Prime Specialization in Higher Genus I

A classical conjecture predicts how often a polynomial in Z[T ] takes prime values. The natural analogous conjecture for prime values of a polynomial f(T ) ∈ κ[u][T ], where κ is a finite field, is false. The conjecture over κ[u] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Möbius function at the values of f on κ[u] when f ∈ κ[u][T ], where p is the characteristic of κ. In this paper, for p 6= 2 we extend the Möbius periodicity results for κ[u] – the affine κ-line – to the case when f has coefficients in the coordinate ring A of any higher-genus smooth affine κ-curve with one geometric point at infinity. The basic strategy is to pull up results from the genus-0 case by means of well-chosen projections to the affine line. Our techniques can also be used to prove nontrivial properties of a correction factor in the conjecture on primality statistics for values of f ∈ A[T ] on A, even as f and κ vary.