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arXiv · 1701.06822

Primes in short intervals on curves over finite fields

Efrat Bank·Tyler Foster
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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.

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Efrat Bank, Tyler Foster. 2017-01-24. Primes in short intervals on curves over finite fields. https://arxiv.org/abs/1701.06822

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