Serre weight conjectures for ${\mathrm{GSp}}_4$

Abstract

We prove the weight part of Serre's conjecture for Galois representations valued in ${\mathrm{GSp}}_4$ that are tamely ramified with explicit genericity at places above $p$ as conjectured by Herzig--Tilouine and Gee--Herzig--Savitt. This improves on arXiv:2304.13879 where an inexplicit genericity hypothesis was required. As an application, we prove a modularity lifting theorem for ${\mathrm{GSp}}_4$ under similar assumptions.

arXiv comment43 pages; revised after referee report; main results now hold under the usual residual automorphy assumption instead of potentially diagonalizably residual automorphy; added section 2.3 and 3.6; Comments are welcome!