For some years now, my collaborators and I have been computing the mod p reductions of two-dimensional local p-adic modular Galois representations at good primes p, using the p-adic and mod p Local Langlands Correspondences. A complete description of the reduction is now available for all weights and primes p at least 5, for slopes at most 2.
Two recent general patterns have emerged from these computations: (1) a zig-zag conjecture that describes the reductions for small half-integral slopes in the tricky case of exceptional weights, and, (2) a pattern showing that some of the reductions in slope v + 1 are related to the reductions in slope v by twisting.
In this talk, we recall some evidence towards (1). We also describe a recent breakthrough proving pattern (2) under an assumption on the weight. The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms (joint work with Arvind Kumar). Finally, we show that the seemingly unrelated patterns (1) and (2) are compatible.