Consider, on the one hand, algebraic integers $\alpha$ and their rational approximants to within a varying exponent $\kappa > 2$; and on the other hand, smooth projective geometrically irreducible curves $C$ of fixed genus $g > 1$ over a varying number field $K$. The following remarkable feature on the number of solutions is shared by the quantitative Roth and Faltings theorems.

  • In Roth's theorem, for any choice of archimedean place (complex embedding of $\overline{\mathbb{Q}}$), the (logarithmic) heights of the small solutions of $|\alpha - p/q| < q^{-\kappa}$ --- those which have too small a height to be of any use in the proof by contradiction --- are bounded linearly in the absolute height $h(\alpha)$ and the chosen valuation $|\alpha|$ of $\alpha$, but neither on the degree $d:=\deg{\alpha} = [\mathbb{Q}(\alpha):\mathbb{Q}]$ nor on the approximation exponent $\kappa$. By Northcott's theorem, the number of small solutions is bounded in terms of $d$ and $h(\alpha)$ alone; and by Mumford's height gap principle, it can also be shown to be at most $C_3(\kappa)(1+\log^+{h(\alpha)}+\log^+{|\alpha|})$. On the other hand, the bound for the number of large solutions depends on $d$ and $\kappa$ alone, but not on $h(\alpha)$ nor $|\alpha|$. (None of this requires the normalization condition that $\alpha$ be integral, which I make to exclude the trivial counterexamples to my question below.)

  • In Faltings' theorem, as proved by Vojta and Bombieri, the heights of small $K$-rational points are similarly bounded linearly in the height of $C$, but independently of the number field $K$. The bound for the number of large solutions, on the other hand, is independent of the height of $C$, but depends instead exponentially on the rank $r$ of the Mordell-Weil group over $K$ of the Jacobian of $C$. (The dependence being like $(A\log{g}+B)7^r$.)

In the latter case, there is the famous uniformity conjecture of Caporaso-Harris-Mazur which predicts that the number of solutions $C(K)$ should be bounded by a uniform constant only depending on $K$, but completely independent from $C$. (This would be implied by the general Bombieri-Lang conjecture generalizing Mordell's conjecture to higher dimension.) In fact, it follows from work of Remond and David-Philippon that the number of small solutions in $C(K)$, and therefore also the total number of $K$-rational points, would be bounded by $A^r$ with a uniform $A = A(g,[K:\mathbb{Q}]) < \infty$, if only one could minorize the normalized (canonical) height of $C$ in its Jacobian by a uniform positive quantity depending only on $g$. The latter height, if one prefers the formulation in the framework of Arakelov theory, is essentially just the arithmetic self-intersection of the canonical (=relative dualizing) class on the minimal regular geometric model of $C$ over $O_K$, divided as usual by $[K:\mathbb{Q}]$.

David and Philippon produce such an explicit lower bound, which however depends not just on $g$, but also on the minimum of the injectivity radii of the complex tori associated to the Jacobian. The latter dependence arises solely from the error term in the comparison between Weil and Neron-Tate heights, and appears to be an artifice of their method. (The same difficulty comes up in other related problems, such as the conjecture on the uniform boundedness of rational torsion and Lang's conjecture on the uniform minorization of canonical heights of rational points.) The Caporaso-Harris-Mazur conjecture, and to an extent perhaps also the success of Mazur's uniformity theorem for the rational torsion of elliptic curves, strongly suggest that the dependence (though the injectivity radius) on the height of the Jacobian ought to be omitted. As noted in the previous paragraph (see S. David and P. Philippon, Minorations des hauteurs normalisees des sous-varietes de varietes abeliennes), it would follow that the number (not height!) of small solutions should depend on $C$ and $K$ only through $g$, $[K:\mathbb{Q}]$, and the Mordell-Weil rank $r$, but not through the height of $C$ (or that of the Jacobian). Coupled to the Vojta-Bombieri estimate for the number of large solutions noted above, it would furthermore follow that the same bound persists for the total number of solutions.

In the analogy with Roth's theorem, my question is whether there are any reasons to expect (or clear indications to the contrary) any uniformity (in $\alpha$) of the number (not height!) of small solutions in Roth's theorem. For this one clearly has to add the normalization that $\alpha \in \overline{\mathbb{Z}}$ is integral, otherwise there are immediate counterexamples already as $\alpha$ runs through the rationals $\mathbb{Q}$. On the other hand, with the integrality normalization, it is easy to see that as $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq 2$, the (total) number of solutions is indeed bounded by a constant depending only on $\kappa$.

In the analogy considered above, the Mordell-Weil rank $r$ should correspond to the approximation exponent $\kappa$, while $[K:\mathbb{Q}]$ should correspond to the degree $d:=[\mathbb{Q}(\alpha):\mathbb{Q}]$. Could it possibly be, then, that the bound on the number of small solutions in Roth's theorem might depend solely on $d$ and $\kappa$ as $\alpha$ runs through:

  • all algebraic integers of a bouned degree $d$?
  • the ring of integers of the fixed number field $\mathbb{Q}(\alpha)$? (This would correspond to bounds on the number of $K$-rational point that depend additionally on the discriminant of $K$, instead of on the height of $C$.)

If true, this would imply that for algebraic integers $\alpha$ of degree $d$, the total number of exceptions to Roth's inequality depends just on $d$ and $\kappa$, similarly to the analogous conjecture on the number of rational points.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.