Title:  Asymptotics related to prime specialization and average
behavior of the Mobius function over finite fields </br>

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Abstract:  </br>
A few years ago, in joint work with K. Conrad and R. Gross,
some surprising
behavior was discovered for the statistics of prime specialization of
generically inseparable irreducible polynomials
over coordinate rings of curves over finite fields, and it was
heuristically explained via some new
periodicity properties for certain average values of the Mobius
function on such coordinate rings.  More recent work has revealed a
better understanding of the situation, especially a connection with
point-counting over finite fields and the
intrinsic geometric meaning of parameters in some formulas.  This
allows one to prove
appealing results concerning the asymptotic structure of these Mobius
averages (in two natural
senses).  The main techniques are deformation theory and the product
formula for G_m-valued local symbols, together with the Lang-Weil
estimate in families.
I will explain the motivation for these questions and indicate why the
asymptotic results are interesting;
in the remaining time I will say something about the proofs.
Since the case of characteristic 2 requires much heavier geometric
machinery (2-adic rigid and
formal geometry), in this talk I will stick with the case of odd
characteristic.