Lang's height conjecture and Szpiro's conjecture (Q983415)

Let \(K\) be a number field, and let \(E/K\) be an elliptic curve with minimal discriminant \(\mathfrak{D}_{E/K}\). It was conjectured by \textit{S. Lang} [Elliptic curves: Diophantine analysis. New York: Springer-Verlag (1978; Zbl 0388.10001)] that there exist constants \(C_1>0\) and \(C_2\) such that for any non-torsion point \(P\in E(K)\), \[ \hat{h}_E(P)\geq C_1\log\mathrm {Norm}_{K/\mathbb{Q}}\mathfrak{D}_{E/K}-C_2. \] \textit{M. Hindry} and the author [Invent. Math. 93, No. 2, 419--450 (1988; Zbl 0657.14018)] have previously shown that a weak form of Szpiro's conjecture, and hence the \(ABC\)-conjecture of Masser and Oesterlé, implies Lang's conjecture; it is natural to ask if Lang's conjecture is in fact equivalent to Szpiro's conjecture. The purpose of this paper is to derive Lang's conjecture from what seems to be a significant weakening of Szpiro's conjecture. In particular, for any ideal \(\mathfrak{D}=\prod \mathfrak{p}_i^{e_i}\), define the Szpiro ratio to be \[ \sigma(\mathfrak{D})=\frac{\sum e_i\log\text{Norm}_{K/\mathbb{Q}}\mathfrak{p}_i}{\sum \log\text{Norm}_{K/\mathbb{Q}}\mathfrak{p}_i}.\tag{*} \] The weak form of Szpiro's conjecture refered to above is that \(\sigma(\mathfrak{D}_{E/K})\) is bounded by some quantity depending just on \(K\). Now for each integer \(J\geq 0\), the author defines the \(J\)-depleted Szpiro ratio \(\sigma_J(\mathfrak{D})\) to be the smallest value obtained by removing the contributions from up to \(J\) terms in the numerator and denominator of the equation (*). Note that \(\sigma_J(\mathfrak{D})\) is a priori non-increasing in \(J\), and that if a small number of primes contribute a great deal to \(\sigma(\mathfrak{D})\), one can expect \(\sigma_J(\mathfrak{D}_{E/K})\) to be a great deal smaller than \(\sigma(\mathfrak{D})\). The author posits the existence of an integer \(J\) depending on \(K\), such that \(\sigma_J(\mathfrak{D}_{E/K})\) is bounded as \(E/K\) varies, an apparent weakening of Szpiro's conjecture. On this assumption, it is shown that there are constants \(C_1>0\) and \(C_2\) depending just on \(K\) such that for any \(E/K\), \[ \hat{h}_E(P)\geq C_1\max\{h(j_E), \log\text{Norm}_{K/\mathbb{Q}}\mathfrak{D}_{E/K}\}-C_2, \] for non-torsion points \(P\in E(K)\). This is the same strong version of Lang's conjecture proved under the stronger Szpiro conjecture by the author and Hindry [loc. cit.].