Sequences of powers with second differences equal to two and hyperbolicity

Garcia-Fritz, Natalia

doi:10.1090/tran/7040

Cited by 7 publications

(14 citation statements)

References 38 publications

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“…The particular case of Problem 2 when we require the second differences to be equal to two, was studied in the author's thesis [11], and in [12]. However, such a result is not enough to reach the applications obtained in the present paper; for example, it does not have consequences for Mohanty's conjecture, although it has some other applications of independent interest related to sharpness of bounds, cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 85%

“…Since the geometric part of this paper uses the method discussed in the previous paragraph, some of the routinary computations (such as verifying irreducibility and smoothness of varieties, or computing the genus of some curves) will be similar to the analogous steps in the applications considered in [25] and [12]. However, the varieties considered in this work are different, and therefore we have included the computations when necessary.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…However, such a result is not enough to reach the applications obtained in the present paper; for example, it does not have consequences for Mohanty's conjecture, although it has some other applications of independent interest related to sharpness of bounds, cf. [12].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this section we recall the method given in Chapter 3 of [11], which is an extension of the method implicit in Vojta's work [25]. As in [12], we include some improvements to [25] which allow us to obtain better bounds in applications.…”

Section: Outline Of the Methodsmentioning

confidence: 99%

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Quadratic sequences of powers and Mohanty’s conjecture

Garcia-Fritz

2018

Int. J. Number Theory

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We prove under the Bombieri-Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by J. Browkin and J. Brzezinski, and independently by E. Gonzalez-Jimenez and X. Xarles.We also show that under the Bombieri-Lang conjecture for surfaces, for every k ≥ 3 there is an absolute bound on the length of sequences formed by k-th powers with constant second differences. This gives a conditional result on one of Mohanty's conjectures on arithmetic progressions in Mordell's elliptic curves y 2 = x 3 + b. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form y 2 = x k + b.These results are proved by unconditionally finding all curves of genus 0 or 1 on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

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Section: Introduction and Main Resultsmentioning

confidence: 85%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Outline Of the Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Quadratic sequences of powers and Mohanty’s conjecture

Garcia-Fritz

2018

Int. J. Number Theory

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“…The classical analytic approach is not useful for us and we need purely algebraic methods. Fortunately such a study was carried out by Garcia-Fritz [26,27,28] in the context of her generalization of Vojta's explicit version of Bogomolov's approach to quasi-hyperbolicity, see [64,18]. While Garcia-Fritz's work is mostly over C, her algebraic methods easily extend to an arbitrary base.…”

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confidence: 99%

A Chabauty-Coleman bound for surfaces

Caro,

Pasten

2021

Preprint

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Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve C of genus g ≥ 2 defined over a number field F , with Jacobian of rank at most g − 1. Namely, in the case F = Q, if p > 2g is a prime of good reduction, then the number of rational points of C is at most the number of Fp-points plus a contribution coming from the canonical class of C. We prove a result analogous to Coleman's bound in the case of a hyperbolic surface X over a number field, embedded in an abelian variety A of rank at most one, under suitable conditions on the reduction type at the auxiliary prime. This provides the first extension of Coleman's explicit bound beyond the case of curves. The main innovation in our approach is a new method to study the intersection of a p-adic analytic subgroup with a subvariety of A by means of overdetermined systems of differential equations in positive characteristic.

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Families of explicit quasi-hyperbolic and hyperbolic surfaces

Garcia-Fritz

Urzúa

2019

Math. Z.

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We construct explicit families of quasi-hyperbolic and hyperbolic surfaces. This is based on earlier work of Vojta, and the recent expansion and generalization of it by the first author. In this paper we further extend it to the singular case, obtaining results for the surface of cuboids, the generalized surfaces of cuboids, and other families of Diophantine surfaces of general type. In particular, we produce explicit families of smooth complete intersection surfaces of multidegrees (m1, . . . , mn) in P n+2 which are hyperbolic, for any n ≥ 8 and any degrees mi ≥ 2. We also show similar results for complete intersection surfaces in P n+2 for n = 4, 5, 6, 7. These families give evidence for [Dem18, Conjecture 0.18] in the case of surfaces.

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Sequences of powers with second differences equal to two and hyperbolicity

Cited by 7 publications

References 38 publications

Quadratic sequences of powers and Mohanty’s conjecture

Quadratic sequences of powers and Mohanty’s conjecture

A Chabauty-Coleman bound for surfaces

Families of explicit quasi-hyperbolic and hyperbolic surfaces

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