The weight reduction of mod $p$ Siegel modular forms for $GSp_4$

Abstract

In this paper we investigate the (classical) weights of mod p Siegel modular forms of degree 2 toward studying Serre's conjecture for GSp4. We first construct various theta operators on the space of such forms a la Katz and define the theta cycles for the specific theta operators. Secondly we study the partial Hasse invariants on each Ekedahl-Oort stratum and their local behaviors. This enables us to obtain a kind of weight reduction theorem for mod p Siegel modular forms of degree 2 without increasing the level.

Weights of the mod $p$ kernel of the theta operators

“…(2) Yamauchi [31] concluded the similar statements as in the two theorems above for the case of degree 2, without condition on the smallness of k compared with p, but under a certain geometrical non-vanishing condition. The proof is also algebraic geometrical.…”

“…A new feature in the Siegel case is that one should study also vector valued generalizations Θ [j] of theta operators for 0 ≤ j ≤ n; their p-adic properties were given in [9], e.g. Θ [1] maps a Siegel modular form T a F (T )q T to a formal series T T a F (T )q T with coefficients in symmetric matrices of size n. In this paper, we discuss the necessity (as in the one variable cases) of the relation between the weight and the prime p for an element of the mod p kernel of the generalized theta operators Θ [j] , in the case where the weight is small compared with p. We remark that Yamauchi [31] and Weissauer [29] also studied the necessity in the special cases Θ [1] or Θ. Moreover we construct elements of the mod p kernel of Θ [j] from arbitrary modular form.…”

Serre weights for $GSp_4$ over totally real fields

“…Klingen ordinary case. In this case, it follows from Proposition 7.2 of[99] that ρ ρ ≤ b < a ≤ p and ρ 2 ρ * 1 . We may assume * is non-trivial and it belongs toH 1 (Q p , ψ −ε −c ad 0 (ρ 1 )) since the unipotent radical of the Siegel parabolic subgroup has the structure of Sym 2 (St 2 ).…”

“…We regard ρ as a representation to GL 4 (F p ) and denote by ρ ss the semi-simplification of ρ. By Proposition 7.2 of [99] we have five types:…”

“…Borel ordinary case. In this case, it follows from Proposition 7.2 of[99] thatρ ε c ψ 0 ⊗ ρ 1 B 0 2 ψ 1 ε a+b ρ *Here a and for each 0≤ i ≤ 2, ψ i : G Qp −→ F × p is an unramified character and 0 ≤ a, b, c ≤ p − 2. Notice that the class of B belongs to H 1 (Q p , (ψ 1 ε a+b ) −1 ad 0 (ρ 1 )) where the space (ψ 1 ε a+b ) −1 ad 0 (ρ 1 ) is of dimension three.…”