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The number of maximal torsion cosets in subvarieties of tori. (English) Zbl 1427.14051
It is known that if \(V\) is a subvariety of \(\mathbb{G}^n\) then the closure of the set of torsion points of \(\mathbb{G}^n\) contained in \(V\) is a union of torsion cosets: that is, translates of subtori by a torsion point.
It then becomes a question to bound the number of such components, in terms of \(n\) and the degrees of the polynomials defining \(V\). It is of course not a priori clear that such bounds should exist, but this, too, is known.
The history of conjecture and partial results is complicated but two particular conjectures, both concerning hypersurfaces, are proved here. The first, by Ruppert, predicts that if \(f\in\mathbb{C}[x_1,\ldots,x_n]\) is of multidegree \((d_1,\ldots,d_n)\) then the number of isolated torsion points on the subvariety of \(\mathbb{G}_m^n\) that it defines is at most \(c_nd_1\ldots d_n\), where \(c_n\) is a constant depending only on \(n\). The second, by Aliev and Smyth, replaces the product of the degrees with the volume of the Newton polytope of \(f\).
In fact the present paper proves more than that, and also provides results for varieties of any dimension. It is shown that if \(V\subset\mathbb{G}_m^n\) is of dimension \(d\) and defined by polynomials of degree at most \(\delta\) then the number of maximal torsion cosets is bounded by \[ \sum_{j=0}^d ((2n-1)(n-1)(2^{2n}+2^{n+1}-2))^{d(n-j)}\delta^{n-j} \] (and in fact these terms are bounds for the degree in each codimension).
Using an old theorem of F. John [in: Studies Essays. 187–204 (1948; Zbl 0034.10503)] the author is then able to show that if \(f\in\overline{\mathbb{Q}}[x_1,\ldots,x_n]\) and \(\Delta\subset \mathbb{R}^n\) is a convex body containing the support of \(f\), then the number of isolated points on the hypersurface in \(\mathbb{G}^n_m\) defined by \(f\) is bounded by \[ ((2n-1)(n-1)(2^{2n}+2^{n+1}-2))^{n(n-1)}2^nn^{2n}\omega_n^{-1}{\operatorname{vol}}_n(\Delta), \] where \(\omega_n\) is the volume of the unit ball. This proves both the conjecture of Ruppert, by taking \(\Delta=[0,d_1]\times\dots\times[0,d_n]\), and the conjecture of Aliev and Smyth, by taking \(\Delta\) to be the Newton polytope of \(f\).
The method of proof is ingenious: no part of it is completely novel in isolation but it combines many previous approaches in a highly inventive way. Using (but generalising) an idea of F. Beukers and C. J. Smyth [in: Number theory for the millennium I. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21–26, 2000. Natick, MA: A K Peters. 67–85 (2002; Zbl 1029.11009)] \(V\) is replaced by a more convenient variety \(V'\) which is in turn replaced by a hypersurface, using bounds on Hilbert functions due to Chardin in a way similar to what is done by F. Amoroso and E. Viada [Duke Math. J. 150, No. 3, 407–442 (2009; Zbl 1234.11081)]. Then John’s result is used to present the bounds in a concrete form.
MSC:
14G05 Rational points
11G35 Varieties over global fields
11G50 Heights
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
12E12 Equations in general fields
14G25 Global ground fields in algebraic geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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