Arithmetic discriminants and morphisms of curves.(English)Zbl 0982.11035
Let \(C\) be a curve over a number field \(k\), with regular model \(X\) over the ring of integers \(R\) of \(k\). For an algebraic point \(P \in C\), the arithmetic discriminant is defined by a formula analogous to the adjunction formula for the genus of a curve on a surface, \[d_a(P) = (H_P . (\omega_{X/\text{Spec} R} + H_P))/[k(P) : k]\] where \(H_P\) is the arithmetic curve on \(X\) corresponding to \(P\).
Let \(h\) be a Weil height on \(C\) associated to a degree \(1\) divisor; thenP. Vojta [J. Am.Math.Soc.5, 763-804 (1992;Zbl 0778.11037)] proves that \(h_K(P) \leq d_a(P) + \epsilon h(P) + O_\epsilon(1)\) for all \(\epsilon > 0\). Here \(h_K\) is a Weil height associated to the canonical divisor \(K\) on \(C\). Vojta conjectures that one may replace \(d_a(P)\) by \(d(P) = \log |N_{k/{\mathbb Q}} D_{k(P)/k}|/[k(P) : {\mathbb Q}]\), the normalised field discriminant of \(k(P)\).
Vojta’s inequality provides a lower bound for \(d_a(P)\) in terms of \(h_K(P)\) and \(h(P)\). In the paper under review, the authors provide an upper bound, namely \[d_a(P) \leq h_K(P) + 2[k(P) : k] h(P) + O(1).\] This is then used to generalise a result of Vojta’s on the finiteness of the set of points on \(C\) of given degree mapping to points with the same field of definition under a dominant morphism \(f : C \to C'\). Vojta’s original result is the special case \(C' = {\mathbb P}^1\).
The authors proceed to sharpen their upper bound for points in a dense open subset of \(C\) to get \(d_a(P) \leq h_K(P) + (2[k(P) : k] - 2 + \epsilon) h(P) + O_\epsilon(1)\) if \([k(P) : k] \leq g(C)\). This leads to corresponding variants of the results derived from the upper bound.
In a concluding section, the authors show that on bi-elliptic genus \(2\) curves, there are families of quadratic points satisfying \(d_a(P) = h_K(P) + 2 h(P) + O(1)\) (besides other families, where \(2h(P)\) is replaced by \(0\) or by \(4h(P)\)).
The reader should be warned that there are a few misprints in this paper, namely on page 1925, where the ‘\(\geq\)’ sign in Corollary 2.1 should be a ‘\(>\)’ sign, and on page 1926, where the formulas involving the ramification divisor \(R_f\) are erroneous.
Let \(h\) be a Weil height on \(C\) associated to a degree \(1\) divisor; thenP. Vojta [J. Am.Math.Soc.5, 763-804 (1992;Zbl 0778.11037)] proves that \(h_K(P) \leq d_a(P) + \epsilon h(P) + O_\epsilon(1)\) for all \(\epsilon > 0\). Here \(h_K\) is a Weil height associated to the canonical divisor \(K\) on \(C\). Vojta conjectures that one may replace \(d_a(P)\) by \(d(P) = \log |N_{k/{\mathbb Q}} D_{k(P)/k}|/[k(P) : {\mathbb Q}]\), the normalised field discriminant of \(k(P)\).
Vojta’s inequality provides a lower bound for \(d_a(P)\) in terms of \(h_K(P)\) and \(h(P)\). In the paper under review, the authors provide an upper bound, namely \[d_a(P) \leq h_K(P) + 2[k(P) : k] h(P) + O(1).\] This is then used to generalise a result of Vojta’s on the finiteness of the set of points on \(C\) of given degree mapping to points with the same field of definition under a dominant morphism \(f : C \to C'\). Vojta’s original result is the special case \(C' = {\mathbb P}^1\).
The authors proceed to sharpen their upper bound for points in a dense open subset of \(C\) to get \(d_a(P) \leq h_K(P) + (2[k(P) : k] - 2 + \epsilon) h(P) + O_\epsilon(1)\) if \([k(P) : k] \leq g(C)\). This leads to corresponding variants of the results derived from the upper bound.
In a concluding section, the authors show that on bi-elliptic genus \(2\) curves, there are families of quadratic points satisfying \(d_a(P) = h_K(P) + 2 h(P) + O(1)\) (besides other families, where \(2h(P)\) is replaced by \(0\) or by \(4h(P)\)).
The reader should be warned that there are a few misprints in this paper, namely on page 1925, where the ‘\(\geq\)’ sign in Corollary 2.1 should be a ‘\(>\)’ sign, and on page 1926, where the formulas involving the ramification divisor \(R_f\) are erroneous.
Reviewer: Michael Stoll (Bonn)
MSC:
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 11J25 | Diophantine inequalities |
| 14G05 | Rational points |
| 14G25 | Global ground fields in algebraic geometry |
| 11G50 | Heights |
| 14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |
| 14H25 | Arithmetic ground fields for curves |
Citations:
Zbl 0778.11037Cite
Full Text:DOI
References:
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