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The minimal regular model of a Fermat curve of odd squarefree exponent and its dualizing sheaf. (English) Zbl 1477.14044

The paper under review is partly based upon an earlier paper by W. G. McCallum [Prog. Math. 26, 57–70 (1982; Zbl 0542.14019)]. In this paper, McCallum analyzed the case when the degree of the Fermat curve is an odd integer \(p\) over \(K=\mathbb{Q}_{p}(\xi)\), where \(\xi=\exp(\frac{2\pi}{p})\) is the \(p\)-th root of unity.
In this situation, the Fermat curve \(F_{K}:x^{p}+y^{p}=1\) has bad reduction at the prime \((\pi)=1-\xi\) of \(\mathcal{O}_{K}=\mathbb{Z}_{p}(\xi)\) lying over \(p\). To analyze this situation carefully McCallum introduced an auxillary curve \(C_{s,K}\): \[ y^{p}=x^{s}(1-x),\ s=1, 2,\dots, p-2 \] The analysis for \(F_{K}\) can thus be recovered from the analysis of \(C_{s,K}\) via the map from \(C_{s,K}\) to \(F_{K}\): \[ f_{s, K}: (x,y)\rightarrow (x^p, x^s y) \] Much of the analysis relies on an auxillary polynomial \(\phi(x,y)=((x+y)^{p}-x^p-y^p)/p\).
The final result can be summarized as following:
Theorem (page 59 of [loc. cit.]). The curve \(F_{K}\) has a regular model \(F\) whose geometric special fibre \(\overline{F}_{0}\) has configuration in diagram 3 (p. 69 of [loc. cit.]). The vertical components are parametrized by the roots of \(\phi(x,-y)\) as follows:
The components labelled \(0, 1, \infty\) correspond to the factors \(x,x-y, y\) of \(\phi(x,-y)\).
The components labelled \(\alpha\) correspond to double factors \((x+\alpha y)^2\) of \(\phi(x,-y)\), \(\alpha\in k\).
The components labelled \(\beta\) correspond to simple factors \((x+\beta y)\) of \(\phi(x,-y)\), \(\beta\not \in k\).

The method involved in the proof is rather standard by working with the projective normalization of the curve \(C_{s,K}\) and \(F_{K}\) and repeatedly blowing up the singular points. McCallum apparently had further plans to study the Jacobian associated to \(F_{K}\) itself, as the map \(f_{s,K}\) induces an isogeny on the Jacobians. This leads to his later paper on the Tate-Shafarevich group of Fermat curves.
The present paper continues the investigation of the Fermat curve when \(N\) is a square-free odd integer. While the methodologies are similar by utilizing the map \(\phi\), the analysis has become much more involved. For example to obtain the normalization, the standard blow-up of \(\mathcal{X}\) has become much more complicated (see the proof of Proposition 3.7, for example). The main result is summarized in Theorem 3.13, which gives a description of the geometric special fibres associated to the local minimal model. The global minimal model is then “glued” from the local minimal model.
The main interest of current paper stems from the following result:
Theorem. Let \(N>0\) be an odd square-free integer with at least two prime factors, and let \(\mathfrak{F}^{\min}_{N}\) be the minimal regular model of the Fermat curve \(F_{N}\) over \(\textrm{Spec}(\mathbb{Z}[\xi_{N}])\). Then the arithmetic self-intersection number of its dualizing sheaf equipped with the Arakelov metric satisfies \[ \omega^2_{\mathfrak{F}^{\min}_{N}}\le (2g-2)[\mathbb Q(\xi_{N}):\mathbb Q](k_1 \log(N)+k_2)+(2g-2)\sum_{p|N}\frac{\phi(N)}{\phi(p)}(\cdots) \log(p) \] where the expression in \((\cdots)\) is a function of order \(O(\log\log(N))\).
This is a great improvement over McCallum’s result as it connects the work on local minimal models to (global) arithmetic intersection theory. However, the reader should be aware that the authors’ claim that “[…]While no bounds in complete families are known to date…” held only true at the time of submission (June 2016). By 1994–1995 C. Soulé [Invent. Math. 116, No. 1–3, 577–599 (1994; Zbl 0834.14013)] had shown that such estimates are possible if one can estimate norm of sections of \(H^{1}(X, L^{-1})\), and by 2017 based upon previous work of K. Yamaki, R. Wilms [“The delta invariant in Arakelov geometry”, https://hdl.handle.net/20.500.11811/6757; Invent. Math. 209, No. 2, 481–539 (2017; Zbl 1390.14073)] derived effective upper bounds for the case of hyper-elliptic curves (thus resolving the case for \(g=2\) completely).
The authors’ result thus provides effective upper bounds for Fermat curves (the first author and U. Kühn [“On the arithmetic self-intersection numbers of the dualizing sheaf for Fermat curves of prime exponent”, Preprint, arXiv:0906.3891] provided similar results for the case when \(N=p\)). The main questions now left are the following:
For Fermat curve of low genus (\(g=3\), for example) – how does the current result on lower/upper bounds expand our current knowledge of arithmetic intersection theory? Can we say more about other curves in \(M_{g}\)?
It was announced by M. van Frankenhuysen [J. Number Theory 95, No. 2, 289–302 (2002; Zbl 1083.11042)] that providing similar upper bounds for the self-intersection number would suffice to prove Vojta’s height inequality (Corollary 2.3).

I have followed up with a personal email to Prof. Vojta, who confirmed this should be correct. In the same paper Machiel van Frankenhuysen claimed that a proof of Vojta’s height inequality will imply the ABC conjecture.
Thus, the main issue is whether the current upper bounds suffice to prove Vojta’s height inequality via a comparison of heights by translating \(\omega^2\) to the degree associated to a section of a line bundle over the curve. Traditionally this is done via Noether’s formula in Faltings-Riemann-Roch. If this is the case and we can “fill in details” of each step, then this may lead to potentially interesting results.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
58J52 Determinants and determinant bundles, analytic torsion
13D45 Local cohomology and commutative rings
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References:

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