Hindry and Silverman's Paper

We now state Szpiro's conjecture over a number field. Let $K$ be a number field, $\Delta = \Delta_{E/K}$ the minimal discriminant and $f= f_{E/K}$ the conductor, and let

$$\sigma_{E/K} = \frac{\log\vert N^K_\mathbb{Q}\Delta_{E/K}\vert} {\log\vert N^K_\mathbb{Q}f_{E/K}\vert}$$

be the Szpiro ratio.

Conjecture 3.1 (Szpiro)   Given $\varepsilon >0$ there are only finitely many $E/K$ with $\sigma_{E/K}\geq 6+\varepsilon$. In particular, $\sigma_{E/K}$ is bounded.

This conjecture is true if one restricts to curves with potential good reduction everywhere (with `` $\varepsilon =0$'').

One can also look at modular degrees of coverings of $E/\mathbb{Q}$ by $X_0(N)$. Any polynomial bound on the modular degree in terms of the conductor would imply Szpiro's conjecture. It's possible to get exponential bounds (see work of Stewart and Yu).

Theorem 3.2 (Hindry and Silverman (1988), Invent. 93)   There are explicit constants $c_1, c_2>0$ such that for all number fields $K$, all elliptic curves $E/K$, and all nontorsion points $P\in E(K)$, we have

$$\hat{h}(P) \geq c_1^{-[K:\mathbb{Q}]}\cdot c_2^{-\sigma_{E/K}} \cdot \log\vert N^K_\mathbb{Q}\Delta_{E/K}\vert.$$

In particular, Lang's conjecture is ``as easy as ABC''. We also get a uniformity result for integral points on curves.

Corollary 3.3   Let $K$ be a number field. Then there exists $C(K)$ such that if $E/K$ is a ``minimal'' elliptic curve, then for any finite set of places of $K$ (including the archimedian places) then we have

$$\vert E(R_S) \vert \leq C(K)^{\vert S\vert + (1+\rank E(K))\sigma_{E/K}}.$$

In particular, for $K=\mathbb{Q}$ and $S=\{\infty\}$, the number of integral points on $E$ is bounded only in terms of the rank of $E(\mathbb{Q})$ (assuming that the $\sigma$'s are bounded).