the modular invariant.In the special case when G is the normalizer of a split (or non-split) Cartan subgroup ofGL ( Z /N Z ), the curve X G is denoted by X split ( N ) (or X nonsplit ( N ), respectively). In this note wefocus more precisely on the case when G is the normalizer of a split Cartan subgroup of GL ( Z /p Z )for p a prime number, i.e. G is conjugate to the set of diagonal and anti-diagonal matrices mod p ,and we prove the following theorem. Theorem 1.1
There exists an absolute effective constant C such that for any prime number p and any P ∈ Y split ( p )( Z ) we have log | j ( P ) | ≤ πp / + 6 log p + C . This is proved in Section 4 by a variation of the method of Runge, after some preparation inSection 2 and 3. The terms 2 πp / and 6 log p seem to be optimal for the method. The constant C may probably be replaced by o (1) when p tends to infinity.We apply Theorem 1.1 to the arithmetic of elliptic curves. Serre proved [22] that for any ellipticcurve E without complex multiplication (CM in the sequel), there exists p ( E ) > p > p ( E ) the natural Galois representation ρ E,p : Gal( ¯ Q / Q ) → GL( E [ p ]) ∼ = GL ( Z /p Z )is surjective. Masser and W¨ustholz [13], Kraus [8] and Pellarin [20] gave effective versions ofSerre’s result; for more recent work, see, for instance, Cojocaru and Hall [5, 6].Serre asked whether p can be made independent of E : does there exist an absolute constant p such that for any non-CM elliptic curve E over Q and any prime p > p the Galois representation ρ E,p is surjective?
1e refer to this as “Serre’s uniformity problem”. The general guess is that p = 37 would probablydo. The group GL ( Z /p Z ) has the following types of maximal proper subgroups: normalizersof (split and non-split) Cartan subgroups, Borel subgroups, and “exceptional” subgroups (thosewhose projective image is isomorphic to one of the groups A , S or A ). To solve Serre’s uniformityproblem, one has to show that for sufficiently large p , the image of the Galois representation isnot contained in any of the above listed maximal subgroups. (See [14, Section 2] for an excellentintroduction into this topic.) Serre himself settled the case of exceptional subgroups, and the workof Mazur [16] on rational isogenies implies Serre uniformity for the Borel subgroups, so to solveSerre’s problem we are left with the Cartan cases. Equivalently, one would like to prove that, forlarge p , the only rational points of the modular curves X split ( p ) and X nonsplit ( p ) are the cusps andCM points, in which case we will say that the rational points are trivial .In the present note we solve the split Cartan case of Serre’s problem. Theorem 1.2
There exists an absolute constant p such that for p > p every point in X split ( p )( Q ) is either a CM point or a cusp. In other words, for any non-CM elliptic curve E over Q and any prime p > p the image of theGalois representation ρ E,p is not contained in the normalizer of a split Cartan subgroup.Several partial results in this direction were available before. In [19, 21] it was proved, by verydifferent techniques, that X split ( p )( Q ) is trivial for a (large) positive density of primes; but themethods of loc. cit. have failed to prevent a complementary set of primes from escaping them. In[1] we allowed ourselves to consider Cartan structures modulo higher powers of primes, and showedthat, assuming the Generalized Riemann Hypothesis, X split ( p )( Q ) is trivial for large enough p .Regarding possible generalizations, note that the Runge’s method applies to the study ofintegral points on an affine curve Y , defined over Q , if the following Runge condition is satisfied:Gal( ¯ Q / Q ) acts non-transitively on the set X \ Y , ( R )where X is the projectivization of Y . The Runge condition is satisfied for the curve X split ( p )because it has two Galois orbits of cusps over Q . Runge’s method also applies to other mod-ular curves such as X ( p ), but, unfortunately, it does not work (under the form we use) with X nonsplit ( p ), because all cusps of this curve are conjugate over Q and the Runge condition fails.Moreover, we need a weak version of Mazur’s method to obtain integrality of rational points, andthis is believed not to apply to X nonsplit ( p ) (see [4]). Several other applications of our techniquesare however possible, and at present we work on applying Runge’s method to general modularcurves over general number fields, see [1, 2]. Acknowledgments
We thank Daniel Bertrand, Imin Chen, Henri Cohen, Bas Edixhoven, Lo¨ıcMerel, Joseph Oesterl´e, Federico Pellarin, Vinayak Vatsal and Yuri Zarhin for stimulating discus-sions and useful suggestions.
Convention
Everywhere in this article the O ( · )-notation, as well as the Vinogradov notation“ ≪ ” implies absolute effective constants. As above, we denote by H the Poincar´e upper half-plane and put ¯ H = H ∪ Q ∪ { i ∞} . For τ ∈ H we, as usual, put q = q τ = e πiτ . For a rational number a we define q a = e πiaτ . Let a = ( a , a ) ∈ Q be such that a / ∈ Z , and let g a : H → C be the corresponding Siegel function [9, Section 2.1]. Then we have the following infinite product presentation for g a [9, page 29]: g a ( τ ) = − q B ( a ) / e πia ( a − ∞ Y n =0 (cid:0) − q n + a e πia (cid:1) (cid:0) − q n +1 − a e − πia (cid:1) , (1)2here B ( T ) = T − T + 1 / g a ◦ γ = g a γ · (a root of unity) for γ ∈ SL ( Z ) , (2) g a = g a ′ · (a root of unity) when a ≡ a ′ mod Z . (3)Remark that the root of unity in (2) is of order dividing 12, and in (3) of order dividing 2 N ,where N is the denominator of a (the common denominator of a and a ). (For (2) use properties K 0 and
K 1 of loc. cit., and for (3) use
K 3 and the fact that ∆ is modular of weight 12.)Moreover, g a ◦ γ = g a · (a root of unity) for γ ∈ Γ( N ) , (4)the root of unity being of order dividing 12 N , because g N a is a modular function on Γ( N ) byTheorem 1.2 from [9, page 31].The following is immediate from (1). Proposition 2.1
Assume that ≤ a < . Then for τ ∈ H satisfying | q τ | ≤ . we have log | g a ( τ ) | = 12 B ( a ) log | q | + log (cid:12)(cid:12) − q a e πia (cid:12)(cid:12) + log (cid:12)(cid:12) − q − a e − πia (cid:12)(cid:12) + O ( | q | ) (where we recall that, all through this article, the notation O ( · ) as well as ≪ imply absoluteeffective constants). For a ∈ Q \ Z Siegel’s function g a is algebraic over the field C ( j ): this again follows fromthe fact that g N a is Γ( N )-automorphic, where, as above, N is the denominator of a . Since g a isholomorphic and does not vanish on the upper half-plane H (again by Theorem 1.2 of loc. cit.),both g a and g − a must be integral over the ring C [ j ]. Actually, a stronger assertion holds. Proposition 2.2
Both g a and (1 − ζ N ) g − a are integral over Z [ j ] . Here N is the denominatorof a and ζ N is a primitive N -th root of unity. This is, essentially, established in [9], but is not stated explicitly therein. Therefore we brieflyindicate the proof here. A holomorphic and Γ( N )-automorphic function f : H → C admits theinfinite q -expansion f ( τ ) = X k ∈ Z a k q k/N . (5)We call the q -series (5) algebraic integral if the following two conditions are satisfied: the negativepart of (5) has only finitely many terms (that is, a k = 0 for large negative k ), and the coefficients a k are algebraic integers. Algebraic integral q -series form a ring. The invertible elements of this ringare q -series with invertible leading coefficient. By the leading coefficient of an algebraic integral q -series we mean a m , where m ∈ Z is defined by a m = 0, but a k = 0 for k < m . Lemma 2.3
Let f be a Γ( N ) -automorphic function regular on H such that for every γ ∈ Γ(1) the q -expansion of f ◦ γ is algebraic integral. Then f is integral over Z [ j ] . Proof
This is, essentially, Lemma 2.1 from [9, Section 2.2]. Since f is Γ( N )-automorphic, theset { f ◦ γ : γ ∈ Γ(1) } is finite. The coefficients of the polynomial F ( T ) = Q ( T − f ◦ γ ) (where theproduct is taken over the finite set above) are Γ(1)-automorphic functions with algebraic integral q -expansions. Since they have no pole on H , they belong to C [ j ] and even to ¯ Z [ j ], where ¯ Z is thering of all algebraic integers, because the coefficients of their q -expansions are algebraic integers.It follows that f is integral over ¯ Z [ j ], hence over Z [ j ]. (cid:3) roof of Proposition 2.2 The function g N a is automorphic of level N and its q -expansion isalgebraic integral (as one can easily see by transforming the infinite product (1) into an infiniteseries). By (2), the same is true for for every ( g a ◦ γ ) N . Lemma 2.3 now implies that g N a isintegral over Z [ j ], and so is g a .Further, the q -expansion of g a is invertible if a / ∈ Z and is 1 − e ± πia times an invertible q -series if a ∈ Z . Hence the q -expansion of g − a is algebraic integral when a / ∈ Z , and if a ∈ Z the same is true for (cid:0) − e ± πia (cid:1) g − a . In the latter case N is the exact denominator of a , whichimplies that (1 − ζ N ) / (cid:0) − e ± πia (cid:1) is an algebraic unit. Hence, in any case, (1 − ζ N ) g − a hasalgebraic integral q -expansion, and the same is true with g a replaced by g a ◦ γ for any γ ∈ Γ(1)(we again use (2) and notice that a and a γ have the same order in ( Q / Z ) ). Applying Lemma 2.3to the function (cid:0) (1 − ζ N ) g − a (cid:1) N , we complete the proof. (cid:3) In this section we define a special “modular unit” (in the spirit of [9]) and study its asymptoticbehavior at infinity. With the common abuse of speech, the modular invariant j , as well as the othermodular functions used below, may be viewed, depending on the context, as either automorphicfunctions on the Poincar´e upper half-plane, or rational functions on the corresponding modularcurves.Since the root of unity in (3) are of order dividing 2 N , where N is a denominator of a , thefunction g N a will be well-defined if we select a in the set (cid:0) N − Z / Z (cid:1) . Thus, fix a positiveinteger N and for a non-zero element a of ( N − Z / Z ) put u a = g N a . After fixing a choicefor ζ N in C (for instance ζ N = e iπ/N ), the analytic modular curve X ( N )( C ) := ¯ H / Γ( N ) hasa modular model over Q ( ζ N ), parameterizing isomorphism classes of generalized elliptic curvesendowed with a basis of their N -torsion with determinant 1. As already noticed, the function u a is Γ( N )-automorphic and hence defines a rational function on the modular curve X ( N )( C ); infact, it belongs to the field Q ( ζ N ) (cid:0) X ( N ) (cid:1) . The Galois group of the latter field over Q ( j ) isisomorphic to GL ( Z /N Z ) / {± } , and we may identify the two groups to make the Galois actioncompatible with the natural action of GL ( Z /N Z ) on ( N − Z / Z ) in the following sense: forany ¯ σ ∈ Gal (cid:0) Q (cid:0) X ( N ) (cid:1)(cid:14) Q ( j ) (cid:1) = GL ( Z /N Z ) / {± } and any non-zero a ∈ ( N − Z / Z ) we have u ¯ σ a = u a σ , where σ ∈ GL ( Z /N Z ) is a pull-back of ¯ σ . Notice that u a = u − a , which follows from (2).For the proof of the statements above the reader may consult [9, pp. 31–36], and especiallyTheorem 1.2, Proposition 1.3 and the beginning of Section 2.2 therein.From now on we assume that N = p ≥ G is the normalizerof the diagonal subgroup of GL ( F p ). In this case there are two Galois orbits of cusps over Q , thefirst being the cusp at infinity, which is Q -rational (we denote it by ∞ ), and the second consisting ofthe ( p − / P , . . . , P ( p − / ), which are defined over the real cyclotomicfield Q ( ζ p ) + . According to the theorem of Manin-Drinfeld, there exists U ∈ Q ( X G ) such that theprincipal divisor ( U ) is of the form m (cid:0) ( p − / · ∞ − ( P + · · · + P ( p − / ) (cid:1) with some positiveinteger m . Below we use Siegel’s functions to find such U explicitly with m = 2 p ( p − Remark 3.1 (a) The general form of units we build is more ripe for generalization, but in thepresent case, using the Q -isomorphism between X split ( p ) and X ( p ) /w p , our unit couldprobably be expressed in terms of (products of) modular forms of shape ∆( nz ).(b) The assumption p ≥ p = 2.Denote by p − F × p the set of non-zero elements of p − Z / Z . Then the set A = (cid:8) ( a,
0) : a ∈ p − F × p (cid:9) ∪ (cid:8) (0 , a ) : a ∈ p − F × p (cid:9) is G -invariant. Hence the function U = Y a ∈ A u a Q ( X G ). In particular, viewed as a function on H , it is Γ-automorphic, where Γis the pullback to Γ(1) of G ∩ SL ( F p ).More generally, for c ∈ Z put β c = (cid:18) c (cid:19) , U c = U ◦ β c = Y a ∈ Aβ c u a (so that U = U ).Let D be the familiar fundamental domain of SL ( Z ) (that is, the hyperbolic triangle withvertices e πi/ , e πi/ and i ∞ , together with the geodesic segments [ i, e πi/ ] and [ e πi/ , i ∞ ]) and D + Z the union of all translates of D by the rational integers. Recall also that j denotes themodular invariant. Lemma 3.2
For any P ∈ Y G ( C ) there exists c ∈ Z (in fact, even c ∈ { , . . . , ( p − / } ) and τ ∈ D + Z such that j ( τ ) = j ( P ) and U c ( τ ) = U ( P ) . Proof
Let τ ′ ∈ H be such that j ( τ ′ ) = j ( P ) and U ( τ ′ ) = U ( P ). There exists β ∈ Γ(1) such that β − ( τ ′ ) ∈ D . Now observe that the set (cid:8) β , . . . , β ( p − / (cid:9) is a full system of representatives of thedouble classes Γ \ Γ(1) / Γ ∞ , where Γ ∞ is the subgroup of Γ(1) stabilizing ∞ . Hence we may write β = γβ c κ with γ ∈ Γ, c ∈ { , . . . , ⌊ p/ ⌋} and κ ∈ Γ ∞ . Then τ = κβ − ( τ ′ ) is as desired. (cid:3) Proposition 3.3
For τ ∈ H such that | q τ | ≤ /p we have (cid:12)(cid:12) log | U c ( τ ) | − ( p − log | q τ | (cid:12)(cid:12) ≤ π p log | q − τ | + O ( p log p ) (6) if p | c , and (cid:12)(cid:12) log | U c ( τ ) | + 2( p −
1) log | q τ | (cid:12)(cid:12) ≤ π p log | q − τ | + O ( p ) (7) if p ∤ c . For the proof of Proposition 3.3 we need an elementary, but crucial lemma.
Lemma 3.4
Let z be a complex number, | z | < , and N a positive integer. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 log (cid:12)(cid:12) − z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π | z − | + O (1) . (8) Proof
We have (cid:12)(cid:12) log | z | (cid:12)(cid:12) ≤ − log(1 − | z | ) for | z | <
1. Applying this with − z k instead of z ,we conclude that it suffices to bound − P ∞ k =1 log(1 − q k ) with q = | z | . Since (8) is obvious for | z | ≤ /
2, we may assume that 1 / ≤ q < . (9)Put τ = log q/ (2 πi ). Then − ∞ X k =1 log | − q k | = 124 log q − log | η ( τ ) | , where η ( τ ) is the Dedekind η -function. Since | η ( τ ) | = | τ | − / | η ( − τ − ) | , we have − ∞ X k =1 log | − q k | = −
124 log | Q | + 124 log q + 12 log | τ | − ∞ X k =1 log | − Q k | (10)with Q = e − πiτ − = e π / log q . The first term on the right of (10) is exactly ( π / / log | z − | , thesecond term is negative, the third term is again negative (here we use (9)), and the infinite sumis O (1), again by (9). The lemma is proved. (cid:3) roof of Proposition 3.3 Write q = q τ . Recall that for a rational number α we define q α = e πiατ . For a ∈ Q / Z we denote by ˜ a the lifting of a to the interval [0 , τ ∈ H satisfying | q | ≤ . | U c ( τ ) | = 6 p X a ∈ Aβ c B (˜ a ) log | q | + 12 p X a ∈ Aβ c (cid:16) log (cid:12)(cid:12) − q ˜ a e πia (cid:12)(cid:12) + log (cid:12)(cid:12) − q − ˜ a e − πia (cid:12)(cid:12)(cid:17) + O ( p | q | ) . (11)The rest of the proof splits into two cases and relies on the identity N − X k =1 B (cid:18) kN (cid:19) = − ( N − N .
The first case: p | c In this case Aβ c = A . Hence X a ∈ Aβ c B (˜ a ) = p − X k =1 B (cid:18) kp (cid:19) + ( p − B (0) = ( p − p . (12)Further, X a ∈ Aβ c (cid:16) log (cid:12)(cid:12) − q ˜ a e πia (cid:12)(cid:12) + log (cid:12)(cid:12) − q − ˜ a e − πia (cid:12)(cid:12)(cid:17) = 2 p − X k =1 log (cid:12)(cid:12) − q k/p (cid:12)(cid:12) + log (cid:12)(cid:12)(cid:12)(cid:12) − q p − q (cid:12)(cid:12)(cid:12)(cid:12) + log p. (13)Lemma 3.4 with z = q /p implies that p − X k =1 log (cid:12)(cid:12) − q k/p (cid:12)(cid:12) ≤ π p log | q − | + O (1) . Also, log | − q p | ≪ | q | p and log | − q | ≪ | q | . Combining all this with (11), (12) and (13), weobtain (6). The second case: p ∤ c In this case Aβ c = { ( a,
0) : a ∈ p − F × p } ∪ { ( a, ab ) : a ∈ p − F × p } , where b ∈ Z satisfies bc ≡ p . Hence X a ∈ Aβ c B (˜ a ) = 2 p − X k =1 B (cid:18) kp (cid:19) = − p − p . Further, X a ∈ Aβ c (cid:16) log (cid:12)(cid:12) − q ˜ a e πia (cid:12)(cid:12) + log (cid:12)(cid:12) − q − ˜ a e − πia (cid:12)(cid:12)(cid:17) = 2 p − X k =1 log (cid:12)(cid:12) − q k/p (cid:12)(cid:12) + 2 p − X k =1 log (cid:12)(cid:12) − ( q /p e πib/p ) k (cid:12)(cid:12) . Again using Lemma 3.4, we complete the proof. (cid:3)
In this section p is a prime number and G is the normalizer of the diagonal subgroup of GL ( Z /p Z ).Define the “modular units” U c as in Section 3. Recall that U = U belongs to the field Q ( X G ).Theorem 1.1 is a consequence of the following two statements.6 roposition 4.1 Assume that p ≥ . For any P ∈ Y G ( C ) we have either log | j ( P ) | ≤ πp / + 6 log p + O (1) or log | j ( P ) | ≤ p − (cid:12)(cid:12) log | U ( P ) | (cid:12)(cid:12) + 2 πp / − p + O (cid:0) (cid:1) . (14) Proposition 4.2
For P ∈ Y G ( Z ) we have ≤ log | U ( P ) | ≤ p log p . Combining the two propositions, we find that for P ∈ Y split ( p )( Z ) we havelog | j ( P ) | ≤ πp / + 6 log p + O (1) , which proves Theorem 1.1 for p ≥ p = 2 as well, but in this case it is easier to appeal to thegeneral Runge theorem: if an affine curve Y , defined over Q , has 2 (or more) rational points atinfinity, then integral points on Y are effectively bounded; see, for instance, [3, 11]. Proof of Proposition 4.1
According to Lemma 3.2, there exists τ ∈ D + Z and c ∈ Z with U c ( τ ) = U ( P ) and j ( τ ) = j ( P ). We write q = q τ . Since τ ∈ D + Z , we have j ( τ ) = q − + O (1) , (15)which implies that either log | j ( P ) | ≤ πp / + 6 log p + O (1) or log | q − | ≥ πp / + 6 log p . Inthe latter case we apply Proposition 3.3. When p ∤ c it yields (cid:12)(cid:12)(cid:12)(cid:12) log | q | + 12( p −
1) log | U c ( τ ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ π p p − πp / + 6 log p ) + O (1) = 2 πp / − p + O (1) , which, together with (15), implies the result. In the case p | c Proposition 3.3 gives (cid:12)(cid:12)(cid:12)(cid:12) log | q | − p − log | U c ( τ ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ π p ( p − (2 πp / + 6 log p ) + O (1) = O (1) , which implies even a better bound than needed. (cid:3) Proof of Proposition 4.2
Since U belongs to Q ( X G ) and has no pole or zero outside the cusps, U ( P ) is a non-zero rational number. Let ζ = ζ p be a primitive p -th root of unity. Since U is a prod-uct of 24 p ( p −
1) Siegel functions, Proposition 2.2 implies that both U and (1 − ζ ) p ( p − U − areintegral over Z [ j ]. Hence, for P ∈ Y G ( Z ) both the numbers U ( P ) and (1 − ζ ) p ( p − U ( P ) − are al-gebraic integers. Since U ( P ) ∈ Q × , it is a non-zero rational integer; in particular, log | U ( P ) | ≥ U ( P ) divides (1 − ζ ) p ( p − . Taking the Q ( ζ ) / Q -norm, we see that U ( P ) p − divides p p ( p − . This proves the proposition. (cid:3) First of all, recall the following integrality property of the j -invariant. Theorem 5.1 (Mazur, Momose, Merel).
For a prime p = 11 or p ≥ , the j -invariant j ( P ) of any non-cuspidal point of X split ( p )( Q ) belongs to Z . This is a combination of results of Mazur [16], Momose [18] and Merel [17]. For more detailssee the Appendix (Section 6), where we give a short unified proof.Denote by h( α ) the absolute logarithmic height of an algebraic number α . If α is a non-zerorational integer, then h( α ) = log | α | ; it follows in particular from the above that, if E is an ellipticcurve over Q endowed with a normalizer of split Cartan mod p with p ≥
17, then h( j E ) = log | j E | .In view of Theorem 5.1, Theorem 1.2 is a straightforward consequence of Theorem 1.1 and thefollowing proposition, whose proof will be the goal of this section.7 roposition 5.2 There exists an absolute effective constant κ such that the following holds.Let p be a prime number, and E a non-CM elliptic curve over Q , endowed with a structure ofnormalizer of split Cartan subgroup in level p . Then h( j E ) = log | j E | ≥ κp. (16)The proof of Proposition 5.2 relies on Pellarin’s refinement [20] of the Masser-W¨ustholz famousbound [12] for the degree of the minimal isogeny between two isogenous elliptic curves. Theorem 5.3 (Masser-W¨ustholz, Pellarin)
Let E be an elliptic curve defined over a numberfield K of degree d . Let E ′ be another elliptic curve, defined over K and isogenous to E . Thenthere exists an isogeny ψ : E → E ′ of degree at most κ ( d ) (1 + h( j E )) , where the constant κ ( d ) depends only on d and is effective. Masser and W¨ustholz had exponent 4, and Pellarin reduced it to 2, which is crucial for us (in fact,any exponent below 4 would do). He also gave an explicit expression for κ ( d ). Corollary 5.4
Let E be a non-CM elliptic curve defined over a number field K of degree d , andadmitting a cyclic isogeny over K of degree δ . Then δ ≤ κ ( d ) (1 + h( j E )) . Proof
Let φ be a cyclic isogeny from E to E ′ , and let φ D : E ′ → E be the dual isogeny. Let ψ : E → E ′ be an isogeny of degree bounded by κ ( d ) (1 + h( j E )) ; without loss of generality, ψ may be assumed cyclic. As E has no CM, the composed map φ D ◦ ψ must be multiplication bysome integer, so that φ = ± ψ . (cid:3) Proof of Proposition 5.2
For an elliptic curve E endowed with a structure of normalizer ofsplit Cartan subgroup in level p over Q , write C and C for the obvious two independent p -isogenies defined over the quadratic field K cut-up by the inclusion of the Cartan group mod p inits normalizer. Set E i := E/C i and recall that there is a cyclic p -isogeny over K from E to E ,factorizing as a product of two p -isogenies: ϕ : E → E → E . It follows from Corollary 5.4 that h( j E i ) ≥ κ p for i = 1 , F ( E ) ≤ h F ( E ) + log p , where h F isFaltings’ semistable height . Finally, for any elliptic curve E over a number field we have (cid:12)(cid:12) h( j E ) − F ( E ) (cid:12)(cid:12) ≤ (cid:0) j E ) (cid:1) + O (1) , see [25, Proposition 2.1]. (Pellarin shows that O (1) can be replaced by 47.15, see [20], equation (51)on page 240.) This completes the proof of Proposition 5.2 and of Theorem 1.2. (cid:3) j -Invariant This appendix is mainly of expository nature: following Mazur, Momose and Merel, we show thatrational points on X split ( p ) are, in fact, integral. Theorem 6.1 (Mazur, Momose, Merel).
For a prime p = 11 or p ≥ , the j -invariant j ( P ) of any non-cuspidal point of X split ( p )( Q ) belongs to Z . Recall that h F ( E ) is defined as [ K : Q ] − deg ω , where K is an extension of K such that E has semi-stablereduction at every place of K , and ω is a N´eron differential on E | K ; it is independent of the choice of K and ω . ℓ of the denominator of j ( P ) must either be 2, or p , or satisfy ℓ ≡ ± p .The cases ℓ ≡ ± p and ℓ = p were settled by Momose [18, Proposition 3.1], together withthe case ℓ = 2 when p ≡ ℓ = 2 with p p ≥ X split ( p ) parametrizes (isomorphism classes of) elliptic curves endowedwith an unordered pair of independent p -isogenies. Let P = (cid:0) E, { A, B } (cid:1) be a Q -point on X split ( p ),which we may assume to be non CM. Then the isogenies A and B are defined over a number field K with degree at most 2. Proposition 6.2
Let P = (cid:0) E, { A, B } (cid:1) ∈ X split ( p )( Q ) and K be defined as above. Let O K be itsring of integers. Then we have the following.(a) The curve E is not potentially supersingular at p .(b) The points ( E, B ) and ( E/A, E [ p ] /A ) = ( E/A, A ∗ ) , where A ∗ is the isogeny dual to A ,coincide in the fibers of characteristic p of X ( p ) / O K .(c) The field K is quadratic over Q , and p splits in K . Proof
Parts (a) and (c) are proved in [18], Lemmas 1.3 and 3.2 respectively. Part (b) followsfrom [19, proof of Proposition 3.1]. For the convenience of the reader we here sketch a proof (withsomewhat different (and simpler) arguments).It follows from Serre’s study of the action of inertia groups I p at p on the formal group ofelliptic curves that if E is potentially supersingular then I p (potentially) acts via a “fundamentalcharacter of level 2”, so that the image of inertia contains a subgroup of order 4 or 6 in a nonsplitCartan subgroup of GL( E [ p ]) (see [22, Paragraph 1]). This gives a contradiction with the factthat a subgroup of index 2 in the absolute Galois group of Q preserves two lines in E [ p ], whencepart (a).For (b) we remark that we may assume the schematic closure of A to be ´etale over O (the ringof integers of a completion K P of K at a prime P above p , whose residue field we denote by k P ):indeed, as E is not potentially supersingular at P , at most one line in E [ p ] can be purely radicialover k P . Up to replacing K P by a finite ramified extension, we shall also assume E is semistableover K P . Now E/A is isomorphic over k P to E ( p ) via the Vershiebung isogeny, and the latter isin turn isomorphic to E /k P as E has a model over Z . Moreover the isomorphism between B and E [ p ] /A as K -group schemes induced by the projection E → E/A extends to an isomorphism over O by Raynaud’s theorem on group schemes of type ( p, . . . , p ), as recalled in [18, Proof of Lemma1.3]. It follows that ( E, B ) k P is isomorphic to ( E/A, E [ p ] /A ) k P = ( w p ( E, A )) k P , whence (b).Pushing this reasonning further yields (c). We indeed first note that, by Mazur’s theoremon rational isogenies, K = Q for p ≥
37, and with the above notations (
E, B ) is also equal to σ ( E, A ) for a σ ∈ Gal( K/ Q ). Now ( E, A ) and (
E, B ) define two points in X ( p )( O ). The twocomponents of X ( p ) F p are left stable by Gal( F p / F p ), but switched by w p . Therefore if p does notsplit in K , the fact that σ ( E, A ) k P = ( E, B ) k P = ( w p ( E, A )) k P leaves no other choice to ( E, A )than specializing at the intersection of the components. As those intersection points modularlycorrespond to supersingular curves, this contradicts (a), and completes the proof. (cid:3)
The curve X split ( p ) admits an obvious double covering by the curve X sp . Car . ( p ), parameterizingelliptic curves endowed with an ordered pair of p -isogenies. We denote by w be the generator of theGalois group of this covering, that is, w modularly exchanges the two p -isogenies. In symbols, if (cid:0) E, ( A, B ) (cid:1) is a point on X sp . Car . ( p ), then w (cid:0) E, ( A, B ) (cid:1) = (cid:0) E, ( B, A ) (cid:1) . We recall certain propertiesof the modular Jacobian J ( p ) and its Eisenstein quotient e J ( p ) (see [15]). Proposition 6.3
Let p be a prime number. Then we have the following. a) [15, Theorem 1] The group J ( p )( Q ) tors is cyclic and generated by cl(0 −∞ ) , where and ∞ are the cusps of X ( p ) . Its order is equal to the numerator of the quotient ( p − / .(b) [15, Theorem 4] The group ˜ J ( p )( Q ) is finite. Moreover, the natural projection J ( p ) → ˜ J ( p ) defines an isomorphism J ( p )( Q ) tors → e J ( p )( Q ) . As Mazur remarks, Raynaud’s theorem on group schemes of type ( p, . . . , p ) insures that J ( p )( Q ) tors defines a Z -group scheme which, being constant in the generic fiber, is ´etale out-side 2, and which at 2 has ´etale quotient of rank at least half that of J ( p )( Q ) tors . Proof of Theorem 6.1
For an element t in the Z -Hecke algebra for Γ ( p ), define the mor-phism g t from X smoothsp . Car . ( p ) / Z to J ( p ) / Z which extends the morphism on generic fibers: g t : (cid:26) X sp . Car . ( p ) → J ( p ) Q = (cid:0) E, ( A, B ) (cid:1) t · cl (cid:0) ( E, A ) − ( E/B, E [ p ] /B ) (cid:1) . Let J ( p ) π → e J ( p ) be the projection to the Eisenstein quotient, and ˜ g t := π ◦ g t . One checks that g t ◦ w = − w p ◦ g t and one knows that (1 + w p ) acts trivially on e J ( p ) from [15, Proposition 17.10].Therefore ˜ g t actually factorizes through a Q -morphism from X split ( p ) to e J ( p ), that we extend bythe universal property of N´eron models to a map from X smoothsplit ( p ) / Z to e J ( p ) / Z . We still denotethis morphism by ˜ g t and we put ˜ g = ˜ g .Let P be a rational point on X split ( p ), and ℓ a prime divisor of the denominator of j ( P ). Then P specializes to a cusp at ℓ . Recall that X split ( p ) has one cusp defined over Q (the rational cusp ),and ( p − / Q . We first claim that P specializes to the rationalcusp. Indeed, it follows from Propositions 6.2 (b) that ˜ g ( P )( F p ) = 0( F p ), and by the remark afterProposition 6.3, ˜ g ( P )( Q ) = 0( Q ) (recall p > X split ( p )( C ) map tocl(0 − ∞ ) in J ( p )( C ) (this can be seen with the above modular interpretation of ˜ g t , using thefact that the non-rational cusps specialize at p to a generalized elliptic curve endowed with a pairof ´etale isogenies. Or, if f denotes the map f : X sp . C . ( p ) → X ( p ), ( E, ( A, B )) ( E, A ), onehas g = cl( f − w p f w ), and as f ( c i ) = 0 ∈ X ( p ) for c i a non-rational cusp and w permutes the c i s, one sees that ˜ g ( c i ) = cl(0 − ∞ ). For more details see for instance the proof of Proposition2.5 in [18]). Therefore, for p ≥
11 and p = 13, Proposition 6.3 implies that if P specializes to anon-rational cusp at ℓ then ˜ g ( P ) would not be 0 at ℓ , a contradiction.Now take an ℓ -adically maximal element t in the Hecke algebra which kills the windingideal I e . Again, as t (1 + w p ) = 0, the above morphism g t factorizes through a morphism g + t from X smoothsplit ( p ) / Z to t · J ( p ) / Z . Moreover g + t ( P ) belongs to t · J ( p )( Q ), hence is a torsion point,as t · J ( p ) is isogenous to a quotient of the winding quotient of J ( p ). We see as above by look-ing at the fiber at p that g + t ( P ) = 0 at p , hence generically. We then easily check by using the q -expansion principle, as in [17, Theorem 5], that g + t is a formal immersion at the specialization ∞ ( F ℓ ) of the rational cusp on X split ( p ). This allows us to apply the classical argument of Mazur(see e.g. [16, proof of Corollary 4.3]), yielding a contradiction: therefore P is not cuspidal at ℓ . (cid:3) References [1]
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