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Interpolation and Comparison Methods in the Mean Field Spin Glass Model Read chapter Read first chapter

2014 | OriginalPaper | Chapter

12. Unlikely Intersections and Pell’s Equations in Polynomials

Author : Umberto Zannier

Published in: Trends in Contemporary Mathematics

Publisher: Springer International Publishing

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Abstract

This short paper surveys around themes related to a conjecture of R. Pink which, roughly speaking, generalises in particular the well-known Manin-Mumford conjecture (a theorem of M. Raynaud) to abelian varieties varying in families. In a series of works in collaboration with D. Masser we have established this for pencils of abelian surfaces, and in further work with D. Bertrand, Masser and A. Pillay we have considered other commutative algebraic groups of dimension 2.
We shall briefly discuss this progress, and present some applications to solvability of Pell’s equations \(X^{2} -\mathit{DY }^{2} = 1\) in polynomials \(X = x(t),Y = y(t)\), where D = D(t) is also a polynomial. This is analogue to the classical one for integers, and was studied already by Abel. In this context solvability is no longer ensured by simple conditions on D and may be considered ‘exceptional’.
In this paper we shall let D(t) vary in a pencil, and for instance we shall point out how our results on Pink’s conjectures imply that for \(D(t) = t^{6} + t+\lambda\) the Pell’s equation is solvable nontrivially only for finitely many complex λ.

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previous chapter Integral Hodge Classes, Decompositions of the Diagonal, and Rationality Questions
next chapter Birational Geometry of Projective Varieties and Directed Graphs
Footnotes
1
This was strengthened later by G. Maurin and in further joint work with Bombieri, P. Habegger, Masser.
 
2
However one should take into account that some relevant work has been done very recently.
 
3
For the first two cases, the treatment of this transition occupies a substantial part of the corresponding papers, whereas for the ‘simple’ case this is the object of work still in progress, also with P. Corvaja.
 
4
It is a remarkable content of these papers that estimates for torsion points on subvarieties of tori lead to important information about Betti numbers of abelian covers of a compact variety.
 
5
The problem of solving the equation in rationals is rather easier: projection to a rational line from the rational point (1, 0) allows to parametrize rationally over \(\mathbb{Q}\) the corresponding hyperbola.
 
6
All of this easily follows from the cyclic structure of solutions, recalled below. For instance, if \(x^{2} -\mathit{Dy}^{2} =\mu \in k^{{\ast}}\), one may ‘square’ to get \(((x^{2} + \mathit{Dy}^{2})/\mu )^{2} - D(2\mathit{xy}/\mu )^{2} = 1\); see also Proposition 12.1 and the observations which follow. Having a non-square μ ∈ k ∗ is somewhat analogue to the so-called ‘negative’ Pell’s equation in the classical case, i.e. with \(\mu = -1\).
 
7
We think here of algebraically closed k; see Remark 12.3(iv) for number fields. This unboundedness is not free of interest in itself and for instance yielded the first counterexample to a certain ‘plausible’ conjecture in model theory of constructive algebra; this is due to L. van den Dries, Ka. Schmidt and H. Schoutens, see [24].
 
8
At least for large enough degy, one may find them even within this family: see [30], p. 92.
 
9
A converse does not hold, as shown e.g. by \(D(t) = t^{2} + 3\); a proof is possible using the Pell’s equation.
 
10
One may also expand \(\sqrt{D(t)}\) at t = ∞ and impose a suitable vanishing at ∞ of \(x + \sqrt{\mathit{Dy}}\) through a linear system in the coefficients; again, this leads to seemingly complicated determinantal varieties.
 
11
All of this is also related to the so-called abc-inequality for polynomials: solutions to a Pell’s equation yield instances in which the bound is attained up to a summand ≤ d − 1; see [29]. For d = 1 it is attained exactly, the map is fully Belyi and boils down to Chebyshev polynomials, as in one of the above examples.
 
12
A formal proof, found in conversation with Corvaja, has not yet been written down and shall hopefully appear in a future paper.
 
13
The above mentioned facts about density suggests that one cannot prove this using merely the complex topology. It seems likely that not even the p-adic ones suffice.
 
14
If m δ = 0 in J, then some function \(\varphi = x + \mathit{uy} \in k[t,u]\setminus k\) has divisor m δ, hence x 2 −Dy 2 is a nonzero constant μ; considering \(\mu ^{-1}\varphi ^{2}\) in place of \(\varphi\), we may achieve μ = 1; see also footnote 6 above.
 
15
In doing this for the Pell’s equation, it is computationally convenient to use the proposition twice, i.e. to perform the torsion checking modulo p through the Pell’s equation, using continued fractions.
 
16
This kind of argument might not work generally, but one can show that it is always effectively possible to decide about this obstruction.
 
17
Note that in case (a) the set of multiples is finite, so (a) and (b) could be fused in a single condition.
 
18
It is possible to decide effectively about this obstruction as well, using deep tools, like estimates for isogeny degrees of Masser and G. Wüstholz [14]. A direct argument seems not to be available.
 
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Metadata
Title
Unlikely Intersections and Pell’s Equations in Polynomials
Author
Umberto Zannier
Copyright Year
2014
Book
Trends in Contemporary Mathematics

Print ISBN: 978-3-319-05253-3

Electronic ISBN: 978-3-319-05254-0

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-05254-0_12

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