THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED

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1 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED LEVENT ALPOGE ABSTRACT We prove that, when elliptic curves E/Q are ordered by height, the average number of integral points # EZ is bounded, and in fact is less than 66 and at most 8 9 on the minimalist conjecture By EZ we mean the integral points on the corresponding quasiminimal Weierstrass model E A,B : y = x 3 + Ax + B with which one computes the naïve height The methods combine ideas from work of Silverman, Helfgott, and Helfgott- Venkatesh with work of Bhargava-Shankar and a careful analysis of local heights for most elliptic curves The same methods work to bound integral points on average over the families y = x 3 + B, y = x 3 + Ax, and y = x 3 D x CONTENTS Introduction Acknowledgements 4 3 Notation, previous results, and outline of the argument 5 3 Notation 5 3 Previous results 6 33 Detailed sketch of proof of boundedness 7 4 Proof of Theorem 9 4 Restricting to a subfamily and handling small points 9 4 Local heights and a gap principle 3 43 Decomposing the set of integral points into classes: I IV 5 44 I is small: multiples of rational points are rarely integral 7 45 II is small: integral points repel in the Mordell-Weil lattice 46 III is small and IV is empty: an explicit bivariate Roth s Lemma 47 Conclusion of proof 30 5 Proof of Theorem and its corollaries 3 5 y = x 3 + B 3 5 y = x 3 + Ax y = x 3 D x 39 Appendix A Optimizing the bound for Theorem 4 References Mathematics Subject Classification G05 primary, D45, 4G05 secondary Key words and phrases integral points, elliptic curves, Siegel s theorem, Mumford gap principle, spherical codes

2 LEVENT ALPOGE INTRODUCTION The question of counting the number of integral solutions to an equation of shape y = x 3 + Ax + B goes back at least to Fermat, who, on considering this question for specific A and B eg one of his challenge problems to the English was to find all integral solutions to y = x 3, developed his method of descent Fermat also applied this method to show certain such equations had no nontrivial rational solutions famously, y = x 3 x, showing that is not the area of a right triangle with rational sides, leading to the question of counting the number of rational solutions to such equations as well This last question has seen great progress Certainly the number of solutions is either infinite or finite, and density considerations [5] imply that 0% of curves with finitely many rational points have any at all Recent work of Bhargava-Shankar [5] and Bhargava- Skinner-Zhang [9] implies that, in fact, both possibilities infinitely many and none at all occur with positive probability This agrees with the expectation derived from the Birch and Swinnerton-Dyer conjecture of each possibility occurring with probability one half the minimalist conjecture of Goldfeld and Katz-Sarnak Progress has also been made for equations of shape y = fx with f Z[x] of fixed odd degree g + > 3 Here, by Faltings s theorem, one cannot have infinitely many solutions, and indeed one expects none with probability In fact Poonen-Stoll [3], building on work of Bhargava-Gross [3], were able to prove that such a curve has no rational solutions with probability at least g + 0 g, which is quite close to for g very large But the analogous question for integral points on elliptic curves does not yield to these methods By a theorem of Siegel there are only finitely many solutions to y = x 3 + Ax + B if A and B are such that the discriminant of the cubic, 4A 3 7B, is nonzero, so that the equation defines an elliptic curve Therefore we are in a situation like that of Poonen- Stoll/Bhargava-Gross, and similarly we expect to have no integral solutions with probability But despite the expected paucity of curves with integral points, until now it was not known whether the average number of integral points on elliptic curves is bounded In this paper we show that it is indeed bounded in fact, by 66 Let us now be more precise An elliptic curve E/Q has a unique Weierstrass model of the form E A,B : y = x 3 + Ax + B, where A and B are such that p 4 A = p 6 B and 4A 3 7B 0 Given a Weierstrass model, we define the set of integral points on the curve as E A,B Z := {x, y Z y = x 3 + Ax + B}, and write # E A,B Z for its cardinality To produce probabilistic statements, we need a notion of density We write HE A,B := max4 A 3, 7B 6 for the naïve height of E A,B Note that our normalization is slightly different from that of Bhargava-Shankar Given a family F of elliptic curves and a function f on this family, we define Avg fe := E F T E F,HE T E F,HE T fe Indeed, this expectation dates back at least to 986: see page 69 of the first edition of Silverman s Arithmetic of Elliptic Curves [35]

3 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 3 Thus for instance Bhargava-Shankar [5] have shown that lim sup T Avg 5 ranke 6 E F T for F the family of all elliptic curves Let now F universal be the family of all elliptic curves, F A=0 be the family of Mordell curves y = x 3 + B B sixth-power free, F B=0 be the family of curves y = x 3 + Ax A fourth-power free, and F congruent be the family of congruent number curves y = x 3 D x D squarefree With this notation in hand, we may state our main result Theorem Let k 0 Let F = F universal, F A=0, F B=0, or F congruent Then: lim sup T Avg # EZ k O k lim sup Avg 3 k ranke, E F T T E F T where the implied constant is effective and absolute Work of Bhargava-Shankar [5] implies that, for F = F universal, lim sup T Avg E F T universal 5 ranke 6, whence the right-hand side of the theorem is when k = and indeed when k log 5 log 3 = 4649 Hence eg the proportion of curves with at least n integral points is on 4649 For this family we optimize our bound to get: Theorem When all elliptic curves E/Q are ordered by height, the average number of integral points # EZ is less than Moreover, if the minimalist conjecture is true, may be replaced by 8 9 That is, lim sup T Avg E F T universal # EZ < , and this upper bound may be replaced by 8 9 if the minimalist conjecture holds3 In Section 53 we describe how to extend work of Heath-Brown in [7] to prove that, for F = F congruent, From this it follows that: lim sup T Avg E F T congruent k ranke O log k Corollary 3 When the congruent number curves E : y = x 3 D x D Z + squarefree are ordered by height, the k-th moment of the number of integral points # EZ k is bounded above by O k, where the implied constant is effective and absolute In particular, the proportion of curves with at least n integral points decays like n Ωlog n Here by the minimalist conjecture we mean not only that the ranks of elliptic curves in F T universal are distributed 50/50 between 0 and in the limit T, but also the same statement for the subfamily of A, B, mod 3 Otherwise 8 should be replaced by another constant smaller than 9 3 This 8 results from being unable to rule out the possibility of almost every rank one curve having an integral 9 generator in the subfamily A, B, mod 3

4 4 LEVENT ALPOGE In Section 5 we describe how to extend work of Kane [4] and Kane-Thorne [5] to prove that, for F = F B=0, there is a very large we will quantify this in the proof subfamily F B=0 F B=0 for which Avg k ranke O log k From this it will follow that: E F T B=0 Corollary 4 When the curves E : y = x 3 + Ax A Z + fourth-power free are ordered by height, the k-th moment of the number of integral points # EZ k is bounded above by O k, where the implied constant is effective and absolute In particular, the proportion of curves with at least n integral points decays like n Ωlog n The subfamily F B=0 will essentially be the subfamily determined by the conditions that A be almost squarefree, have a number of prime factors bounded above by a large constant times log log A the expected number, and not be a multiple of a modulus supporting a character with a problematic Siegel zero Finally, in the case of F = F A=0, work of Ruth [3] bounds the average of # Sel E, but a bound on the average of 3 ranke is not yet known 4 Having stated our main results, let us now detail the organization of the paper In Section 3 we set notation, state previous results towards these theorems, and give a detailed argument leaving inessential details to references to Section 4 along the way towards Theorem, proving boundedness by O rather than an explicit constant We do this because the length of the argument in Section 4 potentially obscures the main ideas, which are already present in the proof of boundedness In Section 4 we prove Theorem, leaving the discussion of the optimization of our bounds to Appendix A In Section 5 we then prove Theorem for the remaining three families by following the general method used to prove Theorem We also prove Corollaries 3 and 4 by adapting the methods of Heath-Brown and Kane-Thorne to control sizes of Selmer groups in these families Finally, in Appendix A we provide details of the optimization for Theorem ACKNOWLEDGEMENTS Theorem, without the explicit constant, was the subject of my senior thesis at Harvard, supervised by Jacob Tsimerman I would like to thank him and Arul Shankar for suggesting the problem and for all their patience I would also like to thank Henry Cohn for his help with bounds on spherical codes and for allowing me to use a program he wrote to optimize linear programming upper bounds for codes on RP n I would further like to thank Manjul Bhargava, Peter Bruin, Noam Elkies, John Cremona, Roger Heath-Brown, 4 However, to show boundedness of the average of # EZ over this family, one may proceed as follows Integral points on y = x 3 + B give solutions to 4a 3 + 7b = 08B via a := 3x, b := y, whence also binary cubics x 3 +axy +by 3 with discriminant 08B Now, by Davenport-Heilbronn, the number of binary cubic forms fx, y with discriminant X, when taken up to GL Z equivalence, is X See eg Theorem 5 of [8] It therefore suffices to show that there are many forms f of shape x 3 + axy + by 3 in each equivalence class If an equivalence class has no such forms, then we are done Otherwise, we need only check that there are many γ GL Z that take x 3 + axy + by 3 to a form of shape x 3 + a xy + b y 3 Note that, given such a p q γ =:, f γ, 0 =, so that fp, r = Moreover, the condition that the x r s y term be zero gives a cubic or linear equation in q depending on whether or not r = 0 upon imposing ps qr = ± Hence the number of such γ is at most six times the number of solutions of fp, r = with p, r Z But, by Thue s theorem in the strengthened form of eg Bennett who gives an upper bound of 0 in [], this is uniformly bounded, completing the argument

5 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 5 Harald Helfgott, Emmanuel Kowalski, Barry Mazur, Joseph Silverman, Katherine Stange, and Yukihiro Uchida for answering questions related to this work Finally, I would like to thank Michael Stoll for pointing out a mistake in a previous version of this paper 3 NOTATION, PREVIOUS RESULTS, AND OUTLINE OF THE ARGUMENT 3 Notation Let us now set notation By f θ g we will mean that there exists some positive constant C θ > 0 depending only on θ such that f C θ g pointwise If θ is omitted ie we write f g, then the implied constant will be absolute By f θ g we will mean f θ g and f θ g By O θ g we will mean a quantity which is θ g, and by Ω θ g we will mean a quantity that is θ g By o we will mean a quantity that approaches 0 in the relevant limit which will always be unambiguous By f = og we will mean f = o g, and by f g we will mean f = + og We will write a, b for the greatest common divisor of two integers a, b Z, ωn for the number of prime factors of n, v p for the p-adic valuation, v for the absolute value at a place v of a number field K normalized so that the product formula holds, and hx for the absolute Weil height of x Q ie, hx := w [K w : Q v ] [K : Q] log + x w, the sum taken over all places w of K with v := w Q and log + a := maxlog a, 0 Similarly Hx := exphx will denote the multiplicative Weil height of x Q Note that, for a b Q in lowest terms, H a b = max a, b Given a rational point P = x, y on E A,B : y = x 3 + Ax + B, hp and HP will denote hx and Hx, respectively h n P ĥp := lim n 4 n will denote the canonical height of P, with Néron local heights ˆλ v such that ˆλ v = ĥ We will similarly write v λ v := log + v By or A,B we will mean 64A 3 + 7B, the discriminant of E A,B We will write N A,B for the conductor of E A,B, defined by N A,B = p p ep, with e p = if p has multiplicative reduction at p, and otherwise e p with equality if p, 3 The definitions of e and e 3 are more complicated, but we will only use that e 8 and e 3 5 By ψ n P we will mean the n-th division polynomial of E A,B, with zeroes at the nonidentity n-torsion points and of homogeneous degree n when x is given degree, y degree 3, A degree, and B degree 3 Note that multiplication by n is then given by np = xp ψ n P ψ n+ P ψ n P ψ n P, ψ n P 4 In general ψ n+ P is a polynomial of degree n + n in x, A, B with leading coefficient in x equal to n +, and ψ n P is y times a polynomial in x, A, B of degree n with leading coefficient in x equal to n By homogeneity, both these polynomials in x have no

6 6 LEVENT ALPOGE term of one degree less in x ie, they are of the form c d x d + c d x d + + c 0 Finally, we will abuse the word average to mean limsup of the average throughout 3 Previous results Now fix A and B for which A,B 0 The first general result bounding integral points on the curve E A,B is Siegel s famous finiteness theorem: Theorem 5 Siegel E A,B Z is finite Next Baker, as an application of his theory of linear forms in logarithms, gave an effective upper bound on the heights of the integral points on E A,B : Theorem 6 Baker, [] Write H := HE A,B Let P E A,B Z Then: xp e 07 H 07 This of course gives a bound on the number of integral points on E A,B As in the case of Roth s theorem in Diophantine approximation, effectively bounding the number of solutions is much easier than bounding their heights Indeed, Siegel s argument was already effective, and Silverman and Hindry-Silverman were the first to use it to give an explicit upper bound They obtained: Theorem 7 Silverman, [33] # E A,B Z O ranke A,B+ω In fact, one can further reduce ω to ω ss, the number of primes of semistable bad reduction Theorem 8 Hindry-Silverman, [] where # E A,B Z O ranke A,B+σ EA,B, σ EA,B := log A,B log N A,B is the Szpiro ratio of E A,B here N A,B is the conductor of E A,B Conjecturally the Szpiro ratio is at most 6+o This is equivalent to the ABC conjecture In any case, the implied constants in both theorems are on the order of 0 0, even if one uses recent improvements to the arguments in Hindry-Silverman namely, Petsche s [30] improved lower bound on the canonical height of a nontorsion rational point on E A,B, one cannot reduce the constants to below this order of magnitude On the other hand it is quite easy to show that most curves have Szpiro ratio at most, say, 00, so one might think that this makes the second bound amenable to averaging But finiteness of the average of 0 0 rankea,b is far out of the reach of current techniques 5 Recent spectacular results of Bhargava-Shankar which will feature centrally in this argument have proven that the average of 5 rankea,b is finite it is at most 6, and this is the extent of current techniques Specifically, Bhargava-Shankar have shown: Theorem 9 Bhargava-Shankar, [6, 7, 4, 5] Let n =, 3, 4, or 5 Then when all elliptic curves E/Q are ordered by height, the average size of the n-selmer group Sel n E is σn, the sum of divisors of n 5 Heath-Brown [9] has proved, assuming the Grand Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture, that the proportion of curves with rank R is R ΩR, whence we may average 0 0 ranke A,B Thus our result follows from combining this theorem of Heath-Brown with the work of Hindry-Silverman for curves of nonnegligible conductor, and the pointwise bound of Helfgott-Venkatesh stated below for those curves of negligible conductor

7 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 7 Note that n ranke # Sel n E via Galois cohomology, whence the average of n ranke is at most σn for n 5 Another result crucial to us is the pointwise bound of Helfgott-Venkatesh, who obtain: Theorem 0 Helfgott-Venkatesh, [] # E A,B Z O ω log 34 ranke A,B From this it follows that see Lemma 4: Corollary Avg # EZ ɛ T ɛ E F T universal To the author s knowledge, except for a potentially small improvement eg exp O log T log log T instead of T ɛ, this is the best result derivable directly from the literature in this direction 6 This sort of result will allow us to restrict our attention to subfamilies of density T Ω, which will be quite useful in what follows 33 Detailed sketch of proof of boundedness Let us now give an argument proving Theorem without an explicit constant To lower the constant to 66 we will have to be much more careful Sketch of proof that lim sup T Avg T E F # EZ < The first thing to note is that universal the size of F T universal is T 5 Indeed, the bound HE A,B T is equivalent to the bounds A T and B T 3 By Corollary, we may restrict to any subfamily of density at least T Ω 7 Fix a δ > 0 We will restrict to the subfamily F F universal with: A T δ, B T 3 δ A, B T δ v p pvp T δ 8 On this subfamily we break the integral points into three classes: where: EZ = EZ small EZ medium EZ large, EZ small := {P EZ hp 5 δ log T }, EZ medium := {P EZ 5 δ log T < hp δ log T }, EZ large := {P EZ δ log T < hp } We will call these the small, medium, and large ranges, respectively By explicit counting, we obtain the bound A T,B T 3 # E A,BZ small T 5 δ 9 Therefore the small range does not contribute to the average 6 There has been extensive work by Heath-Brown [8], Bombieri-Pila [0], and others on bounding the number of rational points of small height, but this does not improve the above bound 7 See Lemma 4 8 To see that this has the desired density, see Lemma 5 9 See the proof of the second part of Lemma 6

8 For θ 0 = π 3 Oδ, this tells us that An, θ 0 33 n 8 LEVENT ALPOGE To bound the points in the medium range, we prove a gap principle analogous to the Mumford gap principle for rational points on higher genus curves which seems to have first appeared in work of Silverman [33] and Helfgott [0] 0 Lemma Helfgott-Mumford gap principle Let P, R EZ medium EZ large Let θ P,R be the angle between them in the Mordell-Weil lattice EQ/tors EQ Z R with respect to the canonical height Then: cos θ P,R max hp hr, hr hp + Oδ Therefore, via P ĥp P, the number of points P EZ medium with canonical height in the range [X, + δx] is ArankE, θ 0, where θ 0 = π 3 Oδ, and An, θ is the maximal number of unit vectors in Rn with pairwise angles at least θ It is a well-known problem in the theory of sphere packing to provide a good upper bound for this quantity For our purposes we will be interested in an upper bound for large n, and one is provided by the work of Kabatiansky-Levenshtein: Theorem 3 Kabatiansky-Levenshtein, [3] [ + sin θ + sin θ An, θ exp n sin θ log sin θ sin θ sin θ log ] sin θ + o sin θ once δ Therefore the number of integral points with canonical height in the interval [X, +δx] is 33 ranke Since we can cover EZ medium with Oδ such intervals, we obtain the bound # EZ medium δ 33 ranke Since, by Bhargava-Shankar, the average of ranke is bounded over this family, the medium range contributes Oδ to the average Finally, to the large range The claim is that there are Oδ log δ 33 ranke many points of EZ large in each coset of EQ/3EQ To see this, let R be a minimal element with respect to height of EZ large in its coset modulo 3 By the same argument as for the medium range, there are Oδ log δ 33 ranke integral points P with hp < δ hr For those points P R mod 3 with hp δ hr, we write P =: 3Q + R with Q EQ Then since P is very close to in the Archimedean topology, Q must be very close to a solution of 3 R = R as well That is, xq must be very close to an x R Q solving x3 R = xr After making this precise, we find that: 9 log xq x R Oδ hq 0 The difficulty in proving this in fact lies in handling the error term, which relies in a careful estimation of the difference between the Weil and canonical height on this curve This is the reason for restricting to the subfamily F : the difference between the two heights is much better controlled in this case See Lemma 9 See 45 and take C, D δ

9 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 9 for some such R Therefore xq x R HQ 9 +Oδ Thus xq is a Roth-type approximation to x R Moreover since hxq = hq δ h R = δ hx R, we see that xq is a large rational approximation, in the sense of Bombieri-Gubler [] As they prove, there are only O such approximations once δ Therefore each coset modulo 3 contributes at most Oδ log δ 33 ranke to # EZ large, whence we obtain the bound # EZ large δ log δ 399 ranke Again by Bhargava-Shankar the average of 4 ranke is bounded, so that the large range contributes Oδ log δ to the average Therefore, in sum, we have found that the average is at most Oδ log δ for any δ sufficiently small Choosing such a δ then gives the result Having given a sketch of an argument proving the weaker theorem that the limsup of the average is bounded, let us now give the full proof of Theorem 4 PROOF OF THEOREM We will follow the structure of the argument given in the previous section reasonably closely, deviating only in the specific details of the application of sphere-packing bounds for numerical reasons, and in being entirely explicit We work only with the average ie, k = there are only a few modifications required for the case of general k, and they are all clear Proof of Theorem As noted, # F T universal T 5 To obtain a good estimate on the difference between the Weil height and the canonical height, we will restrict to a subfamily F F T universal which omits a set of density OT c for some positive c > 0 The following lemma shows that we may do this 4 Restricting to a subfamily and handling small points Lemma 4 Let G F T universal Then, for all ɛ > 0, E G # EZ # F T universal # G # F T universal Proof By Hölder s inequality, it suffices to show that Avg # EZ 000 exp E F T universal By Helfgott-Venkatesh Theorem, we have that # EZ 000 log T 000 E F T universal We apply the crude bound ωn See eg their 63 Ω log T exp O log log T O E F T universal log T log log T O ω E 35 ranke log n log log n and Bhargava-Shankar to conclude

10 0 LEVENT ALPOGE Fix a δ > 0 to be chosen later We will take δ independent of T Let us apply this to first restrict to the subfamily F F T universal defined by the conditions: A T δ B T 3 δ, and B is not a square 3 A, B T δ 4 T 6 δ 5 p pvp T 4δ To see that we may, we prove: Lemma 5 Let G be the complement of F in F T universal Then: # G # F T universal T Ωδ Proof It suffices to impose each condition one by one and check that we throw out a density T Ωδ subset at each step For the first and second conditions this is immediate For the third condition, the number of A T, B T 3 with A, B > T δ is at most T n T 3 n T 5 δ T δ <n T So we may assume the first, second, and third conditions Given these, for the fourth condition, if < T 6 δ, then A = 7 4 B + OT 6 δ 3 = 3 B T 6 δ 3 + O 3 B 4 3 = 3 B 3 + O T δ 3 3 Therefore the number of A, B with A,B < T 6 δ is 3 T 5 δ 3 B T 3 T δ Finally, for the fifth condition given the other four, the argument will be a bit longer Our strategy will be to show that we may take the radical of to be reasonably large, and then we will establish that we may take to not have any nonnegligible square divisors Then we may bound the nonsquarefree part of in terms of square divisors of only, which thus forces it to be small We first show that we may assume the conductor of E A,B is at least T 408 To see this, by Theorem 45 of Helfgott-Venkatesh [], the number of curves of conductor N is N 04 Therefore the number of A, B with conductor at most T 408 is T < T 4999, giving the claim Now note that E A,B has additive reduction at p if and only if p A, B Therefore N A,B p p rad T δ, p,3,p p,3,p A,B where radn := p n p is the radical of n Therefore rad T 405 once δ

11 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED Let us now show that we may assume that if n and n T 99 then n T δ To see this, note that n implies that 64A 3 43B mod n The first claim is that, for fixed A, the number of B T 3 solving this equation modulo n is O ωn A 3, n + T 3, where x α, y β := p x,y pmaxαvpx,βvpy Indeed, at a prime power p e with p > 3, the number of square roots of 4A3 7 it is instead at most 3 3v 3 A n is at most p 3vpA by Hensel s lemma When p = 3 for the same reason, but the implied constant is different Similarly at p = it is 3v A Moreover if 3v p A e, then the number of solutions for B is instead at most const p e, with const and equal to if p > 3 Therefore the number of solutions modulo m is O ωm p A,m p minvpa, 3 vpm Hence the number of B T 3 such that n A,B is O ωn A 3, n + T 3 But then the number of A T, B T 3 for which there exists an n with T δ < n T 99 is at most = # {B T 3 : n A,B } A T B T 3 T δ <n T 99,n A,B T δ <n T 99 A T + T 3, n n T δ <n T 99 O ωn 3 n A T A 3 By examining the residue of the relevant Dirichlet series at s =, one finds that A T A 3, n O ωn p vpn p n 3 T We will again use the bound O ωn ɛ n ɛ to conclude that our sum is at most ɛ T +ɛ p vpn 3 + T 3 p n n T δ <n T 99 T 399+ɛ + T 5 δ 3, as desired Therefore we may assume that the only square divisors n of with n T 99 are smaller than T δ But now p p and p p vp are square divisors of

12 LEVENT ALPOGE Moreover, p p and p p vp divide rad T 95 Therefore these square divisors must both be of size at most T δ! Hence since p pvp divides p vp p T 4δ, we are done p p We will further restrict to a subfamily of curves with no small integral or rational points Specifically, let F F be the subfamily defined by the conditions: E A,B Q tors = 0 If P E A,B Z, then hp > 5 δ log T 3 If Q E A,B Q, then hq > δ log T Let us now prove that we may restrict to this subfamily Lemma 6 Let G be the complement of F in F Then # G # F T Ωδ Proof Theorem in Harron-Snowden [5] allows us to impose the first condition For the second condition, the number of A T, B T 3 such that there is at least one integral point P E A,B Z with hp 5 δ log T is at most # {x, y, A, B Z 4 : x T 5 δ, A T, B T 3, y = x 3 + Ax + B} = # {x, y, A, B : x 0 0 T, A T, B T 3, y = x 3 + Ax + B} + # {y, A, B : A T, B T 3, y = x 3 + Ax + B} 0 0 T x T 5 δ To bound the first sum, note that, given x, y, A, B = y x 3 Ax is determined Moreover y x 3 + T x + T 3 T 3, so that y T 3 Therefore the number of x, y, A, B is at most T T 3 T = T 45 For the second sum, note that in this range y x 3 T x + T 3 T x, whence y x 3 Now, if y, A, B and y, A, B lie in the solution set and without loss of generality y, y > 0, then so that y y = xa A + B B, y y T x + T 3 T x 3 x Therefore the number of y for which there exist A, B making x, y, A, B a solution is at most + T x

13 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 3 Next, given x and y, if x, y, A, B and x, y, A, B are solutions, then A A x = B B, so that A A T 3 x, whence the number of A for which there exists a B making x, y, A, B a solution is at most + T 3 x Putting these together, the second sum is bounded above by # {y, A, B : A T, B T 3, y = x 3 + Ax + B} 0 0 T x T 5 δ + T + T 3 x x 0 0 T x T 5 δ T 5 δ, as desired Finally, for the third condition, note, as above, that the number of A T, B T 3 such that there is at least one rational point Q = x d, y d 3 EA,B Q with hq δ log T is at most # x, y, d, A, B : y = x 3 + Ad 4 x + Bd 6, x T δ, d T 4 δ, A T, B T 3 } Note that if x, y, d, A, B is a solution, then y T 3 d 3 Moreover, x, y, d, A determines B Hence this count is at most: T δ T 3 3 T 4 3δ T 4 δ T = T 5 3δ, whence we are done 4 Local heights and a gap principle The purpose of restricting to this subfamily is to be able to give a very strong estimate on the difference between the Weil and canonical heights on the curves in this family Specifically, Lemma 7 Let E F Let h, ĥ be the Weil and canonical heights on E A,B, respectively Let Q EQ Then ĥq hq = log + 6 xq + 6 log log+ xq + Oδ log T In particular, hq ĥq + Oδ log T and, if xq 6, ĥq hq = Oδ log T Proof Write ĥ h = v ˆλ v λ v, where λ v := log + v, ˆλ v are the Néron local heights, and v runs over the places of Q At a prime p, 3 of good reduction, by eg Theorem 4 3 in [34], the local heights are equal At a prime p of additive reduction so p A, B or at p = or 3, by the same theorem we see that 0 ˆλ p λ p 6 log p 3 Note: our normalization differs from Silverman s by a factor of

14 4 LEVENT ALPOGE Since p A, B implies p, we see that p vp 6 p vp T 4δ, p 6A,B whence the sum of these contributions is 0 p 6A,B p ˆλ p λ p δ log T At a prime p, 3 of multiplicative reduction, by Chapter III Theorem 5 of [6], since v p = whence α = 0 in Lang s notation, we see that ˆλ p λ p = 6 log p Finally, at the infinite place, since je A,B T Oδ, by combining Proposition 54 and 3 of [34] we find that ˆλ Q λ Q = log + 6 xq log + xq + Oδ log T Summing these all up and using the product formula gives the result Given that the Weil and canonical heights are so close, we may now prove a bound on the angle between two integral points by proving a corresponding bound with Weil heights replacing canonical heights Specifically, Lemma 8 Helfgott-Mumford gap principle, cf [0] Let E F Let P R EZ with hp hr recall that automatically hp, hr > 5 δ log T Then: ĥp + R hp + hr + O Proof Write P =: X, Y and R =: x, y with X x Note that since X, x T 5 δ, we have that Y X 3 and y x 3 Now xp + R = Y y X x X x = X x + Xx Y y + AX + x + B X x The numerator has absolute value at most X x by hypothesis The denominator has absolute value at most X Therefore, since cancelling common factors will only make the numerator and denominator smaller, we see that hp + R hp + hr + O If xp + R 6, then this completes the proof, by Lemma 7 Otherwise, write xp + Q = W Z in lowest terms Then ĥp + R = hp + R + 6 log log+ xp + R + Oδ log T = maxlog W, log Z + log T maxlog W log Z, 0 + Oδ log T = log T + log Z + Oδ log T Since as we saw Z X, we find that ĥp + R log T + hr + Oδ log T Observing that hp 5 δ log T finishes the result This results in a lower bound on the angle of integral points close in absolute value:

15 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 5 Lemma 9 Let E F Let P R EZ Let θ P,R be the angle between P and R in the Euclidean space EQ Z R Then: cos θ P,R max hp hr, hr + Oδ hp Proof By definition, ĥp + R ĥp ĥr cos θ P,R = ĥp ĥr By Lemma 7 and the fact that hp, hr > 5 δ log T, we find that ĥp + R hp hr cos θ P,R = + Oδ hp hr Applying Lemma 8 then concludes the argument 43 Decomposing the set of integral points into classes: I IV Fix now a parameter D > We will take D to be in the end Let so that D := D + D + 4, D D = D 4 Fix E F Let r := ranke Note that we may assume r > 0 since EQ tors = 0 and so # EZ = 0 if r = 0 So choose P,, P r EQ such that P 0 has minimal canonical height recall that E has no rational torsion and P i has minimal canonical height among points not inside span Z P,, P i + 3EQ Note that since it follows that ĥp i ± P j ĥp maxi,j P i, P j It follows that, for any ɛ i = ±, k ĥ ɛ i P i i= ĥp mini,j k k i + ĥp i 4 i= Next note that P,, P r is an F 3 -basis for EQ/3EQ Given Q EQ, write iq := min{i Q span Z P,, P i + 3EQ} ie, iq is the least i for which Q is congruent to an element of the Z-span of P,, P i modulo 3 Note that i = 0 implies Q is a multiple of 3 Write H := max 5 Oδ log T, ĥp, where, say, the implied constant is larger than one plus twice the implied constants in Lemma 7, and H i := max ĥpi, D H i 4 4 For instance, the condition hp > D H i implies hp > D i j+ĥp j for every j i, and it also implies hp > D i 5 Oδ log T

16 6 LEVENT ALPOGE Then if r > write EZ = 3EQ EZ r {P EZ, H i ĥp D H i } i= r {P EZ, ip = i, ĥp > D H i } i= =: I r i= II i D r i= III i D, Note that our notation I, II, III is slightly different from the outline, since we have already gotten rid of small points In words, what we have done is broken EZ into multiples of rational points which will be easy to handle 5, points of medium height in their respective cosets, and then points of large height in their respective cosets The curves with points of small height have already been thrown out Note that this decomposition is complete because if P EZ lies outside the union, then ip =: i > 0 and ĥp D H i, so ĥp < H i Therefore, since ip = i, we must have H i = D H i by minimality of P i Thus ˆP < H i Proceeding inductively, we eventually find that ˆP < 5 Oδ log T, contradicting hp > 5 δ log T combined with Lemma 7 Let us further write III i D = =: a {,0,} i :a i>0 a {,0,} i :a i>0 {P III i D, P III i, a D i a j P j mod 3} j= In words, we are breaking the points of large height into their congruence classes modulo 3 Since we will be counting points and their negatives together below, we have forced a i > 0 rather than a i 0 Given a {, 0, } i with a i 0, we will write R a := i j= a jp j Let us further break III i, a D into a set we will show is empty and a set to which we can apply Roth-like 5 In the rank case all points are multiples of a rational point, so in some sense EZ =: I would be consistent notation here, but we have not bothered because it would be unnecessarily confusing

17 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 7 techniques Specifically, write { III i, a D = P III i, a D :!Q EQ : P = 3Q + R a; R EQ with 3 R = R a, we have xq x R > { P III i, a D :!Q EQ : P = 3Q + R a; R EQ:3 R= R a =: IV i, a D III i, a, R D 3 R= R a xq x R } min xq x R 3 R = R a, R R } min xq x R 3 R = R a, R R In words, we have written P III i, a D as P = 3Q + R a, and split the points up based on the element of the nine-element set 3R that Q is close to IVi, a D is the set of points with Q not close to any point in 3R, which will be empty once D is sufficiently large This is because xp is large, so P is close to the origin, so that Q is close to such a solution 44 I is small: multiples of rational points are rarely integral Let us now begin bounding the sizes of each of the sets I,, IV The sets I and II D require almost no work The following lemma expresses the fact that rational points rarely have integral multiples: in the rank one case, at worst one has the generator and its negative as integral points via the theory of lower bounds on linear forms in elliptic logarithms, and in the higher rank case no triple of a rational point is integral on a curve in our family Lemma 0 Let E F Then: # EZ when r =, and I = otherwise Before we prove this lemma, we will prove a preparatory lemma on the coefficients of the division polynomials of E Recall that the denominator of the multiplication-by-n map, ψ n P, is homogeneous in x, A, B of degree n with the usual grading Write ψ n P =: c f x fx A f A B f B, f N 3 :f x+f A +3f B =n with c f Z The claim is that these c f do not grow too fast as f x decreases More precisely, Lemma c f n O O log n n f x It is a theorem of Lang that c f O n in general which is only weaker for f x on, but this is not enough for our purposes Proof of Lemma Write We will show that n mod ψ n P =: y f N 3 :f x+f A +3f B = n 4 C f n O log n O n 4 C f x fx A f A B f B f x,

18 8 LEVENT ALPOGE from which the bound for c f follows That is, we will show that there are absolute constants K, K, and K 3 such that, for all f, n f x log n C f K n K 4 K 3 43 First choose K > so large that for n 0 0 the bound C f K holds Take K = Take K 3 so large that 0 0 Kn 3 4K+0 log 9 log n < K3 for all n > 0 0 The bound will then follow by induction Specifically, recall the recursive formulas for the division polynomials: for odd indices, ψ m+ = ψ m+ ψm 3 ψ m ψm+, 3 and, for even indices, ψ m = ψm ψm+ψm ψ m ψ y m+ So suppose we have proved 43 for all n < n From the recursions and induction it follows immediately that the leading coefficient of ψ n is n, which satisfies the claimed bound since K >, K = Hence we may assume n f x < 4 For n of the form n =: 4m +, using the recursive formula, we find that 3 ψ 4m+ = ψ m ψm+ 3 ψm+ ψm + x 3 + Ax + B y y Expanding and applying the induction hypothesis, we find that the coefficient of x fx A f A B f B in ψ 4m+ is, in absolute value, at most a sum of at most n 6 terms corresponding to decompositions f = e + + e 4, each at most But log m + log n log 9, so that 00K 4 n 4K K log m+ 8m +4m f x 3 log m + log n log 9 log n Inserting this into the inequality and using f x < log C f K n K n K 3 n 4 f x n 4, we find that [ 00Kn 3 3K+6 K log 9 log n 3 and the factor in brackets is smaller than by hypothesis For n not congruent to mod 4 the argument is exactly the same, using the other recursive relation when n is even This finishes our preparations Let us now prove Lemma 0 Proof of Lemma 0 For the first bound, note that if np is integral for some n, then P must be integral To see this, write P = x d, y d in lowest terms and suppose d > Then 3 since xnp = xψ np ψ n+ P ψ n P ψ n P ],

19 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 9 is the quotient of two homogeneous polynomials of degree n and n, respectively again, x, y, A, B are given degrees, 3,, and 3, respectively with the numerator having leading term x n, we see that, on clearing denominators, xnp = xn + dz, dz which is not an integer since x, d = by hypothesis So if P = P is not integral we are done for the rank case If it is integral, then the claim is that none of its multiples np, n >, are also integral Indeed, since P is integral, we find that hp > 5 δ log T since E F Let us first show that np is not integral for < n O log T Of course it suffices to show that the denominator d n in lowest terms of xnp is larger than for these n But Lemma 9 of [36] or, equivalently, Proposition 43 in [8] allows us to do this Indeed, we find that log d n log ψ n P n 4 log log ψ np 3n log T O By Lemma, the coefficient of x k is at most n O O log n T n k Hence since xp T 5 δ is much larger than T, we find that ψ n P is dominated by its top term Specifically, for n O log T, so that ψ n P xp n n O O log n T Ω xp n, log d n n hp 3n log T O 9 Oδ log T O, which is positive Thus d n > and so xnp is not integral for n O log T This in fact completes the first estimate since it shows that no integral point is thrice a rational point in general as well for this application we could simply use Lang s coefficient bound, of course Thus it remains to show that np is not integral for n O log T This will follow from David s bounds on linear forms in elliptic logarithms in fact we will show that np is not integral for n log T log log T log log log T To do this we apply the Corollary of equation 6 in [4] Let us translate their notation into ours Recall that, for us, r =, so that their C Moreover, since our curves have no torsion, their g = Their N is our n Their µ = log max A, B 3 log T + O They define the real period ω to be ω := ρ dx x3 + Ax + B, where ρ R is the largest real solution of ρ 3 + Aρ + B = 0 Let us show that T ω T +Oδ Let ρ, ρ C be the other two roots Since A and B satisfy T Oδ A, B 3 T,

20 0 LEVENT ALPOGE it follows by the reverse triangle inequality that the same bounds hold for ρ, ρ, and ρ Now the integral over [0 0 T, is T since x 3 +Ax+B x 3 there Hence, since the integrand is positive, the lower bound on ω follows For the upper bound, we split into cases If ρ, ρ are not real, then Reρ = Reρ = ρ and Imρ = Imρ = ρ ρ In this case, on ρ, 0 0 T x 3 + Ax + B x ρ ρ ρ If ρ, ρ are real, then on ρ, 0 0 T x 3 + Ax + B = x ρx ρ x ρ x ρρ ρ ρ ρ Since the discriminant of x 3 +Ax+B is T 6 Oδ, applying Mahler s bound on the bottom of page 6 in [9], in both cases it follows that x 3 + Ax + B x ρ T Oδ on the interval Hence the integral over the interval is 0 0 T ρ dx 0 0 T x3 + Ax + B T +Oδ ρ dx x ρ T +Oδ, completing the argument It follows that their c T Oδ Note also that their h log T The bound ρ, ρ, ρ T implies that their ξ 0 T Finally, we turn to the expression 3π u ω Imτ defining their log V Since we may take τ in the classical fundamental domain for SL Z acting on the upper half plane, we have Imτ Now, u, the elliptic logarithm of our P = P =: ξ, η, satisfies u = dx ω x3 + Ax + B ξ T +Oδ Thus But xp T 5 δ implies that this is Therefore their log V satisfies ξ 3π u ω Imτ u ω xp T +Oδ T 3+Oδ log V ĥp Finally, their λ = ĥp in the rank one case This completes the translation of their notation Their Corollary now reads since certainly any integral point P satisfies the hypothesis of their Proposition, which is xp T xp is positive since xp 3 + AxP + B is: Corollary Cf equation 6 of [4] For E F of rank one and generator P = P, if np is integral and n, then n log T log nlog log n It follows that, if np is integral, then n log T log log T log log log T Since we have already shown that if n > then n O log T, this completes the argument Note that we have now completely handled the cases of ranke = 0 or Hence from now on we may assume ranke

21 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 45 II is small: integral points repel in the Mordell-Weil lattice Let < J < be a parameter which we will choose at the end J will depend on r for r Write J =: cos θ We encode the fact that integral points repel in the Mordell-Weil lattice with the following lemma Lemma 3 # II i D log D log J max # S S RP r : v w S, v,w cos θ+oδ We will bound the maximum occurring in this bound with a bound on codes in RP n via linear programming techniques for n and a simpleminded volume estimate for n Proof It suffices to prove that the number of points with height in an interval [m, M] is log M m log J max # S S RP r : v w S, v,w cos θ+oδ To see this, note that [m, M] log M m log J [mj i, mj i+ ], i= so that it suffices to prove this bound for an interval of the form [m, J m] But now if hr hp J hr, then by Lemma 8, cos θ P,R J + Oδ = cos θ + Oδ Therefore the map {P EZ : hp [m, J m]}/± RP r via ±P {±P ĥp } the projection to RP r of the nonzero point P R r = EQ Z R is injective since cos θ P,R < if P R once δ J Moreover the image satisfies the condition that for every v w in the image, v, w = cos θ v,w cos θ + Oδ, as desired This completes the proof of the second bound i, a, R 46 III is small and IV is empty: an explicit bivariate Roth s Lemma For III D and we will follow Siegel s proof of his finiteness theorem Write C := 5 δ D, so that IV i, a D for every P III i, a D Lemma 4 Let P III i, a D Then: we have hp > C log T Note also that hr a, ĥr a D + Oδ hp

22 LEVENT ALPOGE Proof Observe that i i ĥ a j P j i j + ĥp i j= j= i i j + ĥp D i j+ j= i = j j D + OδhP, j= where the first step follows from 4 and the second follows from the definition of III i D, plus Lemma 7 But in general k lx l x x, l= so that we find that hr a D + Oδ hp by 4 and Lemma 7, as desired Having established this estimate, let us now prove: Lemma 5 Let P III i, a D Then: log 3 R= R a xq x R hp 9 log T max hp, 9 D Oδ, and Proof Observe that + D + Oδ hp 9hQ D Oδ = log xp hp = log x3q + R a h3q + R a Let us examine the numerator and denominator of this expression 44

23 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 3 First, the denominator Note that hp = h3q + R a ĥ3q + R a Oδ log T = ĥ3q + R a Oδ 3 ĥq ĥr a Oδ 3 hp hq Oδ, D where we have used the triangle inequality for ĥ and Lemma 4 Therefore hp 3 hq + D + Oδ The same argument works to prove that hq hp D Oδ 3 as well This proves the second statement of the Lemma Now we move to the numerator in 44 Observe that Note also that 9 x3q = xqψ 3Q ψ Qψ 4 Q ψ 3 Q 3 R= R a xq x R = ψ 3 Q xq ψ Qψ 4 Q ψ 3 Q xr a = ψ 3 Q x3q xr a, since both are polynomials in xq of degree 9 with leading coefficient 9 and roots exactly at xq = x R for some R with 3 R = R a But then, since in general xw + Z = yw yz xw + xzxw xz xw xz, we have that x3q + R a 9 xq x R 3 R= R a = ψ 3 Q 4 y3q yr a x3q + xr a x3q xr a Now from the equation y = x 3 + Ax + B, we find that y x + T 3 xr a exphr a hp exp + Oδ D = xp D +Oδ in general Also,

24 4 LEVENT ALPOGE Therefore, as we saw in the proof of Lemma 8, by writing out the numerator and denominator, x3q = xp R a xr a since xr a is much smaller than xp in absolute value But if xq 0 0 T, then x3q xq since it is a quotient of two polynomials dominated by their leading terms Therefore we find that in general xq, x3q xr a + T and so yq, y3q xr a + T 3 Therefore x3q + R a 9 xq x R 3 R= R a Written another way, = ψ3 Q 4 y3q yr a x3q + xr a x3q xr a xr a + T 9 max T 9, xp 9 +Oδ D log x3q + R a log 3 R= R a xq x R Therefore, returning to 44, we find that: + max 9 log T, 9hP D + Oδ + O xq log 3 R= R a x R 9 log T + max hp hp, 9 D + Oδ This completes the proof Let us now show that, once D is suitably chosen, IV i, a D is empty Recall that C = 5 δ D Lemma 6 Suppose Then IV i, a D = 576 C D + max C, 9 D < Proof Suppose P IV i, a D Then, by definition, xq x R min x R x R 9 R R,3 R=3 R = R a 3 R= R a Now, as we saw in the previous lemma, as polynomials in xq, 9 xq x R = ψ 3 Q xq ψ Qψ 4 Q ψ 3 Q xr a 3 R= R a This is homogeneous of degree 9 in xq, xr, A, B when the variables are given degrees,,, 3, respectively Therefore the coefficients of xq in the first two terms namely, ψ 3 Q xq ψ Qψ 4 Q are bounded in absolute value by T 8 Thus the polynomial has naïve height, in the sense of Bugeaud and Mignotte [], at most 8 log T +hr a To see this, clear the denominator of xr a so that the polynomial is an integral polynomial and

25 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 5 then the estimate is clear Therefore by the estimate on page 6 of Mahler [9], we find that min R R,3 R=3 R = R a x R x R T 64 HR a 8, and hence that 3 R= R a log xq x R 576 log T + 7hR a + O Therefore by Lemma 5 it follows that 576 log T + 7hR a + O 9 log T + max hp hp, 9 D + Oδ Applying Lemma 4, we see that 576 log T hp log T D + max hp, 9 D + Oδ The desired contradiction now follows once δ by using the inequality hp > C log T i, a, R Finally we will bound the size of III D for D suitably chosen The idea here is that, roughly, we have obtained the inequality log 3 R= R a xq x R hq 449, and now Q is very close to some R Therefore, roughly, this tells us that xq x R HQ 448, and so xq is a Roth-type rational approximation to x R But Roth s theorem requires many such rational approximations to reach a contradiction, and hence provides a poor bound on their number for our purposes In fact xq is also a Siegel-type rational approximation, in the sense that x R is of degree 9 over Q, and deg x R = 8 = 44 < 448 Moreover xq has very large height compared to x R, so if we are very careful with how we prove Siegel s theorem on Diophantine approximation namely, via Roth s lemma for bivariate polynomials, we will be able to conclude So let c < be another parameter which we will choose such that c Given c and D, we may bound the size of III i, a, R D as follows Lemma 7 Suppose s Z + is such that c 3 κ s κ + κ s D i, a, R Then # III D s 9 + κ + c > i, a, R Proof Let P III D Note that, for all R R such that 3 R = R a, xq x R > x R x R by the triangle inequality Therefore xq x R 3 R = R a, R R 3 R = R a, R R x R x R

26 6 LEVENT ALPOGE By a bound of Mahler the last line on page 6 of [9], x R x R T 56 HR a 7 3 R = R a, R R Hence, by Lemma 5, log xq x R hp 9 max C, 9 D 56 C 7 D + Oδ Next, applying the second part of Lemma 5, we therefore find that log xq x R 9 7 hq max C, 7 D 504 C 63 D + Oδ + D + Oδ 45 Write Then κ := 9 7 max C, 7 D 504 C 63 + D + Oδ D xq x R HQ κ That is, xq Q is a rational approximation to x R Q with exponent κ Moreover, hq 9 hp D Oδ D Oδ ĥr a 9 D Oδ ĥ R D Oδ h R 46 so that xq is a large rational approximation of x R as well To bound the number of these, we will run through the usual argument for Siegel s theorem on Diophantine approximation via Roth s lemma, except we will be explicit and careful in our bounds Write α := x R whence deg α 9 and α T + xr a and let us suppose there were s + such approximations ie λ i λ j satisfying: λ i Q, λ i T + xr a, 3 λ i α Hλ i κ, 4 hλ i D Oδ hα, 5 hλ i C 9 D Oδ log T Let us also suppose, without loss of generality, that Hλ s+ Hλ s Hλ Note that, by rationality of the β i we have that Hence ie, Hλ i Hλ i λ i λ i Hλ i κ Hλ i Hλ i κ hλ i κ hλ i + O

27 THE AVERAGE NUMBER OF INTEGRAL POINTS ON ELLIPTIC CURVES IS BOUNDED 7 Therefore hλ s+ κ s hλ + Os Hence λ s+ and λ are very far apart in height, and it is these rational approximations that we will use We will write β := λ s+ and β := λ Now let d > d Z + be such that d hβ d hβ d 6 We will take d, d at the end of the argument, so any error terms suppressed by factors of d or d will be negligible Let t := c 3 and let deg α K := deg α d d t + t d + d = d d c cd cd d d c + Oδ once d, d c,δ An application of Siegel s lemma gives us the following: Claim 8 There is a nonzero p Z[x, y] such that for all nonnegative integers k, l with and such that k x l ypα, α = 0 k d + l d t, Hp OH R d +d c Oδ Proof of Claim We apply Siegel s lemma in the form of Bombieri-Gubler Lemma 9 [] Indeed, we are imposing the conditions 0 i d,0 j d a ij α i+j k l d d k k = 0 on the coefficients a ij Z of P But recall that we have the relation with den α deg α = fα fz := den z deg α den gz, and gz Q[z] the minimal polynomial of α here den is the least positive integer such that den g Z[z] Multiplying our relations through by den d+d deg α+ and repeatedly applying this relation reduces us to forcing deg α times as many conditions but now with integral coefficients for each condition with coefficients in Qα Importantly, since the coefficients of fz Z[z] are all of absolute value at most OHα and we apply the relation d + d times, the resulting linear conditions on a ij have coefficients bounded in absolute value by O d+d H R d+d, where we get an O d+d H R d+d from the α i+j k l terms, and an O d+d from the binomial coefficients and the sum 6 Of course infinitely many such d and d exist if hβ hβ is irrational, but since we do not require d, d =, such d, d exist in case the ratio of heights is rational as well!