Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Barroero, Fabrizio; Capuano, Laura

doi:10.1112/plms.12289

Cited by 13 publications

(28 citation statements)

References 54 publications

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“…Indeed, for instance, in [MZ15] and [MZ20], the authors show that a generic curve in B 2d , where d ≥ 3, contains at most finitely many complex points that correspond to Pellian polynomials. One can see also [BMPZ16] and [Sch19] for the non-squarefree case and [BC20] for similar results for the generalized Pell-Abel equation.…”

Section: Introductionsupporting

confidence: 60%

Betti maps, Pell equation in polynomials and almost Belyi maps

Barroero,

Capuano,

Zannier

2021

Preprint

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We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation A 2 − DB 2 = 1, with A, B, D ∈ C[t] and certain ramified covers P 1 → P 1 arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Covaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials D that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann Existence Theorem associates to the above-mentioned covers certain permutation representations: we are able to characterize the representations corresponding to "primitive" solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations when D has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.

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Section: Introductionsupporting

confidence: 60%

Betti maps, Pell equation in polynomials and almost Belyi maps

Barroero,

Capuano,

Zannier

2021

Preprint

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“…After intermediate progress, this result, if we limit to the ground field Q, was recently extended by F. Barroero and L. Capuano, to cover not merely torsion points but also linear relations. They prove in [2], Thm. 1.1, a result which immediately implies the following: Theorem 2.4.…”

Section: 7mentioning

confidence: 86%

“…1.1, a result which immediately implies the following: Theorem 2.4. [ [2], Thm. 1.2] Let A → C be an abelian scheme over a(n affine) curve C defined over a number field and let σ : C → A be a section whose image is not contained in any proper group subscheme.…”

Section: 7mentioning

confidence: 99%

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Finiteness theorems on elliptical billiards and a variant of the Dynamical Mordell-Lang Conjecture

Corvaja,

Zannier

2021

Preprint

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We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in (0, π), there are only finitely many cases for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results have their origin in 'relative' cases of the Manin-Mumford conjecture, and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell-Lang conjecture. In turn, this embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former; here we shall offer an explicit example related to diophantine equations in algebraic tori.

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“…We remark that numerous variations on the theme of Theorem 6 have been studied by Masser and Zannier [41, 37, 39, 40], Barroero and Capuano [4, 2, 3] and Schmidt [49]. These include very interesting applications to the solvability of Pell’s equation in polynomials and to integrability in elementary terms.…”

Section: Diophantine Applicationsmentioning

confidence: 97%

Point counting for foliations over number fields

Binyamini

2022

Forum of Mathematics, Pi

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Let ${\mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${\mathbb K}$ . For an algebraic $V\subset {\mathbb M}$ over ${\mathbb K}$ , write $\delta _{V}$ for the maximum of the degree and log-height of V. Write $\Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${\mathcal B}$ of a leaf ${\mathcal L}$ . We prove effective bounds on the geometry of the intersection ${\mathcal B}\cap V$ . In particular, when $\operatorname {codim} V=\dim {\mathcal L}$ we prove that $\#({\mathcal B}\cap V)$ is bounded by a polynomial in $\delta _{V}$ and $\log \operatorname {dist}^{-1}({\mathcal B},\Sigma _{V})$ . Using these bounds we prove a result on the interpolation of algebraic points in images of ${\mathcal B}\cap V$ by an algebraic map $\Phi $ . For instance, under suitable conditions we show that $\Phi ({\mathcal B}\cap V)$ contains at most $\operatorname {poly}(g,h)$ algebraic points of log-height h and degree g. We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections $P,Q$ of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever $P,Q$ are simultaneously torsion their order of torsion is bounded effectively by a polynomial in $\delta _{P},\delta _{Q}$ ; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given $V\subset {\mathbb C}^{n}$ , there is an (ineffective) upper bound, polynomial in $\delta _{V}$ , for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.

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Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Cited by 13 publications

References 54 publications

Betti maps, Pell equation in polynomials and almost Belyi maps

Betti maps, Pell equation in polynomials and almost Belyi maps

Finiteness theorems on elliptical billiards and a variant of the Dynamical Mordell-Lang Conjecture

Point counting for foliations over number fields

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