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The leading constant for rational points in families

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arxiv 2210.13559 v5 pith:C4GRGKKE submitted 2022-10-24 math.NT math.AG

Daniel Loughran , Nick Rome , Efthymios Sofos This is my paper · ORCID
classification math.NTmath.AG
keywords numberrationalasymptoticsconicsconjectureconstantcountingdiagonal
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We prove asymptotics for Serre's problem on the number of diagonal planar conics with a rational point and use this to put forward a new conjecture on counting the number of varieties in a family which are everywhere locally soluble.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An improved large sieve for quadratic characters via Hooley neutralisers and its applications

    math.NT 2025-06 conditional novelty 7.0 of 10

    Using Hooley neutralisers, a large sieve for quadratic characters is improved for multiplicatively weighted sequences, with applications to hyperbolic-region character sums.

  2. Rational points in a family of conics over $\mathbb{F}_2(t)$

    math.NT 2024-12 accept novelty 7.0 of 10

    Over F_2(t), the number of parameters y of height 2^M for which the conic x0^2+x0x1+yx1^2=t x2^2 has a rational point is asymptotically c 2^{2M}/M^{1/2}, with an explicit Euler product constant c.

  3. Solubility of a family of conics with polynomial coefficients in many variables

    math.NT 2025-11 reject novelty 6.0 of 10

    An asymptotic for the density of soluble conics with polynomial coefficients is proposed, but the sign decomposition (6.5) that bridges the circle-method counts to the true count is false.

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