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Masser and Oesterl´e[9] (with some exponent), such a proof would be of interest. It is the purpose of this note to explain how the pigeonhole argument in [11] may be combined with the Fourier averaging methods in [5] to prove Lang’s height conjecture using a weaker from of Szpiro’s con- jecture. Roughly speaking, the “prime-depleted” version of Szpiro’s conjecture that we require allows one to discard a bounded number of primes from D(E/K) and F(E/K) before comparing them. We briefly summarize the contents of this paper. In Section 1 we de- scribe the prime-depleted Szpiro conjecture and prove that it implies Lang’s height conjecture. Section 2 contains various elementary prop- erties of the prime-depleted Szpiro ratio. Finally, in Section 3 we state a prime-depleted ABC-conjecture and show that it is a consquence of the prime-depleted Szpiro conjecture.

1. The Prime-Depleted Szpiro Conjecture We begin with some definitions. Definition. Let D be an integral ideal and factor D = pei as a product of prime powers. We write ν(D) for the number of factors in the product, i.e., ν(D) is the number of distinct prime ideals dividingQ D. The Szpiro ratio of D is the quantity

log N QD e log N Qp σ(D)= K/ = i K/ i . log N p log N p K/Q i i P K/Q i (If D = (1), we set σ(D) = 1.)Q More generally,P for any integer J ≥ 0, the J-depleted Szpiro ratio of D is defined as follows:

ei σJ (D) = min σ pi . I⊂{1,2,...,ν(D)} #I≥ν(D)−J Yi∈I  Thus σJ (D) is the smallest value that can be obtained by removing up to J of the prime powers divided D before computing the Szpiro ratio. We note that if ν(D) ≤ J, then σJ (D) = 1 by definition. Example 3. log 1728 log 27 σ (1728) = ≈ 4.16, σ (1728) = =3, σ (1728) = 1. 0 log 6 1 log 3 2 Conjecture 4 (Prime-Depleted Szpiro Conjecture). Let K/Q be a number field. There exist an integer J ≥ 0 and a constant C3, de- pending only on K, such that for all elliptic curves E/K,

σJ D(E/K) ≤ C3.
Lang’s Height Conjecture and Szpiro’s Conjecture 3

It is clear from the definition that σ0(D) = σ(D). An elementary argument (Proposition 9) shows that the value of σJ decreases as J increases,

σ0(D) ≥ σ1(D) ≥ σ2(D) ≥··· . Hence the prime-depleted Szpiro conjecture is weaker than the classical version, which says that σ0 D(E/K) is bounded independent of E. Before stating our main result, we need one further definition.
Definition. Let E/K be an elliptic curve defined over a number field. The height of E/K is the quantity

h(E/K) = max h j(E) , log NK/QD(E/K) . For a given field K, there are only finitely
many elliptic curves E/K of bounded height, although there may be infinitely many elliptic curves of bounded height defined over fields of bounded degree [13]. We now state our main result, which implies that the J-depleted Szpiro conjecture implies Lang’s height conjecture. Theorem 5. Let K/Q be a number field, let J ≥ 1 be an integer, let E/K be an elliptic curve, and let P ∈ E(K) be a nontorsion point. There are constants C1 > 0 and C2, depending only on [K : Q], J, and the J-depleted Szpiro ratio σJ D(E/K) , such that ˆ h(P ) ≥C1h(E/K
) − C2. In particular, the depleted Szpiro conjecture implies Lang’s height con- jecture.

Remark 6. As in [10], it is not hard to give explicit expressions for C1 and C2 in terms of [K : Q], J, and σJ D(E/K) , but we will not do so here. In terms of the dependence on the Szpiro ratio, probably the best that comes out of a careful working of the proof
is something like cJ C1 ≫≪ σJ D(E/K) for some absolute constant c. But until the (depleted)
Szpiro conjecture is proven or a specific application arises, such explict expressions seem of limited utility. Proof. We refer the reader to [14, Chapter 6] for basic material on canonical local heights on elliptic curves. Replacing P with 12P , we may assume without loss of generality that the local height satisfies 1 λˆ(P ; v) ≥ log N D(E/K) 12 K/Q 4 JOSEPH H. SILVERMAN for all nonarchimedean places v at which E does not have split multi- plicative reduction. We factor the discriminant D(E/K) into a product

D(E/K)= D1D2 with ν(D2) ≤ J and σJ D(E/K) = σ(D1). We also choose an integer M ≥ 1 whose value will be specified
later, and for convenience we let d = [K : Q]. Using a pigeon-hole principle argument as described in [11], we can find an integer k with 1 ≤ k ≤ (6M)J+d such that for all 1 ≤ m ≤ M we have ˆ ∞ λ(mkP ; v) ≥ c1 log max |j(E)|v, 1 − c2 for all v ∈ MK , ˆ N D −1 0 p D λ(mkP ; v) ≥ c3 log K/Q (E/K) v for all v ∈ MK with v | 2. (Here and in what follows, c ,c ,... are absolute positive constants.) 1 2 Roughly speaking, we need to force J + d local heights to be positive for all mP with 1 ≤ m ≤ M, which is why we may need to take k as large as O(M)J+d. We next use the Fourier averaging technique described in [5]; see also [6, 10]. Let pv | D1 be a prime at which E has split multiplicative reduction. The group of components of the special fiber of the N´eron model of E at v is a cyclic group of order

nv = ordv D(E/K) , and we let 0 ≤ av(P ) < n be the component
that is hit by P . (In prac- tice, there is no prefered orientation to the cyclic group of components, so av(P ) is only defined up to ±1. This will not affect our computa- tions.) The formula for the local height at a split multiplicative place (due to Tate, see [14, VI.4.2]) says that 1 a (P ) λˆ(P ; v) ≥ B v log N pnv . 2 n K/Q v  v  In this formula, B(t) is the periodic second Bernoulli polynomial, equal 2 1 to t − t + 6 for 0 ≤ t ≤ 1 and extended periodically modulo 1. As in [5], we are going to take a weighted sum of this formula over mP for 1 ≤ m ≤ M. The periodic Bernoulli polynomial has a Fourier expansion 1 e2πint 1 ∞ cos(2πnt) B(t)= = . 2π2 n2 π2 n2 n≥1 n=1 Xn6=0 X Lang’s Height Conjecture and Szpiro’s Conjecture 5

We also use the formula (Fej´er kernel)

M m 1 M 2 1 1 − cos(mt)= eimt − . M +1 2(M + 1) 2 m=1   m=0 X X

Hence M m 1 − λˆ(mP ; v) M +1 m=1 X   M m 1 ma (P ) ≥ 1 − B v log N pnv M +1 2 n K/Q v m=1 v X     M m 1 ∞ cos(2πnma (P )/n ) = 1 − v v M +1 2π2 n2 m=1 n=1 X   X 1 ∞ 1 M m 2πnma (P ) = 1 − cos v 2π2 n2 M +1 n n=1 m=1 v X X     1 ∞ 1 1 M 2 1 = e2πinmav (P )/nv − . 2π2 n2 2(M + 1) 2 n=1 m=0 ! X X

We split the sum over n into two pieces. If n is a multiple of nv, then the quantity between the absolute value signs is equal to M + 1, and if n is not a multiple of nv, we simply use the fact that the absolute value is non-negative. This yields the local estimate

M m 1 − λˆ(mP ; v) M +1 m=1 X   1 ∞ (M + 1)2 1 ∞ 1 ≥ − log N pnv 4π2(M + 1) (nn )2 4π2 n2 K/Q v n=1 v n=1 ! X X (M + 1) 1 = − log N pnv . 24n2 24 K/Q v  v  We next sum the local heights over all primes dividing D1,

M m 1 − λˆ(mP ; v) M +1 v 1 m=1 pX|D X   1 (M + 1) ≥ − nv log NK/Qpv. 24 nv pXv|D1   6 JOSEPH H. SILVERMAN

We set

N −1 N M +1=2 nv log K/Qpv nv log K/Qpv, pXv|D1 pXv|D1 which gives the height estimate

M m 1 1 − λˆ(mP ; v) ≥ n log N p M +1 24 v K/Q v v 1 m=1 v 1 pX|D X   pX|D 1 −1 = log N D(E/K) . 24 K/Q v pv|D1 X

We also need to estimate the size of M. This is done using the elemen- tary inequality

n n n 2 −1 aixi aixi ≥ ai , (1) i=1 i=1 i=1 X X  X  valid for all ai, xi > 0. (This is a special case of Jensen’s inequality, applied to the function 1/x.) Applying (1) with xi = nv and ai = log NK/Qpv yields

2 nv log NK/Qpv pv|D1 2 2 M +1 ≤ 2  X  =2σ(D1) =2σJ D(E/K) . log N p  K/Q v   pv|D1 
 X    In particular, the chosen value of M is bounded solely in terms of σJ D(E/K) . We now combine the estimates for the local heights to obtain
M m 1 − hˆ(mkP ) M +1 m=1 X   M m ≥ 1 − + λˆ(mkP ; v) M +1 ∞ m=1 v X   v∈XMK p |DX(E/K) M m = + + 1 − λˆ(mkP ; v) ∞ M +1 v 1 v 2 m=1 v∈XMK pX|D pX|D  X   Lang’s Height Conjecture and Szpiro’s Conjecture 7

M m ≥ 1 − c1 log max |j(E)|v, 1 − c2 ∞ M +1 v∈MK m=1   X X 
1 −1 + log N D(E/K) 24 K/Q v pv|D1 X M m −1 + 1 − c log N D(E/K) M +1 3 K/Q v pv|D2 m=1   X X ≥ c4h j(E) + c5 log ND(E/K) − c6 . In the last line we have
used the fact that D(E/K)j(E) is integral, so N D −1 log max |j(E)|v, 1 + log K/Q (E/K) v ≥ h j(E) . ∞ v∈M pv|D1D2 XK  X
On the other hand, M m M m 1 − hˆ(mkP )= 1 − m2k2hˆ(P ) M +1 M +1 m=1 m=1 X   X   k2M(M + 1)(M + 2) = hˆ(P ). 12 Adjusting the constants yet again yields N D ˆ c7h j(E) + c8 log K/Q (E/K) − c9 c10h(E/K) − c9 h(P ) ≥ 2 3 ≥ 2 3 .
k M k M J+d Since M depends only on σJ D(E/K) and since k ≤ (6M) , this gives the desired lower bound for hˆ(P ). 
Remark 7. As in [10], a similar argument can be used to prove that #E(K)tors is bounded by a constant that depends only on [K : Q], J, and σJ D(E/K) .

2. Some elementary
properties of the prime-depleted Szpiro ratio We start with an elementary inequality.

Lemma 8. Let n ≥ 2, and let α1,...,αn and x1,...,xn be positive real numbers, labeled so that αn = max αi. Then α x + ··· + α x α x + ··· + α x 1 1 n n ≥ 1 1 n−1 n−1 , x1 + ··· + xn x1 + ··· + xn−1 with strict inequality unless α1 = ··· = αn. 8 JOSEPH H. SILVERMAN

n n Proof. Let A = i=1 αixi and X = i=1 xi. Then

A(X − xPn) − (A − αnxn)X =(PαnX − A)xn n

= (αn − αi)xi xn ≥ 0. (2) i=1 X  Hence A A − α x ≥ n n , (3) X X − xn and since the xi are assumed to be positive, inequalities (2) and (3) are strict unless the αi are all equal.  We apply the lemma to prove some basic properties of the J-depleted Szpiro ratio. Proposition 9. Let J ≥ 1. (a) For all integral ideals D,

σJ−1(D) ≥ σJ (D). Further, the inequality is strict unless D has the form D = Ie for a squarefree ideal I. (b) Assume that ν(D) ≥ J. Then there exists an ideal d | D satisfying

ν(d)= J and σJ (D)= σ(D/d). (c) Let p be a prime ideal and D an ideal with p ∤ D. Then

e σJ (D) σJ (D) ≥ σJ (p D) ≥ . log NK/Qp (d) Let p be a prime ideal and let D an arbitrary ideal (so p is allowed to divide D). Then

e σJ (D) (log NK/Qp)σJ (D) ≥ σJ (p D) ≥ . log NK/Qp

ei Proof. (a) Write D = pi . To ease notation, we let N Q qi = log K/Qpi.

If ν(D) ≤ J − 1, then σJ−1(D) = σJ (D) = 1, so there is nothing to prove. Assume now that ν(D) ≥ J. Let I ⊂ 1, 2,...,ν(D) be a set of indices with #I ≥ ν(D) − (J − 1) satisfying 

σJ−1(D)= eiqi qi. Xi∈I  Xi∈I Lang’s Height Conjecture and Szpiro’s Conjecture 9

Let k ∈ I be an index satisfying ek = max{ei : i ∈ I}. Then Lemma 8 with αi = ei and xi = qi yields

eiqi eiqi i∈I i∈I,i6=k σJ−1(D)= X ≥ X ≥ σJ (D). qi qi i∈I X i∈XI,i6=k Further, Lemma 8 says that the inequality is strict unless all of the ei are equal, in which case D is a power of a squarefree ideal. e e (b) If D = I is a power of a squarefree ideal, then σJ (D) = σ(D/c ) for every ideal c | I satisfying ν(c) = J, so the assertion to be proved is clear. We may thus assume that D is not a power of a squarefree ideal. Suppose in this case that σJ (D) = σ(D/d) for some d | D with ν(d) ≤ J − 1. Then

σJ−1(D) ≤ σ(D/d)= σJ (D), contradicting the fact proven in (a) that σJ−1(D) > σJ (D) (strict in- equality). (c) We always have e σJ (p D) ≤ σJ−1(D), e since in computing σJ (p D), we can always remove p and J − 1 other primes from D. If this inequality is an equality, we’re done. Other- e wise the value of σJ (p D) is obtained by removing J primes from D. Continuing with the notation from (a) and letting q = log NK/Qp, this means that there is an index set I with #I ≥ ν(D) − J such that

eq + eiqi q + eiqi i∈I i∈I q + X σJ (D)= X ≥ X = , q + qi q + qi q + Y Xi∈I Xi∈I where to ease notation, we write X and Y for the indicated sums. e If Y = 0, then also X = 0 and ν(D) ≤ J, so σJ (p D) equals either e or 1. In either case, it is greater than σJ (D) = 1. So we may assume that Y > 0, which implies that Y ≥ log 2. We observe that

eiqi X i∈I = X ≥ σJ (D). Y qi Xi∈I 10 JOSEPH H. SILVERMAN

Hence X 1+ q/X σ (D) σ (D) σ (D)= · ≥ J ≥ J . J Y 1+ q/Y 1+ q/Y 3q (The final inequality is true since q ≥ log 2 and Y ≥ log 2.) This proves that σJ (D) is greater than the smaller of σJ−1(D) and σJ (D)/3q. But from (a) we have σJ−1(D) ≥ σJ (D), so the latter is the minimum. i ′ ′ (d) Let D = p D with p ∤ D . Then writing q = log NK/Qp as usual and applying (c) several times, we have e e+i ′ ′ i ′ σJ (p D)= σJ (p D ) ≤ σJ (D ) ≤ qσJ (p D )= qσJ (D). Similarly σ (D′) σ (piD′) σ (D) σ (peD)= σ (pe+iD′) ≥ J ≥ J = J . J J q q q 

3. The Prime-Depleted Szpiro and ABC Conjectures In this section we describe a prime-depleted variant of the ABC- conjecture and show that it is a consequence of the prime-depleted Szpiro conjecture. For ease of notation, we restrict attention to K = Q and leave the generalization to arbitrary fields to the reader. Conjecture 10 (Prime-Depleted ABC-conjecture). There exist an in- teger J ≥ 0 and a constant C5 such that if A,B,C ∈ Z are integers satisfying A + B + C =0 and gcd(A,B,C)=1, then

σJ (ABC) ≤ C5. The classical ABC-conjecture (with non-optimal exponent) says that σ(ABC) is bounded, which is stronger than the prime-depleted version, since σJ (ABC) is less than or equal to σ(ABC). Proposition 11. If the prime-depleted Szpiro conjecture is true, then the prime-depleted ABC-conjecture is true. Proof. We suppose that the prime-depleted Szpiro conjecture is true, say with J primes deleted. Let A,B,C ∈ Z be as in the statement of the depleted ABC-conjecture. We consider the Frey curve E : y2 = x(x + A)(x − B). An easy calculation [15, VIII.11.3] shows that the minimal discrimi- nant of E is either 24(ABC)2 or 2−8(ABC)2, so in any case we can Lang’s Height Conjecture and Szpiro’s Conjecture 11 write D(E/Q)=2e(ABC)2 for some exponent e ∈ Z. Then using Proposition 9 we find that 2 e 2 σJ (ABC) 2σJ (ABC) σJ D(E/Q) = σJ 2 (ABC) ≥ = . log 2
log 2

Hence the boundedness of σJ D(E/Q) implies the boundedness of σ (ABC).  J
Remark 12. The Szpiro and ABC-conjectures have many important consequences, including asymptotic Fermat (trivial), a strengthened version of Roth’s theorem [1, 16], the infinitude of non-Wieferich primes [12], non-existence of Siegel zeros [4], Faltings’ theorem (Mordell con- jecture) [3, 16],. . . . (For a longer list, see [8].) It is thus of interest to ask which, if any, of these results follows from the prime-depleted Szpiro conjecture. As far as the author has been able to determine, the answer is none of them, which would seem to indicate that the prime-depleted Szpiro conjecture is qualitatively weaker than the orig- inal Szpiro conjecture.

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