


Date 28 September 2022
Time 45 pm
Venue: Ramanujan Hall
Speaker: Prof. Eknath Ghate, TIFR, Mumbai
Title: Semistable representations as limits of crystalline representations
Abstract: We construct an explicit sequence of crystalline representations
converging to a given irreducible twodimensional semistable
representation of the Galois group of Q_p. The convergence takes place in
the blowup space of twodimensional trianguline representations studied
by Colmez and Chenevier. It is connected to a classical formula going back
to Greenberg and Stevens expressing the Linvariant as a logarithmic
derivative.
Our convergence result can be used to compute the reductions of any
irreducible twodimensional semistable representation in terms of the
reductions of certain nearby crystalline representations of exceptional
weight. For instance, using our zigzag conjecture on the reductions of
crystalline representations of exceptional weights, we recover completely
the work of BreuilMezard and GuerberoffPark on the reductions of
irreducible semistable representations of weights at most p+1, at least
on the inertia subgroup. As new cases of the zigzag conjecture are
proved, we further obtain some new information about the reductions for
small odd weights.
Finally, we use the above ideas to explain away some apparent violations
to local constancy in the weight of the reductions of crystalline
representations of small weight that were noted in our earlier work and
which provided the initial impetus for this work.
This is joint work with Anand Chitrao and Seidai Yasuda.