Primes in short intervals on curves over finite fields

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. CONTENTS

Correlations between primes in short intervals on curves over finite fields

“…In recent work [2], Bary-Soroker and the first author use their work with Rosenzweig [3] on the asymptotic distribution of primes inside short intervals in F q [t] to establish a natural counterpart, over the field F q (t), to the still unsolved Hardy-Littlewood conjecture. In [4], the two present authors show how to extend the results of [3] to give asymptotic distributions of primes in short intervals on the complement of a very ample divisor in a smooth, geometrically irreducible projective curve C over a finite field F q . In the present paper, we apply ideas from [2] and [4] to prove a natural counterpart to the Hardy-Littlewood conjecture on the complement of a very ample divisor in a smooth, geometrically irreducible projective curve C over a finite field F q .…”

“…In [4], the two present authors show how to extend the results of [3] to give asymptotic distributions of primes in short intervals on the complement of a very ample divisor in a smooth, geometrically irreducible projective curve C over a finite field F q . In the present paper, we apply ideas from [2] and [4] to prove a natural counterpart to the Hardy-Littlewood conjecture on the complement of a very ample divisor in a smooth, geometrically irreducible projective curve C over a finite field F q .…”

“…By the same reasoning as in [4,Proposition 4.3.3], it suffices to prove that det(u, v) cannot be 0 identically on U ξη .…”

“…As in the proof of [4,Proposition 4.3.3], choose an effective very ample divisor E 0 satisfying (3), define m 0 def = dim H 0 C, O(E 0 ) − 1, and let C ⊂ P m 0 be the closed embedding determined by E 0 .…”

“…The latter provides an asymptotic formula for the number of F q -valued points a ∈ A m+1 (F q ) for which each of the elements L 1 (F a ), ..., L n (F a ) generates a prime ideal in R. The formulation of Proposition 3.1.4 should be viewed as an explicit Chebotarev theorem. Our proof makes use of [1, Appendix A and Theorem 3.1] and [4,Proposition 5.1.4].…”