Serre weights for $GSp_4$ over totally real fields

Abstract

We prove a variant of Serre's weight conjecture for ρ : Gal(F /F ) −→ GSp 4 (Fp) for any totally real field F and any rational prime p > 2 by using automorphic lifting techniques developed by Barnet-Lamb, Gee, Geraghty, and Taylor. The formulation of our Serre conjecture is done by following Toby Gee's philosophy. Applying these results to the case when F = Q with a detailed study of potentially diagonalizable, crystalline lifts with some prescribed properties, we also define classical (naive) Serre's weights. This weight would be the minimal weight among possible classical weights in some sense which occur in candidates of holomorphic Siegel Hecke eigen cusp forms of degree 2 with levels prime to p. The main task is to construct a potentially diagonalizable automorphic lift for ρ by assuming only the adequacy condition. The main theorems in this paper also extend many results obtained by Gee and Geraghty [6] for potentially ordinary lifts and Gee and Geraghty for companion forms [36].

A new theta cycle for $GSp_4$ and an Edixhoven type theorem

“…We will give details of the cycle for any f satisfying some condition to guarantee the non-vanishing of Θ i (f ) for any positive integer i. Combining with the automorphy lifting theorems due to Gee-Geraghyty [9] which are extended by the author [22] we prove the following:…”

“…For a character χ : G Qp −→ F × p , a class of the Galois cohomology H 1 (G Qp , χ) is said to be très ramifiée (ramified) if χ = ε and it is a très ramifiée (ramified) class in the sense of Definition 9.10 of [22]. If a class is neither très ramifiée nor ramified, then we say it peu ramifiée.…”

“…Then we have the dual basis e * ur , e * ram of H 1 (G Qp , F(ε)). For each class α of H 1 (G Qp , M ) we also define Since τ 1 is liftable to a cyrtalline extension since ρ 1 is irreducible by Proposition 9.3.2 of [22].…”

“…Then the classical Serre weight will be closely related to the Hodge-Tate weights of a potentially diagonalizable crystalline lift of ρ| G Qp which preserves * -ordinary. In [22], the classical Serre weights are defined but not explicit in some cases. Here we give a precise definition as in [17], [4].…”

“…This is a preliminary to define the classical Serre weights. Foreach character χ : G Qp −→ F × p and each class α ∈ H 1 (G Qp , χ), put (4.1)r χ,α := p − 1 α is très ramifiée or ramified 0 otherwise (see Definition 9.14 of[22]). For integers i, j and a prime p, we define an element of Z 2 byδ p ij := (p − 1, p − 1) if i = j (0, 0) otherwise.Let F be a finite extension ofF p .…”