Correlation between primes in short intervals on curves over finite fields
In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.
I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting.
I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, a Chebotarev density type theorem.
This is a joint work with Tyler Foster.