Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty

Ellenberg, Jordan S.; Hast, Daniel Rayor

doi:10.48550/arxiv.1706.00525

Cited by 5 publications

(5 citation statements)

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“…(Lorenzini-Tucker [12] had actually already used primes of bad reduction, but they required a regular model of the curve in question, which led to non-uniform bounds.) We mention also the recent results of Ellenberg and Hast [6], which reproves Faltings' theorem for superelliptic curves using non-abelian Chabauty, and those of Katz-Rabinoff-Zureick-Brown [9], which use Stoll's ideas and tropical geometry to prove uniform bounds for general curves satisfying a rank hypothesis.…”

Section: Background and Main Theoremsmentioning

confidence: 95%

Rank-Favorable Bounds for Rational Points on Superelliptic Curves of Small Rank

Kantor¹

2017

Preprint

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Let C be a curve of genus at least three defined over a number field, and let r be the rank of the rational points of its Jacobian. Under mild hypotheses on r, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on C by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on r: when r is small, there should be fewer points on C. In a 2013 paper, Stoll established such a "rank-favorable" bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results to superelliptic curves, noting in the process some differences that ought to inform uniformity conjectures for general curves. Our results have stark implications for bounding numbers of rational points, since r is expected to be small for "most" curves.

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Section: Background and Main Theoremsmentioning

confidence: 95%

Rank-Favorable Bounds for Rational Points on Superelliptic Curves of Small Rank

Kantor¹

2017

Preprint

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“…In [EH18], the dimension hypothesis is proved for a smooth projective hyperbolic curve X over Q such that there exists a smooth projective hyperbolic curve Y /Q with CM Jacobian and a dominant map f : XQ → YQ (corresponding to case (4) of Situation 1.4). We now verify Lemma 5.1 in this setting.…”

Section: Curves Dominating a Curve With CM Jacobianmentioning

confidence: 99%

Functional transcendence for the unipotent Albanese map

Hast

2019

Preprint

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We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields.

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“…Coates and Kim [16] proved that when X/Q is a curve whose Jacobian is isogenous to a product of CM abelian varieties, for n sufficiently large, X(Q p ) n is finite. Recently, Ellenberg and Hast [23] used this to give a new proof of finiteness of X(Q) of any solvable Galois cover X of P 1 (which, for instance, includes the class of superelliptic curves). Even in these cases, it is not clear how to actually compute X(Q p ) n .…”

Section: Introductionmentioning

confidence: 99%