A height inequality for rational points on elliptic curves implied by the abc-conjecture
AA HEIGHT INEQUALITY FOR RATIONAL POINTS ON ELLIPTICCURVES IMPLIED BY THE ABC-CONJECTURE
ULF K ¨UHN, J. STEFFEN M ¨ULLER
Abstract.
In this short note we show that the uniform abc -conjecture over numberfields puts strong restrictions on the coordinates of rational points on elliptic curves. Forthe proof we use a variant of the uniform abc -conjecture over number fields formulated byMochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferichprimes. Introduction If E/ Q is an elliptic curve in Weierstraß form and P ∈ E ( Q ) \ { O } , where O is the pointat infinity, then it is well known that we can write P = (cid:18) a P d P , b P d P (cid:19) , where a P , b P , d P ∈ Z satisfy gcd( d P , a P b P ) = 1.The structure of the denominators d P has been studied, for instance, by Everest-Reynolds-Stevens [ERS07], and has recently received increasing attention in the context of ellipticdivisibility sequences first studied by Ward [War48], see for instance [EEW01] or [Rey12b]and the references therein. In this paper we make the following conjecture, where rad( n )denotes the product of distinct prime divisors of an integer n . Conjecture 1.1.
Let E/ Q be an elliptic curve in Weierstraß form. For all (cid:15) > thereexists a constant c (cid:15) such that max (cid:26)
12 log | a P | , log | d P | (cid:27) ≤ (1 + (cid:15) ) log rad( d P ) + c (cid:15) for all P ∈ E ( Q ) \ { O } .Remark . A strong form of Siegel’s Theorem implies the weaker inequality12 log | a P | ≤ (1 + (cid:15) ) log | d P | + O (1) , see for example [Sil86, Example IX.3.3]. Date : November 3, 2018. a r X i v : . [ m a t h . N T ] N ov ULF K ¨UHN, J. STEFFEN M ¨ULLER
Our main theorem relates Conjecture 1.1 to the uniform abc -conjecture over number fields.
Theorem 1.3.
Conjecture 1.1 follows from the uniform abc -conjecture over number fields.Remark . It is straightforward to generalize both Conjecture 1.1 and Theorem 1.3 toarbitrary number fields. For ease of notation we restrict to the rational case here.
Remark . Mochizuki [Moc12] has recently announced a proof of the uniform abc -conjectureover number fields.We now list some consequences of Conjecture 1.1.
Proposition 1.6.
Suppose that Conjecture 1.1 holds and let E/ Q be an elliptic curve inWeierstraß form. Then the set of all P ∈ E ( Q ) \ { O } such that the squarefree part of d P is bounded is finite. If the bound on the squarefree part of d P in Proposition 1.1 is 1, then we get a conditionalproof of Siegel’s Theorem that there are only finitely many integral points on E , and wecan also deduce: Corollary 1.7.
Suppose that Conjecture 1.1 holds and let E/ Q be an elliptic curve inWeierstraß form. Then the set of all P ∈ E ( Q ) \ { O } such that d P is a perfect power isfinite.Remark . It is shown in [ERS07, Theorem 1.1] that for a fixed exponent n >
1, thereare only finitely many P ∈ E ( Q ) \ { O } such that d P is an n th power. According to [ERS07,Remark 1.2], the uniform abc -conjecture over number fields implies that for n (cid:29)
0, thereare no P ∈ E ( Q ) \ { O } such that d P is an n th power. Together, these results also implythat the finiteness of the set of P ∈ E ( Q ) \ { O } such that d P is a perfect power is aconsequence of the uniform abc -conjecture over number fields. However, a direct proof ofthe assertion from [ERS07, Remark 1.2] has not been published and, according to Reynolds[Rey12a], is rather complicated.Another application of Conjecture 1.1 concerns elliptic non-Wieferich primes . For a prime p , we define N p := E ( F p ). If P ∈ E ( Q ) is non-torsion, let W E,P := (cid:8) p prime : N p P (cid:54)≡ O (mod p ) (cid:9) be the set of elliptic non-Wieferich primes to base P . The following result is due toSilverman. Theorem 1.9. (Silverman, [Sil88, Theorem 2] ) Assume that the abc -conjecture (over Q )holds. If an elliptic curve E/ Q has j -invariant equal to 0 or 1728 and if P ∈ E ( Q ) isnon-torsion, then (1) |{ p ∈ W E,P : p ≤ X }| ≥ (cid:112) log( X ) + O E,P (1) as X → ∞ . HEIGHT INEQUALITY IMPLIED BY ABC 3
This is the analogue of [Sil88, Theorem 1], giving an asymptotic lower bound (dependent onthe abc -conjecture over Q ) for the number of classical non-Wieferich primes up to a givenbound. In particular, this proves that the abc -conjecture over Q implies the existence ofinfinitely many elliptic non-Wieferich primes to any base P ∈ E ( Q ) if j ( E ) ∈ { , } .See [Vol00] for further results concerning elliptic non-Wieferich primes.If we assume Conjecture 1.1 instead of the abc -conjecture over Q , we can eliminate thecondition on the j -invariant of E . Proposition 1.10.
Assume Conjecture 1.1 and let E/ Q be an elliptic curve. Then (1) holds for every non-torsion P ∈ E ( Q ) . In Section 2 we recall work of Mochizuki from [Moc10], which we use in Section 3 forthe proof of Theorem 1.3. We prove Proposition 1.6 in Section 4 and Proposition 1.10 inSection 5.
Acknowledgements
We thank Joe Silverman for helpful comments and his suggestion that Proposition 1.10should hold. We also thank Jonathan Reynolds for helpful discussions. The second authorwas supported by DFG-grant KU 2359/2-1.2.
Mochizuki’s height inequality
In this section, we define Mochizuki’s log-conductor function [Moc10, §
1] and state hisConjecture 2.3. We assume some familiarity with the basics of Arakelov theory, see forinstance [Sou97].Let K be a number field, let X be a smooth, proper, geometrically connected curve over K and let D be an effective divisor on X . Extend X to a proper normal model X whichis flat over Spec( O K ) and extend D to an effective horizontal divisor D ∈
Div( X ). Wecan define a function log-cond X , D on X ( K ) as follows: Let P ∈ X ( K ) and let F be anumber field containing P . Then P induces a morphism P : Spec( O F ) → Y , where Y isthe normalisation of X ×
Spec( O F ) and we definelog-cond X , D ( P ) := 1[ F : Q ] (cid:91) deg F (( P ∗ D ) red ) ∈ R , where (cid:91) deg F is the arithmetic degree of an arithmetic divisor on Spec( O F ). Remark . Note that up to a bounded function, log-cond X , D only depends on X and D (see [Moc10, Remark 1.5.1]). Remark . Alternatively, we could define the log-conductor function as follows: Extend X to a proper regular model X over Spec( O K ) and D to an effective horizontal divisor ULF K ¨UHN, J. STEFFEN M ¨ULLER
D ∈
Div( X ). Let π : X (cid:48) → X × Spec( O F ) be the minimal desingularization and let P ∈ Div( X (cid:48) ) be the Zariski closure of P . Then we definelog-cond (cid:48)X , D ( P ) := 1[ F : Q ] (cid:88) p ∈ S log N m ( p ) ∈ R , where S is the set of finite primes of F such that the intersection multiplicity ( P , π ∗ D ) p (cid:54) = 0.Then it is easy to see that log-cond X , D = log-cond (cid:48)X , D + O (1).The following variation of Vojta’s height conjecture is due to Mochizuki: Conjecture 2.3. (Mochizuki, [Moc10, § ) Let X be a smooth, proper, geometricallyconnected curve over a number field K . Let D ⊂ X be an effective reduced divisor, U X := X \ supp( D ) , d a positive integer and ω X the canonical sheaf on X . Fix a propernormal model X of X which is flat over Spec( O K ) and extend D to an effective horizontaldivisor D on X . Suppose that ω X ( D ) is ample and let h ω X ( D ) be a Weil height function on X with respect to ω X ( D ) .If (cid:15) > , then there exists a constant c (cid:15),d, X , D such that h ω X ( D ) ( P ) ≤ (1 + (cid:15) ) (cid:0) log disc( k ( P )) + log-cond X , D ( P ) (cid:1) + c (cid:15),d, X , D for all P ∈ X ( K ) such that [ k ( P ) : Q ] ≤ d , where k ( P ) is the minimal field of definitionof P (as a point over Q ).Remark . Mochizuki [Moc10, Theorem 2.1] proves that Conjecture 2.3 follows from theuniform abc -conjecture over number fields.3.
Proof of Theorem 1.3
Proof:
We specialize Conjecture 2.3 to the case K = Q , X = E , d = 1 and D = ( O ). Let P ∈ E ( Q ) \ { O } ; then we have h ω E ( D ) ( P ) = h D ( P ) = max (cid:26)
12 log | a P | , log | d P | (cid:27) + O (1) . In order to compute the log-conductor of P we consider the model X over Spec( Z ) deter-mined by the given Weierstraß equation of E and extend D to D ∈
Div( X ) by taking theZariski closure. Then a prime number p lies in the support of P ∗ D if and only if p | d P and hence we get log-cond X , D ( P ) = log rad( d P ) . Therefore Conjecture 2.3 implies Conjecture 1.1. Using Remark 2.4, this finishes the proofof the theorem. (cid:3)
HEIGHT INEQUALITY IMPLIED BY ABC 5 Proof of Proposition 1.6
Proof:
Let 0 < (cid:15) (cid:28) c (cid:15) be the corresponding constant from Conjec-ture 1.1. Suppose that P ∈ E ( Q ) \ { O } satisfies d P = d (cid:48) P · n (cid:89) i =1 p t i i , where d (cid:48) P is squarefree, p , . . . , p n are primes and t , . . . , t n are integers such that t i > i . Then, according to Conjecture 1.1, we must have n (cid:88) i =1 ( t i − − (cid:15) ) log p i ≤ c (cid:15) + (cid:15) log d (cid:48) P . This implies that if d (cid:48) P is bounded from above, then (cid:80) ni =1 ( t i − − (cid:15) ) log p i is bounded fromabove as well. Hence d P is bounded from above, as is the height of P by Remark 1.2. Thisproves the corollary, as there are only finitely many P of bounded height. (cid:3) Proof of Proposition 1.10
Let E/ Q be an elliptic curve in Weierstraß form and let P ∈ E ( Q ) have infinite order.Note that the only place in Silverman’s proof of Theorem 1.9 where the assumption j ( E ) ∈{ , } is invoked is in the proof of [Sil88, Lemma 13]. For Q ∈ E ( Q ) \ { O } we write d Q = d (cid:48) Q · v Q , where d (cid:48) Q is as in the proof of Proposition 1.6).In order to deduce the statement of [Sil88, Lemma 13], it suffices to show that for all (cid:15) > c (cid:48) (cid:15) such thatlog v nP ≤ (cid:15) log( d nP ) + c (cid:48) (cid:15) for all n ≥
1. If we assume Conjecture 1.1, then we can in fact prove a stronger result:
Lemma 5.1.
Assume Conjecture 1.1. Then for all (cid:15) > there exists a constant c (cid:48) (cid:15) suchthat log( v Q ) ≤ (cid:15) log rad( d Q ) + c (cid:48) (cid:15) . for all Q ∈ E ( Q ) \ { O } .Proof: Let (cid:15) >
0, let (cid:15) (cid:48) = 2 (cid:15) and let c (cid:15) (cid:48) be the constant from Conjecture 1.1. Since d (cid:48) Q = rad( d (cid:48) Q ), Conjecture 1.1 predicts(2) log (cid:18) v Q rad( v Q ) (cid:19) ≤ (cid:15) (cid:48) log rad( d Q ) + c (cid:15) (cid:48) ULF K ¨UHN, J. STEFFEN M ¨ULLER for any Q ∈ E ( Q ) \ { O } . But since, by construction, v Q is not exactly divisible by anyprime number, we also get log rad( v Q ) ≤ log (cid:18) v Q rad( v Q ) (cid:19) . The latter is at most (cid:15) (cid:48) log rad( d Q ) + c (cid:15) (cid:48) by (2). Rewriting (2), we concludelog v Q ≤ (cid:15) (cid:48) log rad( d Q ) + 2 c (cid:15) (cid:48) = (cid:15) log rad( d Q ) + 2 c (cid:15) (cid:48) . (cid:3) References [EEW01] Manfred Einsiedler, Graham Everest, and Thomas Ward. Primes in elliptic divisibility sequences.
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