Key points are not available for this paper at this time.
Let F be a totally real field of even degree in which p splits completely. Let r: GF₄ (Fₚ) be a modular Galois representation unramified at all finite places away from p and upper-triangular, maximally nonsplit, and of parallel weight at places dividing p. Fix a place w dividing p. Assuming certain genericity conditions and Taylor–Wiles assumptions, we prove that the GSp₄ (Fₖ) -action on the corresponding Hecke-isotypic part of the space of mod p automorphic forms on a compact mod center form of GSp₄ with infinite level at w determines r|₆_₅ₖ.
Enns et al. (2024) studied this question.