Let \(f(X)\in {\mathbb Z}[X]\) be an irreducible polynomial of positive degree such that no prime \(p\) divides all the values \(f(n)\), \(n\in {\mathbb Z}\). Is has been conjectured long ago, and it is generally believed, that such a polynomial takes on prime values at infinitely many \(n\in {\mathbb Z}\). Consider now the ring \(\kappa[u]\) in place of \({\mathbb Z}\), where \(\kappa\) is a finite field, and assume that \(f(X)\in \kappa[u][X]\) is an irreducible polynomial of positive degree and such that no irreducible polynomial \(\pi[u]\in\kappa[u]\) divides all specializations of \(f\) at \(X=g\in \kappa[u]\). By analogy, one could conjecture that \(f\) has infinitely many irreducible specializations \(f(g)\). However, this conjecture is not true.
The paper under review gives a historical introduction to these problems and the recent results proved in other joint papers of the author. In B. Conrad, K. Conrad and R. Gross[“Irreducible specialization in genus 0”, submitted] explicit polynomials which disprove the conjecture are given, and these polynomials have the property that, when specialized at \(X=g\), it is not equally likely that \(f(g)\) has an even or an odd number of irreducible factors. Taking this into account, and with the help of many examples and tables, the author then adds some hypotheses on the polynomial \(f\) under which he thinks that the conjecture should be true (there is also a quantitative version of the conjecture). The failure of the original conjecture implies that there exist certain families of elliptic curves whose root number has no uniform distribution. The paper ends by discussing an explicit example given in B. Conrad, K. Conrad and H. A. Helfgott [ Adv. Math. 198, No. 2, 684–731 (2005; Zbl 1113.11033)].
For the entire collection see [Zbl 1078.11002].