# Primes in short intervals on curves over finite fields

## TL;DR

This paper establishes an analogue of the Prime Number Theorem for short intervals on algebraic curves over finite fields, providing a key step towards understanding prime distributions in function fields.

## Contribution

It proves an asymptotic count of irreducible elements in short intervals on curves over finite fields, extending previous results and addressing a conjecture in the function field setting.

## Key findings

- Asymptotic formula for irreducible elements in short intervals
- Uniform results in the large q limit
- Extension of previous theorems to arbitrary genus curves

## Abstract

We prove an analogue of the Prime Number Theorem for short intervals on a smooth projective geometrically irreducible curve of arbitrary genus over a finite field. A short interval "of size E" in this setting is any additive translate of the space of global sections of a sufficiently positive divisor E by a suitable rational function f. Our main theorem gives an asymptotic count of irreducible elements in short intervals on a curve in the "large q" limit, uniformly in f and E. This result provides a function field analogue of an unresolved short interval conjecture over number fields, and extends a theorem of Bary-Soroker, Rosenzweig, and the first author, which can be understood as an instance of our result for the special case of a divisor E supported at a single rational point on the projective line.