Number Theory Seminar

The catalog of special functions in real and complex analysis is largely
constructed by solving ordinary differential equations. In number
theory, solutions of p-adic differential equations also play an
important role; for instance, as discovered by Dwork in the 1960s, zeta
functions of algebraic varieties over finite fields can often be
described in terms of solutions of p-adic differential equations.
However, convergence of these solutions is in many respects a subtler
question than in the archimedean case. We describe an emerging theory of
"Newton polygons" for p-adic differential equations, which combines over
50 years of prior work with some recent innovations introduced in work
of Baldassarri, Poineau, Pulita, and the speaker.