2021/09/28 by Anand Chitrao, Eknath Ghate, Chitrao, Anand +3 · 1 citation
Mathematics · #11F80 (Primary) 14G22 (Secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2109.13676
openalex publication_date 2021/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct an explicit sequence Vkn,an of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation V_k,L of Gal(ℚp/ℚp). The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid analytic setting and may be of independent interest. Also, we recover a formula of Stevens expressing the L-invariant as a logarithmic derivative. Our result can be used to compute the reduction of V_k,L in terms of the reductions of the Vkn,an. For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil-Mézard and Guerberoff-Park computing the reductions of the V_k,L for weights at most p-1 (resp. p+1), at least on the inertia subgroup. In the cases where zig-zag is known, we are further able to obtain some new information about the reductions for small odd weights. Finally, we explain some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight.