How many rational points does a random curve have?

Ho, Wei

doi:10.1090/s0273-0979-2013-01433-2

Cited by 6 publications

(6 citation statements)

References 75 publications

(29 reference statements)

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“…The corresponding question over a number field is also central in arithmetic statistics, but has a rather different flavour. See Ho[3] for a comprehensive survey.…”

mentioning

confidence: 99%

A heuristic for the distribution of point counts for random curves over a finite field

Achter

Erman

Kedlaya

et al. 2015

Phil. Trans. R. Soc. A.

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How many rational points are there on a random algebraic curve of large genus g over a given finite field ? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q +1+1/( q −1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g .

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“…The corresponding question over a number field is also central in arithmetic statistics, but has a rather different flavour. See Ho[3] for a comprehensive survey.…”

mentioning

confidence: 99%

A heuristic for the distribution of point counts for random curves over a finite field

Achter

Erman

Kedlaya

et al. 2015

Phil. Trans. R. Soc. A.

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“…When the representation is coregular, meaning that the ring of invariants Q[V ] G is a polynomial ring, they have developed powerful geometry-of-numbers techniques to count integral orbits of V . Combining orbit parametrisations with these counting techniques has led to many striking results; see [BS15a,BS15b,BS13a,BS13b,BG13,BGW17] for some highlights and [Ho14,Bha14b] for surveys of these results.…”

Section: Contextmentioning

confidence: 99%

Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics

Laga¹

2022

Preprint

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A simply laced Dynkin diagram gives rise to a family of curves over Q and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of these representations.In this paper, we complete Thorne's picture and show that 2-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang.The main innovations are: an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck-Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives.

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“…Recent years have witnessed rapid progress on this flavor of problem; results on ranks include [BS13] (elliptic curves), [BG13] (Jacobians of hyperelliptic curves), and [Tho15] (certain families of plane quartics). Combining these rank results with Chabauty's method and other techniques, several recent results [PS14,Bha13,SW13,BGW13] prove that a strong version of the uniformity conjecture (in that there are no "non-obvious" points) holds for "random curves" in such families; see [Ho14] for a recent survey.…”

Section: Bhargavologymentioning

confidence: 99%

Diophantine and tropical geometry, and uniformity of rational points on curves

Katz¹,

Rabinoff²,

Zureick-Brown³

2018

Algebraic Geometry: Salt Lake City 2015

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We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty-Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of p-adic integration, especially to the comparison of analytic continuations of p-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.

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How many rational points does a random curve have?

Cited by 6 publications

References 75 publications

A heuristic for the distribution of point counts for random curves over a finite field

A heuristic for the distribution of point counts for random curves over a finite field

Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics

Diophantine and tropical geometry, and uniformity of rational points on curves

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