p-adic families of modular forms as constructed first by Hida and in
greater generality by Coleman and their generalizations are a great
tool in modern number theory that appears in different parts of the Langlands program.
In these talks
I will try to explain the geometric construction of these families in the
case of Picard automorphic forms. Geometric means that we will try
to interpolate p-adically the sheaves on the Picard modular surface.
As the local geometry of the surface depends on the behavior of the
prime p in the quadratic field of the Shimura Datum, I will focus in
the case where the prime is inert.
After studying the p-adic geometry of the Picard surface using the
universal family of p-divisible groups that naturally live on it, we will
study the overconvergence of a canonical filtration by analyzing the
variation of the Hodge-Tate maps of these groups. Using this, we can
p-adically interpolate the automorphic (coherent) sheaves on a strict
neighborhood of the mu-ordinary locus, construct a compact p-adic
operator acting on sections of these sheaves and use this datum to
construct the Eignnvariety, a three dimensional p-adic variety that
parametrizes p-adic congruences between overconvergent Picard
If time permit, I will try to explain how we can use this object to
construct classes in some Selmer groups as predicted by the BlochKato
conjecture using a method of Bellaiche and Chenevier.