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) epsilon [u][T] where k is a finite field, is false. The conjecture over k[u] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Mobius function at the values of f on k [u] when f epsilon k[u][T-p], where p is the characteristic of k. In this paper, for p not equal 2 we extend the Mobius periodicity results for k[u]-the affine k-line-to the case when f has coefficients in the coordinate ring A of any higher-genus smooth affine k-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 epsilon A[T-p] on A, even as f and k vary.
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机译：一个经典的推测可以预测Z [T]中的多项式取质数的频率。多项式f（T）epsilon [u] [T]的素数的自然相似猜想是错误的，其中k是有限域。通过引入校正因子对k [u]上的猜想进行修正，该校正因子在f epsilon k [u] [Tp]时在k [u]上的f值处对Mobius函数的意外周期性进行编码，其中p为k的特征在本文中，对于p不等于2的情况，我们将k [u]仿射k线的Mobius周期结果扩展到当f在任何更高阶光滑仿射k曲线的坐标环A中具有系数的情况下在无穷远处有一个几何点。基本策略是通过精心选择的仿射线预测，从0属案例中得出结果。我们的技术还可以用于证明关于f上的f epsilon A [T-p]的值的素数统计猜想中的校正因子的非平凡性质，即使f和k变化也是如此。
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