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

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

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## Publication Date

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## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics

### Citation

*Mathematika: A Journal of Pure and Applied Mathematics*.

*Mathematika: A Journal of Pure and Applied Mathematics*, 2022.

*Mathematika: A Journal of Pure and Applied Mathematics*, Wiley, 2022.

## Published In

## ISSN

## Publication Date

## Publisher

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics