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Geometric generalizations of the square sieve, with an application to cyclic covers

Publication ,  Journal Article
Bucur, A; Cojocaru, AC; Lalín, MN; Pierce, LB
Published in: Mathematika: a journal of pure and applied mathematics
2022

We formulate a general problem: given projective schemes $\mathbb{Y}$ and $\mathbb{X}$ over a global field $K$ and a $K$-morphism $\eta$ from $\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in $\mathbb{X}(K)$ of height at most $B$ have a pre-image under $\eta$ in $\mathbb{Y}(K)$? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a non-trivial answer to the general problem when $K=\mathbb{F}_q(T)$ and $\mathbb{Y}$ is a prime degree cyclic cover of $\mathbb{X}=\mathbb{P}_{K}^n$. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.

Duke Scholars

Published In

Mathematika: a journal of pure and applied mathematics

ISSN

0025-5793

Publication Date

2022

Publisher

Wiley

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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Bucur, A., Cojocaru, A. C., Lalín, M. N., & Pierce, L. B. (2022). Geometric generalizations of the square sieve, with an application to cyclic covers. Mathematika: A Journal of Pure and Applied Mathematics.
Bucur, Alina, Alina Carmen Cojocaru, Matilde N. Lalín, and Lillian B. Pierce. “Geometric generalizations of the square sieve, with an application to cyclic covers.” Mathematika: A Journal of Pure and Applied Mathematics, 2022.
Bucur A, Cojocaru AC, Lalín MN, Pierce LB. Geometric generalizations of the square sieve, with an application to cyclic covers. Mathematika: a journal of pure and applied mathematics. 2022;
Bucur, Alina, et al. “Geometric generalizations of the square sieve, with an application to cyclic covers.” Mathematika: A Journal of Pure and Applied Mathematics, Wiley, 2022.
Bucur A, Cojocaru AC, Lalín MN, Pierce LB. Geometric generalizations of the square sieve, with an application to cyclic covers. Mathematika: a journal of pure and applied mathematics. Wiley; 2022;
Journal cover image

Published In

Mathematika: a journal of pure and applied mathematics

ISSN

0025-5793

Publication Date

2022

Publisher

Wiley

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics