Counting special points: Logic, diophantine geometry, and transcendence theory

Scanlon, Thomas

doi:10.1090/s0273-0979-2011-01354-4

Cited by 20 publications

(13 citation statements)

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“…We briefly mention Pila-Zannier's proof the the Manin-Mumford conjecture [46], Pila's proof of the André-Oort conjecture for modular curves [45] and Masser-Zannier's work on simultaneous torsion points in elliptic families [32]. We refer the reader to [56,48] for a survey. Some of the most striking diophantine applications, particularly around the study of modular curves and Shimura varieties, require the generality of the Pila-Wilkie theorem for R an,exp , i.e.…”

Section: 44mentioning

confidence: 99%

Complex cellular structures

Binyamini

Novikov

2019

Ann. of Math. (2)

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We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For δ ∈ (0, 1) and a complex cell C we define its holomorphic extension C ⊂ C δ , which is again a complex cell. The hyperbolic geometry of C within C δ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved.Using this theory, we refine the Yomdin-Gromov algebraic lemma on C rsmooth parametrizations of semialgebraic sets: we show that the number of C r charts can be taken to be polynomial in the smoothness order r and in the complexity of the set. The algebraic lemma was initially invented in the work of Yomdin and Gromov to produce estimates for the topological entropy of C ∞ maps. For analytic maps our refined version, combined with work of Burguet, Liao and Yang, establishes an optimal refinement of these estimates in the form of tight bounds on the tail entropy and volume growth. This settles a conjecture of Yomdin who proved the same result in dimension two in 1991. A self-contained proof of these estimates using the refined algebraic lemma is given in an appendix by Yomdin.The algebraic lemma has more recently been used in the study of rational points on algebraic and transcendental varieties. We use the theory of complex cells in these two directions. In the algebraic context we refine a result of Heath-Brown on interpolating rational points in algebraic varieties. In the transcendental context we prove an interpolation result for (unrestricted) logarithmic images of subanalytic sets.

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Section: 44mentioning

confidence: 99%

Complex cellular structures

Binyamini

Novikov

2019

Ann. of Math. (2)

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“…We assume for the purposes of this overview that the reader is familiar with this approach, as there are already several good surveys available (e.g. [27]) in addition to the original paper. There are three main sources of ineffectivity in the proof of [26], as follows:…”

Section: 4mentioning

confidence: 99%

Some effective estimates for André–Oort in Y(1)ⁿ

Binyamini

Kowalski

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let X ⊂ Y (1) n be a subvariety defined over a number field F and let (P 1 , . . . , Pn) ∈ X be a special point not contained in a positive-dimensional special subvariety of X. We show that the if a coordinate P i corresponds to an order not contained in a single exceptional Siegel-Tatuzawa imaginary quadratic field K * then the associated discriminant |∆(P i )| is bounded by an effective constant depending only on deg X and [F : Q]. We derive analogous effective results for the positive-dimensional maximal special subvarieties.From the main theorem we deduce various effective results of André-Oort type. In particular we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective André-Oort statement for hypersurfaces defined by polynomials satisfying this condition.

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“…In this section we briefly review the notions of ominimal structures and their properties that we will use later on. For details we refer to [7]and [30] as well as references therein.…”

Section: 1mentioning

confidence: 99%