A zig-zag conjecture and local constancy for Galois representations (Algebraic Number Theory and Related Topics 2018)

Abstract

We make a zig-zag conjecture describing the reductions of irreducible crystalline twodimensional representations of GQp of half-integral slopes and exceptional weights. Such weights are two more than twice the slope mod (p − 1). We explain how zig-zag can be deduced from known results for half-integral slopes at most 3 2 . We then explore the connection between zig-zag and local constancy results in the weight. We show that known cases of zig-zag force local constancy to fail for small weights, and explain how local constancy forces zig-zag to fail for some small weights and half-integral slopes at least 2. However, we expect zig-zag to be qualitatively true in general. We end with some compatibility results between zig-zag and other results.