Title: Prime values of polynomials
Speaker: Professor Brian Conrad
Speaker Info: University of Michigan
Classical probabilistic heuristics due to Hardy and Littlewood (and really Gauss) predict how often one expects an irreducible polynomial over the integers to take on prime values, including the case of several polynomials at once. There are some basic obstructions one must take into account, all of which are local. No cases of such heuristics have ever been proven beyond the case of a single linear polynomial in one variable (essentially the prime number theorem and Dirichlet's theorem), nor have likely counterexamples even been found.Date: Friday, February , 2003
Despite our complete ignorance, one can ask for more. In one direction, one can ask about polynomials in several variables. This turns out to be relatively straightfoward to formulate, except that to make sense of a certain infinite product factor in the asymptotic one needs to use Deligne's generalization of the Riemann Hypothesis for varieties over finite fields. But more interesting phenomena lie just around the corner: there are fruitful analogies between the ring of ordinary integers and the ring of polynomials in one variable over a finite field (the "function field case"), so one can ask whether the classical unproven heuristics have analogues in this setting. Rather amazingly, we'll see that the classical heuristics are not only provably false (for interesting reasons!) in the function field case, but they can be plausibly salvaged in a manner which involves a mixture of algebra and geometry, and required proving some surprising periodicity phenomena with no known classical analogues.
It turns out that the correct conjecture in the function field case involves obstructions which are global and not just local (in contrast to the classical case in integers), and the case of characteristic 2 is particularly vexing. Many illustrative numerical examples will be given, and several concrete open questions will be stated at the end.
No knowledge of algebraic geometry or number theory will be assumed. This is joint work with Keith Conrad and Robert Gross.