Date 28 September 2022

Time 4-5 pm

Venue: Ramanujan Hall

Speaker: Prof. Eknath Ghate, TIFR, Mumbai

Title: Semi-stable representations as limits of crystalline representations

Abstract: We construct an explicit sequence of crystalline representations
converging to a given irreducible two-dimensional semi-stable
representation of the Galois group of Q_p. The convergence takes place in
the blow-up space of two-dimensional trianguline representations studied
by Colmez and Chenevier. It is connected to a classical formula going back
to Greenberg and Stevens expressing the L-invariant as a logarithmic
derivative.

Our convergence result can be used to compute the reductions of any
irreducible two-dimensional semi-stable representation in terms of the
reductions of certain nearby crystalline representations of exceptional
weight. For instance, using our zig-zag conjecture on the reductions of
crystalline representations of exceptional weights, we recover completely
the work of Breuil-Mezard and Guerberoff-Park on the reductions of
irreducible semi-stable representations of weights at most p+1, at least
on the inertia subgroup. As new cases of the zig-zag 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.