Reductions of Galois Representations

for Slopes in (1, 2)

Shalini Bhattacharya, Eknath Ghate

Received: April 27, 2015 Revised: July 20, 2015

Communicated by Don Blasius

Abstract. We describe the semisimplifications of the mod p reductions of certain crystalline two-dimensional local Galois representations of slopes in (1, 2) and all weights. The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences for GL_{2}(ℚ_{p}). We also give a complete description of the submodules generated by the second highest monomial in the mod p symmetric power representations of GL_{2}(𝔽_{p}).

2010 Mathematics Subject Classification: Primary: 11F80

Keywords and Phrases: Reductions of Galois representations, Local Langlands Correspondence, Hecke operators.

1. Introduction

Let p be an odd prime. In this paper we study the reductions of two-dimensional crystalline p-adic representations of the local Galois group G_{ℚ_{p}}. The answer is known when the weight k is smaller than 2p + 1 [E92], [B03a], [B03b] or when the slope is greater than ⌊\frac{k−2}{p−1}⌋ [BLZ04]. The answer is also known if the slope is small, that is, in the range (0, 1) [BG09], [G10], [BG13]. Here we treat the next range of fractional slopes (1, 2), for all weights k ≥ 2. Let E be a finite extension field of ℚ_{p} and let v be the valuation of ℚ̄_{p} normalized so that v(p) = 1. Let a_{p} ∈ E with v(a_{p}) > 0 and let k ≥ 2. Let V_{k,a_{p}} be the irreducible crystalline representation of G_{ℚ_{p}} with Hodge-Tate weights (0, k − 1) such that D_{cris}(V_{k,a_{p}}^{∗}) = D_{k,a_{p}}, where D_{k,a_{p}} = Ee_{1} ⊕ Ee_{2} is the filtered φ-module as defined in [B11, §2.3]. Let V̄_{k,a_{p}} be the semisimplification of the reduction of V_{k,a_{p}}, thought of as a representation over 𝔽̄_{p}. Let ω = ω_{1} and ω_{2} denote the fundamental characters of level 1 and 2 respectively, and let ind(ω_{2}^{a}) denote the representation of G_{ℚ_{p}} obtained by inducing the character ω_{2}^{a} from G_{ℚ_{p^{2}}} . Let unr(x) be the unramified character of G_{ℚ_{p}} taking (geometric) Frobenius at p to x ∈ 𝔽̄_{p}^{∗}. Then, a priori, V̄_{k,a_{p}} is isomorphic either to ind(ω_{2}^{a}) ⊗ unr(λ) or unr(λ)ω^{a} ⊕ unr(μ)ω^{b}, for some a, b ∈ ℤ and λ, μ ∈ 𝔽̄_{p}^{∗}. The former representation is irreducible on G_{ℚ_{p}} when (p + 1) ∤ a, whereas the latter is reducible on G_{ℚ_{p}}. The following theorem describes V̄_{k,a_{p}} when 1 < v(a_{p}) < 2. Since the answer is known completely for weights k ≤ 2p + 1, we shall assume that k ≥ 2p + 2.

Theorem 1.1. Let p ≥ 3. Let 1 < v(a_{p}) < 2 and k ≥ 2p + 2. Let r = k − 2 ≡ b mod (p − 1), with 2 ≤ b ≤ p. When b = 3 and v(a_{p}) = \frac{3}{2}, assume that

(⋆)  v(a_{p}^{2} − \binom{r − 1}{2}(r − 2)p^{3}) = 3.

Then, V̄_{k,a_{p}} has the following shape on G_{ℚ_{p}}:

b = 2 ⟹ \begin{cases}ind(ω_{2}^{b+1}), & if p ∤ r(r − 1) \\ ind(ω_{2}^{b+p}), & if p | r(r − 1), \end{cases}

3 ≤ b ≤ p − 1 ⟹ \begin{cases}ind(ω_{2}^{b+p}), & if p ∤ r − b \\ ind(ω_{2}^{b+1}), & if p | r − b, \end{cases}

b = p ⟹ \begin{cases}ind(ω_{2}^{b+p}), & if p^{2} ∤ r − b \\ unr(\sqrt{−1})ω ⊕ unr(−\sqrt{−1})ω, & if p^{2} | r − b. \end{cases}

Using the theorem, and known results for 2 ≤ k ≤ 2p + 1, we obtain:

Corollary 1.2. Let p ≥ 3. If k ≥ 2 is even and v(a_{p}) lies in (1, 2), then V̄_{k,a_{p}} is irreducible.

It is in fact conjectured [BG15, Conj. 4.1.1] that if k is even and v(a_{p}) is nonintegral, then the reduction V̄_{k,a_{p}} is irreducible on G_{ℚ_{p}}. This follows for slopes in (0, 1) by [BG09]. Theorem 1.1 shows that V̄_{k,a_{p}} can be reducible on G_{ℚ_{p}} for slopes in (1, 2) only when b = p or b = 3 (or both). Since k is clearly odd in these cases, the corollary follows. Let ρ_{f} : Gal(ℚ̄/ℚ) → GL_{2}(E) denote the global Galois representation attached to a primitive cusp form f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{0}(N)) of (even) weight k ≥ 2 and level N coprime to p. It is known that ρ_{f}|_{G_{ℚp}} is isomorphic to V_{k,a_{p}}, at least if a_{p}^{2} ≠ 4p^{k−1}. This condition always holds for slopes in (1, 2) except possibly when k = 4 and a_{p} = ±2p^{\frac{3}{2}} . Since it is expected to hold generally, we assume it. We obtain:

Corollary 1.3. Let p ≥ 3. If the slope of f at p lies in (1, 2), then ρ̄_{f}|_{G_{ℚp}} is irreducible.

Remark 1.4. Here are several remarks.

• Theorem 1.1 treats all weights for slopes in the range (1, 2), subject to a hypothesis. It builds on [GG15, Thm. 2], which treated weights less than p^{2} − p.

Reductions of Galois Representations for Slopes in (1, 2) 945

• The hypothesis (⋆) in the theorem applies only when b = 3 and v(a_{p}) =

\frac{3}{2} and is mild in the sense that it holds whenever the unit \frac{a_{p}^{2}}{p^{3}} and \binom{r−1}{2}(r − 2) have distinct reductions in 𝔽̄_{p}.

• The theorem agrees with all previous results for weights 2 < k ≤ 2p+1 described in [B11, Thm. 5.2.1] when specialized to slopes in (1, 2). It could therefore be stated for all weights k > 2. We note that (⋆) is satisfied for weights k ≤ 2p + 1, except possibly for k = 5.

• When b = p and p^{2} | r − b, the theorem shows that V̄_{k,a_{p}} is always reducible if p ≥ 5 (and under the hypothesis (⋆) when p = 3). This is a new phenomenon not occurring for slopes in (0, 1). When b = 3, v(a_{p}) = \frac{3}{2} and (⋆) fails, we expect that V̄_{k,a_{p}} might also be reducible in some cases, by analogy with the main result of [BG13].

• Fix k, a_{p} and b = b(k) as in Theorem 1.1. Then the theorem implies the following local constancy result: for any other weight k′ ≥ 2p + 2 with k′ ≡ k mod p^{1+v(b)}(p − 1), the reduction V̄_{k′,a_{p}} is isomorphic to V̄_{k,a_{p}}, except possibly if v(a_{p}) = \frac{3}{2} and b = 3. We refer to [B12, Thm. B] for a general local constancy result for any positive slope.

The proof of Theorem 1.1 uses the p-adic and mod p Local Langlands Correspondences due to Breuil, Berger, Colmez, Dospinescu, Paškūnas [B03a], [B03b], [BB10], [C10], [CDP14], [P13], and an important compatibility between them with respect to the process of reduction [B10]. The general strategy is due to Breuil and Buzzard-Gee and is outlined in [B03b], [BG09], [GG15]. We briefly recall it now and explain several new obstacles we must surmount along the way. Let G = GL_{2}(ℚ_{p}), K = GL_{2}(ℤ_{p}) be the standard maximal compact subgroup of G and Z = ℚ_{p}^{∗} be the center of G. Consider the locally algebraic representation of G

Π_{k,a_{p}} = \frac{ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2}}{T − a_{p}} ,

where r = k − 2, ind_{KZ}^{G} is compact induction and T is the Hecke operator, and consider the lattice in Π_{k,a_{p}} given by

(1.1)  Θ_{k,a_{p}} := image (ind_{KZ}^{G}Sym^{r}ℤ̄_{p}^{2} → Π_{k,a_{p}})
≅ \frac{ind_{KZ}^{G}Sym^{r}ℤ̄_{p}^{2}}{(T − a_{p})(ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2}) ∩ ind_{KZ}^{G}Sym^{r}ℤ̄_{p}^{2}} .

It is known that the semisimplification of the reduction of this lattice satisfies Θ̄_{k,a_{p}}^{ss} ≃ LL(V̄_{k,a_{p}}), where LL is the (semisimple) mod p Local Langlands Correspondence of Breuil [B03b]. One might require here the conditions a_{p}^{2} ≠ 4p^{k−1} and a_{p} ≠ ±(1 + p)p^{(k−2)/2}, see [BB10], but these clearly hold if k ≥ 2p + 2 and v(a_{p}) < 2. By the injectivity of the mod p Local Langlands Correspondence, Θ̄_{k,a_{p}}^{ss} determines V̄_{k,a_{p}} completely, and so it suffices to compute Θ̄_{k,a_{p}}.

Let V_{r} = Sym^{r}𝔽̄_{p}^{2} be the usual symmetric power representation of Γ := GL_{2}(𝔽_{p}) (hence of KZ, with p ∈ Z acting trivially). Clearly there is a surjective map

ind_{KZ}^{G}V_{r} ↠ Θ̄_{k,a_{p}}, (1.2)

for r = k − 2. Write X_{k,a_{p}} for the kernel. A model for V_{r} is the space of all homogeneous polynomials of degree r in the two variables X and Y over 𝔽̄_{p} with the standard action of Γ. Let X_{r−1} ⊂ V_{r} be the Γ- (hence KZ-) submodule generated by X^{r−1}Y. Let V_{r}^{∗} and V_{r}^{∗∗} be the submodules of V_{r} consisting of polynomials divisible by θ and θ^{2} respectively, for θ := X^{p}Y − XY^{p}. If r ≥ 2p + 1, then Buzzard-Gee have shown [BG09, Rem. 4.4]:

• v(a_{p}) > 1 ⟹ ind_{KZ}^{G} X_{r−1} ⊂ X_{k,a_{p}},
• v(a_{p}) < 2 ⟹ ind_{KZ}^{G} V_{r}^{∗∗} ⊂ X_{k,a_{p}}.

The proof of Theorem 1.1 for k = 2p+2 is known (cf. [GG15, §2]) and involves slightly different techniques, so for the rest of this introduction assume that r ≥ 2p + 1. It follows that when 1 < v(a_{p}) < 2, the map (1.2) induces a surjective map ind_{KZ}^{G} Q ↠ Θ̄_{k,a_{p}}, where

Q := \frac{V_{r}}{X_{r−1} + V_{r}^{∗∗}} .

To proceed further, one needs to understand the ‘final quotient’ Q. It is not hard to see that a priori Q has up to 3 Jordan-Hölder factors as a Γ-module. The exact structure of Q is derived in §3 to §6 by giving a complete description of the submodule X_{r−1} and understanding to what extent it intersects with V_{r}^{∗∗}. When 0 < v(a_{p}) < 1, the relevant ‘final quotient’ in [BG09] is always irreducible allowing the authors to compute the reduction (up to separating out some reducible cases) using the useful general result [BG09, Prop. 3.3]. When 1 < v(a_{p}) < 2, we show Q is irreducible if and only if

• b = 2, p ∤ r(r − 1) or b = p, p ∤ r − b, and we obtain V̄_{k,a_{p}} immediately in these cases (Theorem 8.1). Generically, the quotient Q has length 2 when 1 < v(a_{p}) < 2. In fact, we show that Q has exactly two Jordan-Hölder factors, say J and J′, in the cases complementary to those above

• b = 2, p | r(r − 1) or b = p, p | r − b, as well as in the generic case

• 3 ≤ b ≤ p − 1 and p ∤ r − b. We now use the Hecke operator T to ‘eliminate’ one of J or J′. Something similar was done in [B03b] and [GG15] for bounded weights. That this can be done for all weights is one of the new contributions of this paper (see §8). It involves constructing certain rational functions f ∈ ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2}, such that (T − a_{p})f ∈ ind_{KZ}^{G}Sym^{r}ℤ̄_{p}^{2} is integral, with reduction mapping to a simple function in say ind_{KZ}^{G} J′ that generates this last space of functions as a Gmodule. As (T −a_{p})f lies in the denominator of the expression (1.1) describing Θ_{k,a_{p}}, its reduction lies in X_{k,a_{p}}. Thus we obtain a surjection ind_{KZ}^{G} J ↠ Θ̄_{k,a_{p}} and can apply [BG09, Prop. 3.3] again. For instance, let

J_{1} = V_{p−b+1} ⊗ D^{b−1} and J_{2} = V_{p−b−1} ⊗ D^{b},

where D denotes the determinant character. Then in the latter (generic) case above, Q ≅ J_{1} ⊕J_{2} is a direct sum. We construct a function f which eliminates J′ = J_{2} so that J = J_{1} survives, showing that V̄_{k,a_{p}} ≅ ind(ω_{2}^{b+p}) (Theorem 8.3). The situation is more complicated when Q has 3 Jordan-Hölder factors, namely J_{0} = V_{b−2} ⊗ D, in addition to J_{1} and J_{2} above. That this happens at all came as a surprise to us since it did not happen in the range of weights considered in [GG15]. We show that this happens for the first time when r = p^{2} − p + 3, and in general whenever

• 3 ≤ b ≤ p − 1 and p | r − b. This time we construct functions f killing J_{0} and J_{1} (except when b = 3, v(a_{p}) = \frac{3}{2} and v(a_{p}^{2} − p^{3}) > 3), so that J_{2} survives instead, and the reduction becomes ind(ω_{2}^{b+1}) (Theorems 8.6, 8.7). Since J_{2} was also the ‘final quotient’ in [BG09], the reduction in these cases is the same as the generic answer obtained for slopes in (0, 1). As a final twist in the tale, we remark that even though one can eliminate all but 1 Jordan-Hölder factor J, one needs to further separate out the reducible cases when J = V_{p−2} ⊗ D^{n}, for some n. This happens in three cases:

• b = 3, p ∤ r − b,

• b = p = 3, p || r − b,
• b = p, p^{2} | r − b.

In §9 we construct additional functions and use them to show that the map ind_{KZ}^{G} J ↠ Θ̄_{k,a_{p}} factors either through the cokernel of T or the cokernel of T^{2} − cT + 1, for some c ∈ 𝔽̄_{p}, and then apply the mod p Local Langlands Correspondence directly to compute V̄_{k,a_{p}}, as was done in [B03b], [BG13]. In the first two cases, we show that the map above factors through the cokernel of T so that the reducible case never occurs. We work under the assumption (⋆), namely if v(a_{p}) = \frac{3}{2}, then v(a_{p}^{2} − \binom{r−1}{2}(r − 2)p^{3}) is equal to 3, which is the generic sub-case (Theorem 9.1). On the other hand, in the third case we show that if p ≥ 5 or if p = 3 = b and (⋆) holds, then the map factors through the cokernel of T^{2} + 1, so that V̄_{k,a_{p}} is reducible and is as in Theorem 1.1 (Theorem 9.2). One of the key ingredients that go into the proof of Theorem 1.1 is a complete description of the structure of the submodule X_{r−1} of V_{r}. We give its structure now as the result might be of some independent interest. To avoid technicalities, we state the following theorem in a weaker form than what we actually prove. Let M := M_{2}(𝔽_{p}) be the semigroup of all 2 × 2 matrices over 𝔽_{p} and consider V_{r} as a representation of M, with the obvious extension of the action of Γ = GL_{2}(𝔽_{p}) on it.

Theorem 1.5. Let p ≥ 3. Let r ≥ 2p + 1 and let X_{r−1} = ⟨X^{r−1}Y ⟩ be the M-submodule of V_{r} generated by the second highest monomial. Then 2 ≤ length X_{r−1} ≤ 4, as an M-module. More precisely, if 2 ≤ b ≤ p − 1, then X_{r−1} fits into the exact sequence of M-modules

V_{p−b+1} ⊗ D^{b−1} ⊕ V_{p−b−1} ⊗ D^{b} → X_{r−1} → V_{b−2} ⊗ D ⊕ V_{b} → 0,

and if b = p, then

V_{1} ⊗ D^{p−1} → X_{r−1} → W → 0,

where W is a quotient of the length 3 M-module V_{2p−1}.

Theorem 1.5 is proved for representations defined over 𝔽_{p} in §3 and §4 using results of Glover [G78]. Here we have stated the corresponding result after extending scalars to 𝔽̄_{p}. We recall that V_{r}^{∗} is the largest singular M-submodule of V_{r} [G78, (4.1)]. It is the M-module structure of X_{r−1} given in the theorem rather than just the Γ-module structure that plays a key role in understanding how X_{r−1} intersects with V_{r}^{∗} and V_{r}^{∗∗}. A more precise description of the structure of X_{r−1} can be found in Propositions 3.13 and 4.9. There we show that the Jordan-Hölder factors in Theorem 1.5 that actually occur in X_{r−1} are completely determined by the sum of the p-adic digits of an integer related to r. As a corollary, we obtain the following curious formula for the dimension of X_{r−1} in all cases.

Corollary 1.6. Let p ≥ 3 and let r ≥ 2p + 1. Write r = p^{n}u, with p ∤ u. Set δ = 0 if r = u and δ = 1 otherwise. Let Σ be the sum of the digits of u − 1 in its base p expansion. Then

dim X_{r−1} = \begin{cases}2Σ + 2 + δ(p + 1 − Σ), & if Σ ≤ p − 1 \\ 2p + 2, & if Σ > p − 1. \end{cases}

2. Basics

2.1. Hecke operator T. Recall G = GL_{2}(ℚ_{p}) and KZ = GL_{2}(ℤ_{p})ℚ_{p}^{∗} is the standard compact mod center subgroup of G. Let R be a ℤ_{p}-algebra and let V = Sym^{r}R^{2} ⊗ D^{s} be the usual symmetric power representation of KZ twisted by a power of the determinant character D (with p ∈ Z acting trivially), modeled on homogeneous polynomials of degree r in the variables X, Y over R. For g ∈ G, v ∈ V, let [g, v] ∈ ind_{KZ}^{G}V be the function with support in KZg^{−1} given by

g′ ↦ \begin{cases}g′g · v & if g′ ∈ KZg^{−1} \\ 0 & otherwise. \end{cases}

Any function in ind_{KZ}^{G}V is a finite linear combination of functions of the form [g, v], for g ∈ G and v ∈ V. The Hecke operator T is defined by its action on these elementary functions via

(2.1) T ([g, v]) = ∑_{λ∈𝔽_{p}} [g ( \begin{matrix}p & [λ] \\ 0 & 1 \end{matrix} ) , v (X, −[λ]X + pY)] + [g ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) , v(pX, Y)] ,

where v = v(X, Y) ∈ V and [λ] denotes the Teichmüller representative of λ ∈ 𝔽_{p}. We will always denote the Hecke operator acting on ind_{KZ}^{G}V for various choices of R = ℤ̄_{p}, ℚ̄_{p} or 𝔽̄_{p} and for different values of r and s by T, as the underlying space will be clear from the context.

2.2. The mod p Local Langlands Correspondence. Let V be a weight, i.e., an irreducible representation of GL_{2}(𝔽_{p}), thought of as a representation of KZ by inflating to GL_{2}(ℤ_{p}) and making p ∈ ℚ_{p}^{∗} act trivially. Let V_{r} = Sym^{r}𝔽̄_{p}^{2} be the r-th symmetric power of the standard two-dimensional representation of GL_{2}(𝔽_{p}) on 𝔽̄_{p}^{2}. The set of weights V is exactly the set of modules V_{r} ⊗ D^{i}, for 0 ≤ r ≤ p − 1 and 0 ≤ i ≤ p − 2. For 0 ≤ r ≤ p − 1, λ ∈ 𝔽̄_{p} and η : ℚ_{p}^{∗} → 𝔽̄_{p}^{∗} a smooth character, let

π(r, λ, η) := \frac{ind_{KZ}^{G}V_{r}}{T − λ} ⊗ (η ∘ det)

be the smooth admissible representation of G, where ind_{KZ}^{G} is compact induction and T is the Hecke operator defined above; T generates the Hecke algebra End_{G}(ind_{KZ}^{G} V_{r}) = 𝔽̄_{p}[T]. With this notation, Breuil’s semisimple mod p Local Langlands Correspondence [B03b, Def. 1.1] is given by:

• λ = 0: ind(ω_{2}^{r+1}) ⊗ η ⟼^{LL} π(r, 0, η),

• λ ≠ 0: (ω^{r+1}unr(λ) ⊕ unr(λ^{−1})) ⊗ η

⟼^{LL} π(r, λ, η)^{ss} ⊕ π([p − 3 − r], λ^{−1}, ηω^{r+1})^{ss},

where {0, 1, …, p − 2} ∋ [p − 3 − r] ≡ p − 3 − r mod (p − 1). It is clear from the classification of smooth admissible irreducible representations of G by Barthel-Livné [BL94] and Breuil [B03a], that this correspondence is not surjective. However, the map “LL” above is an injection and so it is enough to know LL(V̄_{k,a_{p}}) to determine V̄_{k,a_{p}}.

2.3. Modular representations of M and Γ. In order to make use of results in Glover [G78], let us abuse notation a bit and let V_{r} be the space of homogeneous polynomials F (X, Y) in two variables X and Y of degree r with coefficients in the finite field 𝔽_{p}, rather than in 𝔽̄_{p}. For the next few sections (up to §6) we similarly consider all subquotients of V_{r} as representations defined over 𝔽_{p}. This is not so serious as once we have established the structure of X_{r−1} or Q over 𝔽_{p}, it immediately implies the corresponding result over 𝔽̄_{p}, by extension of scalars. Let M be the semigroup M_{2}(𝔽_{p}) under matrix multiplication. Then M acts on V_{r} by the formula

( \begin{matrix}a & b \\ c & d \end{matrix} ) · F (X, Y) = F (aX + cY, bX + dY),

making V_{r} an M-module, or more precisely, an 𝔽_{p}[M]-module. One has to be careful with the notation V_{r} while using results from [G78] as Glover indexed the symmetric power representations by dimension instead of the degree of the polynomials involved. In this paper, V_{r} always has dimension r + 1.

We denote the set of singular matrices by N ⊆ M. An 𝔽_{p}[M]-module V is called ‘singular’, if each matrix t ∈ N annihilates V, i.e., if t · V = 0, for all t ∈ N. The largest singular submodule of an arbitrary 𝔽_{p}[M]-module V is denoted by V^{∗}. Note that any M-linear map must take a singular submodule (of its domain) to a singular submodule (of the range). This simple observation will be very useful for us. Let X_{r} and X_{r−1} be the 𝔽_{p}[M]-submodules of V_{r} generated by the monomials X^{r} and X^{r−1}Y respectively. One checks that X_{r} ⊂ X_{r−1} and are spanned by

the sets {X^{r}, (kX + Y)^{r} : k ∈ 𝔽_{p}} and {X^{r}, Y^{r}, X(kX + Y)^{r−1}, (X + lY)^{r−1}Y :

k, l ∈ 𝔽_{p}} respectively [GG15, Lem. 3]. Thus we have dim X_{r} ≤ p + 1 and dim X_{r−1} ≤ 2p + 2. We will describe the explicit structure of the modules X_{r} and X_{r−1}, according to the different congruence classes a ∈ {1, 2, …, p − 1} with r ≡ a mod (p − 1). It will also be convenient to use the representatives b ∈ {2, …, p − 1, p} of the congruence classes of r mod (p − 1). For s ∈ ℕ, we denote the sum of the digits of s in its base p expansion by Σ_{p}(s). It is easy to see that Σ_{p}(s) ≡ s mod (p − 1), for any s ∈ ℕ. Let us write r = p^{n}u, where n = v(r) and hence p ∤ u. The sum Σ_{p}(u − 1) plays a key role in the study of the module X_{r−1}. For r ≡ a mod (p − 1), observe that the sum Σ_{p}(u − 1) ≡ a − 1 mod (p − 1), therefore it varies discretely over the infinite set {a − 1, p + a − 2, 2p + a − 3, ⋯ }. Let θ = θ(X, Y) denote the special polynomial X^{p}Y − XY^{p}. For r ≥ p + 1, we know [G78, (4.1)]

V_{r}^{∗} := \{F ∈ V_{r} : θ | F \} ≅ \begin{cases}0, & if r ≤ p \\ V_{r−p−1} ⊗ D, & if r ≥ p + 1 \end{cases}

is the largest singular submodule of V_{r}. We define V_{r}^{∗∗}, another important submodule of V_{r}, by

V_{r}^{∗∗} := \{F ∈ V_{r} : θ^{2} | F \} ≅ \begin{cases}0, & if r < 2p + 2 \\ V_{r−2p−2} ⊗ D^{2}, & if r ≥ 2p + 2. \end{cases}

Note that V_{r}^{∗∗} is obviously not the largest singular submodule of V_{r}^{∗}. Next we introduce the submodules

X_{r}^{∗} := X_{r} ∩ V_{r}^{∗}, X_{r}^{∗∗} := X_{r} ∩ V_{r}^{∗∗}, X_{r−1}^{∗} := X_{r−1} ∩ V_{r}^{∗}, X_{r−1}^{∗∗} := X_{r−1} ∩ V_{r}^{∗∗}.

It follows that X_{r}^{∗} and X_{r−1}^{∗} are the largest singular submodules inside X_{r} and X_{r−1} respectively. The group GL_{2}(𝔽_{p}) ⊆ M is denoted by Γ. For r ≥ 2p + 1, we will study the Γ-module structure of

Q := \frac{V_{r}}{X_{r−1} + V_{r}^{∗∗}} .

We will be particularly interested in the bottom row of the following commutative diagram of M-modules (hence also of Γ-modules):

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \frac{X_{r−1}^{∗}}{X_{r−1}^{∗∗}} \ar[d] \ar[r] & \frac{X_{r−1}}{X_{r−1}^{∗∗}} \ar[d] \ar[r] & \frac{X_{r−1}}{X_{r−1}^{∗}} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \frac{V_{r}^{∗}}{V_{r}^{∗∗}} \ar[r] \ar[d] & \frac{V_{r}}{V_{r}^{∗∗}} \ar[d] \ar[r] & \frac{V_{r}}{V_{r}^{∗}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \frac{V_{r}^{∗}}{V_{r}^{∗∗} + X_{r−1}^{∗}} \ar[r] \ar[d] & Q \ar[r] \ar[d] & \frac{V_{r}}{V_{r}^{∗} + X_{r−1}} \ar[d] \ar[r] & 0. \\ & 0 & 0 & 0} (2.2)

Proposition 2.1. Let p ≥ 3 and r ≥ p, with r ≡ a mod (p − 1), for 1 ≤ a ≤ p − 1. Then the Γ-module structure of V_{r}/V_{r}^{∗} is given by

0 → V_{a} → \frac{V_{r}}{V_{r}^{∗}} → V_{p−a−1} ⊗ D^{a} → 0. (2.3)

The sequence splits as a sequence of Γ-modules if and only if a = p − 1.

Proof. For r ≥ p, we obtain that V_{r}/V_{r}^{∗} ≅ V_{a+p−1}/V_{a+p−1}^{∗}, using [G78, (4.2)]. The exact sequence then follows from [B03b, Lem. 5.3]. Note that it must split when a = p − 1, as V_{p−1} is an injective Γ-module. The fact that it is non-split for the other congruence classes can be derived from the Γ-module structure of V_{a+p−1} (see, e.g., [G78, (6.4)] or [GG15, Thm. 5]). □

Proposition 2.2. Let p ≥ 3 and 2p + 1 ≤ r ≡ a mod (p − 1), with 1 ≤ a ≤ p − 1. Then the Γ-module structure of V_{r}^{∗}/V_{r}^{∗∗} is given by

(2.4)  0 → V_{p−2} ⊗ D → \frac{V_{r}^{∗}}{V_{r}^{∗∗}} → V_{1} → 0,  if a = 1,
(2.5)  0 → V_{p−1} ⊗ D → \frac{V_{r}^{∗}}{V_{r}^{∗∗}} → V_{0} ⊗ D → 0,  if a = 2,
(2.6) 0 → V_{a−2} ⊗ D → \frac{V_{r}^{∗}}{V_{r}^{∗∗}} → V_{p−a+1} ⊗ D^{a−1} → 0, if 3 ≤ a ≤ p − 1,

and the sequences split if and only if a = 2.

Proof. We use [G78, (4.1)] to get that V_{r}^{∗}/V_{r}^{∗∗} ≅ (V_{r−p−1}/V_{r−p−1}^{∗}) ⊗ D. Since p ≤ r − p − 1 by hypothesis, we apply Proposition 2.1 to deduce the Γ-module structure of (V_{r−p−1}/V_{r−p−1}^{∗}) ⊗ D. □

The following lemma will be used many times throughout the article to determine if certain polynomials F ∈ V_{r} are divisible by θ or θ^{2}. We skip the proof since it is elementary.

Lemma 2.3. Suppose F (X, Y) = ∑_{0≤j≤r}c_{j} · X^{r−j}Y^{j} ∈ 𝔽_{p}[X, Y] is such that

c_{j} ≠ 0 implies j ≡ a mod (p − 1), for some fixed a ∈ {1, 2, ⋯, p − 1}. Then (i) F ∈ V_{r}^{∗} if and only if c_{0} = c_{r} = 0 and ∑_{j}c_{j} = 0 in 𝔽_{p}.

(ii) F ∈ V_{r}^{∗∗} if and only if c_{0} = c_{1} = c_{r−1} = c_{r} = 0 and ∑_{j}c_{j} = ∑_{j}jc_{j} = 0

in 𝔽_{p}.

2.4. Reduction of binomial coefficients. In this article, the mod p reductions of binomial coefficients play a very important role. We will repeatedly use the following theorem and often refer to it as Lucas’ theorem, as it was proved by E. Lucas in 1878.

Theorem 2.4. For any prime p, let m and n be two non-negative integers with p-adic expansions m = m_{k}p^{k} + m_{k−1}p^{k−1} + ⋯ + m_{0} and n = n_{k}p^{k} + n_{k−1}p^{k−1} + ⋯ + n_{0} respectively. Then \binom{m}{n} ≡ \binom{m_{k}}{n_{k}} · \binom{m_{k−1}}{n_{k−1}} ⋯ \binom{m_{0}}{n_{0}} mod p, with the convention that \binom{a}{b} = 0, if b > a.

The following elementary congruence mod p will also be used in the text. For any i ≥ 0,

∑^{p−1} ; k=0k^{i} ≡ \begin{cases}−1, & if i = n(p − 1), for some n ≥ 1, \\ 0, & otherwise (including the case i = 0, as 0^{0} = 1). \end{cases}

This follows from the following frequently used fact in characteristic zero. For any i ≥ 0,

(2.7)  ∑_{λ∈𝔽_{p}}[λ]^{i} = \begin{cases}p, & if i = 0, \\ p − 1, & if i = n(p − 1) for some n ≥ 1, \\ 0, & if (p − 1) ∤ i, \end{cases}

where [λ] ∈ ℤ_{p} is the Teichmüller representative of λ ∈ 𝔽_{p}. We now state some important congruences, leaving the proofs to the reader as exercises. These technical lemmas are used in checking the criteria given in Lemma 2.3, and also in constructing functions f ∈ ind_{KZ}^{G} Sym^{r}ℚ̄_{p}^{2} with certain desired properties (cf. §7, §8 and §9).

Lemma 2.5. For r ≡ a mod (p − 1), with 1 ≤ a ≤ p − 1, we have

S_{r} := ∑_{\begin{matrix}0 <j < r, \\ j ≡ a mod (p−1) \end{matrix}} \binom{r}{j} ≡ 0 mod p.

Moreover, we have \frac{1}{p}S_{r} ≡ \frac{a − r}{a} mod p, for p > 2.

Lemma 2.6. Let r ≡ b mod (p − 1), with 2 ≤ b ≤ p. Then we have

T_{r} := ∑_{\begin{matrix}0< j < r−1, \\ j≡ b−1 mod (p−1) \end{matrix}} \binom{r}{j} ≡ b − r mod p.

Lemma 2.7. Let p ≥ 3, r ≡ 1 mod (p − 1), i.e., b = p with the notation above. If p | r, then

S_{r} := ∑_{\begin{matrix}1< j < r, \\ j≡ 1 mod (p−1) \end{matrix}} \binom{r}{j} = ∑_{\begin{matrix}0< j < r−1, \\ j≡ 0 mod (p−1) \end{matrix}} \binom{r}{j} ≡ (p − r) mod p^{2}.

3. The case r ≡ 1 mod (p − 1)

In this section, we compute the Jordan-Hölder (JH) factors of Q as a Γ-module, when r ≡ 1 mod (p − 1). This is the case a = 1 and b = p, with the notation above.

Lemma 3.1. Let p ≥ 3, r > 1 and let r ≡ 1 mod (p − 1).

(i) If p ∤ r, then X_{r}^{∗}/X_{r}^{∗∗} ≅ V_{p−2} ⊗ D, as a Γ-module.

(ii) If p | r, then X_{r}^{∗}/X_{r}^{∗∗} = 0.

Proof. (i) Consider the polynomial F (X, Y) = ∑_{k∈𝔽_{p}}(kX + Y)^{r} ∈ X_{r}. We have

F (X, Y) = ∑^{r} ; j=0\binom{r}{j}· ∑_{k∈𝔽_{p}}k^{r−j}·X^{r−j}Y^{j} ≡ ∑_{\begin{matrix}0≤j < r, \\ j≡ 1 mod (p−1) \end{matrix}}−\binom{r}{j}·X^{r−j}Y^{j} mod p.

The sum of the coefficients of F (X, Y) is congruent to 0 mod p, by Lemma 2.5. Applying Lemma 2.3, we get that F (X, Y) ∈ V_{r}^{∗}. As p ∤ r, the coefficient of X^{r−1}Y in F (X, Y) is −r ≢ 0 mod p. Hence F (X, Y) ∉ V_{r}^{∗∗} and so F (X, Y) has non-zero image in X_{r}^{∗}/X_{r}^{∗∗}. For r = 2p − 1, we have 0 ≠ X_{r}^{∗}/X_{r}^{∗∗} ⊆ V_{r}^{∗}/V_{r}^{∗∗} ≅ V_{p−2} ⊗D, which is irreducible and the result follows. If r ≥ 3p − 2, then V_{r}^{∗}/V_{r}^{∗∗} has dimension p + 1, but [G78, (4.5)] implies that dim X_{r}^{∗} ⪇ dim X_{r} ≤ p + 1. So we have 0 ≠ X_{r}^{∗}/X_{r}^{∗∗} ⊊ V_{r}^{∗}/V_{r}^{∗∗}. Now it follows from Proposition 2.2 that X_{r}^{∗}/X_{r}^{∗∗} ≅ V_{p−2} ⊗ D.

(ii) Write r = p^{n}u, where n ≥ 1 and p ∤ u. The map ι : X_{u} → X_{r}, defined by ι(H(X, Y)) := H(X^{p^{n}}, Y^{p^{n}}), is a well-defined M-linear surjection from X_{u} to X_{r}. It is also an injection, as H(X^{p^{n}}, Y^{p^{n}}) = H(X, Y)^{p^{n}} ∈ 𝔽_{p}[X, Y]. Hence the M-isomorphism ι : X_{u} → X_{r} must take X_{u}^{∗}, the largest singular submodule of X_{u}, isomorphically to X_{r}^{∗}. If u = 1, then X_{r}^{∗} ≅ X_{u}^{∗} = 0, so X_{r}^{∗} = X_{r}^{∗∗} follows trivially. If u > 1, then as p ∤ u ≡ r ≡ 1 mod (p − 1), we get u ≥ 2p − 1 and V_{u}^{∗} ≅ V_{u−p−1} ⊗ D. For any F ∈ X_{r}^{∗}, we have F = ι(H), for some H ∈ X_{u}^{∗}. Writing H = θH′ with H′ ∈ V_{u−p−1}, we get F = ι(H) = (θH′)^{p^{n}}. As n ≥ 1, clearly θ^{2} divides F. Therefore X_{r}^{∗} ⊆ V_{r}^{∗∗}, equivalently X_{r}^{∗} = X_{r}^{∗∗}. □

The p-adic expansion of r − 1 will play an important role in our study of the module X_{r−1}. Write

(3.1)  r − 1 = r_{m}p^{m} + r_{m−1}p^{m−1} + ⋯ r_{i}p^{i},

where r_{j} ∈ {0, 1, ⋯, p − 1}, m ≥ i and r_{m}, r_{i} ≠ 0. If i > 0, then we let r_{j} = 0, for 0 ≤ j ≤ i − 1. With the notation introduced in Section 2.3, we have a = 1, so Σ_{p}(r − 1) ≡ 0 mod (p − 1). Excluding the case r = 1, note that the smallest possible value of Σ_{p}(r − 1) is p − 1. Also recall that the dimension of X_{r−1} is bounded above by 2p + 2 and a standard generating set is given by {X^{r}, Y^{r}, X(kX + Y)^{r−1}, (X + lY)^{r−1}Y : k, l ∈ 𝔽_{p}}, over 𝔽_{p}.

Lemma 3.2. For p ≥ 2, if p ≤ r ≡ 1 mod (p − 1) and Σ = Σ_{p}(r − 1) = p − 1, then

∑^{p−1} ; k=0X(kX + Y)^{r−1} ≡ −X^{r} and ∑^{p−1} ; l=0(X + lY)^{r−1}Y ≡ −Y^{r} mod p.

As a consequence, dim X_{r−1} ≤ 2p.

Proof. It is enough to show one of the congruences, since the other will then follow by applying the matrix w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) to it. We compute that

F (X, Y) = ∑^{p−1} ; k=0X(kX+Y)^{r−1} ≡ ∑_{\begin{matrix}0<s<r \\ s≡0 mod (p−1) \end{matrix}}−\binom{r − 1}{s}·X^{s+1}Y^{r−1−s} mod p.

We claim that if 0 < s < r − 1 and s ≡ 0 mod (p − 1), then \binom{r−1}{s} ≡ 0 mod p. The claim implies that F (X, Y) ≡ −\binom{r−1}{r−1} · X^{r} ≡ −X^{r} mod p, as required. \underline{Proof of claim}: Let s = s_{m}p^{m} + ⋯ + s_{1}p + s_{0} be the p-adic expansion of s < r − 1, where m is as in the expansion (3.1) above. Since s ≡ 0 mod (p − 1), we have Σ_{p}(s) ≡ 0 mod (p−1) too. If \binom{r−1}{s} ≢ 0 mod p, then by Lucas’ theorem 0 ≤ s_{j} ≤ r_{j}, for all j. Taking the sum, we get that 0 ≤ Σ_{p}(s) ≤ Σ = p − 1. But since s > 0, Σ_{p}(s) has to be a strictly positive multiple of p − 1, and so it is p − 1. Hence s_{j} = r_{j}, for all j ≤ m, and we have s = r − 1, which is a contradiction. □

We observe that p | r if and only if r_{0} = p − 1 in (3.1). Therefore if Σ = Σ_{p}(r − 1) = r_{0} + ⋯ + r_{m} = p − 1, then the condition p | r is equivalent to r = p. Our next proposition treats the case Σ = p − 1, and to avoid the possibility of p dividing r, we exclude the case r = p. The fact that p ∤ r will be used crucially in the proof. This does not matter, as eventually we wish to compute Q for r ≥ 2p + 1.

Proposition 3.3. For p ≥ 2, if p < r ≡ 1 mod (p − 1) and Σ = Σ_{p}(r − 1) = p − 1, then

(i) X_{r−1} ≅ V_{2p−1} as an M-module, and the M-module structure of X_{r} is

given by

0 → V_{p−2} ⊗ D → X_{r} → V_{1} → 0.

(ii) X_{r−1}^{∗} = X_{r}^{∗} ≅ V_{p−2} ⊗ D and X_{r−1}^{∗∗} = X_{r}^{∗∗} = 0.

(iii) For r > 2p, Q has only one JH factor V_{1}, as a Γ-module.

Proof. It is easy to check that {S(kS + T)^{2p−2}, (S + lT)^{2p−2}T : k, l ∈ 𝔽_{p}} gives a basis of V_{2p−1} over 𝔽_{p}. We define an 𝔽_{p}-linear map η : V_{2p−1} → X_{r−1}, by

η (S(kS + T)^{2p−2}) = X(kX + Y)^{r−1}, η ((S + lT)^{2p−2}T ) = (X + lY)^{r−1}Y,

for k, l ∈ 𝔽_{p}. Note that for r ≤ p^{2} − p + 1, the map η is the same as the one used in the proof of [GG15, Prop. 6]. We claim that η is in fact an M-linear injection. By Lemma 3.2, we have

(3.2)  η(S^{2p−1}) = X^{r}, η(T^{2p−1}) = Y^{r}.

The M-linearity can be checked on the basis elements of V_{2p−1} above by an elementary computation which uses the fact that r − 1 ≡ 0 mod (p − 1) and (3.2), so we leave it to the reader. As a Γ-module, soc(V_{2p−1}) = V_{2p−1}^{∗} ≅ V_{p−2} ⊗ D is irreducible. Therefore if ker η ≠ 0, then it must contain the submodule V_{2p−1}^{∗}. Consider

H(S, T) = ∑^{p−1} ; k=0 ( \begin{matrix}k & 1 \\ 1 & 0 \end{matrix} ) · S^{2p−1} = (S^{p}T − ST^{p})S^{p−2} ∈ V_{2p−1}^{∗}.

By M-linearity, we have η(H) = F (X, Y) ∈ X_{r}^{∗}\X_{r}^{∗∗}, where F is as in the proof of Lemma 3.1 (i). In particular, this shows that H ∉ ker η. As V_{2p−1}^{∗} ⊈ ker η, we have ker η = 0. Thus η : V_{2p−1} → X_{r−1} is an injective M-linear map. By Lemma 3.2, dim X_{r−1} ≤ 2p = dim V_{2p−1}, forcing η to be an isomorphism. Therefore the largest singular submodule X_{r−1}^{∗} inside X_{r−1} has to be isomorphic to V_{2p−1}^{∗} ≅ V_{p−2} ⊗ D, the largest singular submodule of V_{2p−1}. Then Lemma 3.1 (i) implies that X_{r}^{∗} is a non-zero submodule of X_{r−1}^{∗} ≅ V_{p−2} ⊗ D, which is irreducible. So we must have X_{r}^{∗} = X_{r−1}^{∗}. Again by Lemma 3.1 (i), X_{r−1}^{∗∗}(⊇ X_{r}^{∗∗}) is a proper submodule of X_{r−1}^{∗}. Hence X_{r−1}^{∗∗} = X_{r}^{∗∗} = 0. Since dim(X_{r−1}/X_{r−1}^{∗}) = p + 1 = dim(V_{r}/V_{r}^{∗}), the rightmost module in the bottom row of Diagram (2.2) is 0. As the dimension of X_{r−1}^{∗}/X_{r−1}^{∗∗} is p − 1, the leftmost module must have dimension 2. It has to be V_{1}, as the short exact sequence (2.4) does not split for p ≥ 3. For p = 2 and r ≥ 5, the only twodimensional quotient of V_{r}^{∗}/V_{r}^{∗∗} is V_{1}, as one checks that V_{r}^{∗}/V_{r}^{∗∗} ≅ V_{1} ⊕ V_{0}. Hence we get Q ≅ V_{1} as a Γ-module. □

The next lemma about the dimension of X_{r−1} is a special case of Lemma 4.2, proved at the beginning of Section 4.

Lemma 3.4. For p ≥ 2, suppose p ∤ r ≡ 1 mod (p−1). If Σ = Σ_{p}(r−1) > p−1, then dim X_{r−1} = 2p + 2.

Lemma 3.5. For any r, if dim X_{r−1} = 2p + 2, then dim X_{r} = p + 1.

Proof. Suppose X_{r} has dimension smaller than p + 1. Then the standard spanning set of X_{r} is linearly dependent, i.e., there exist constants A, c_{k} ∈ 𝔽_{p}, for k ∈ {0, 1, …, p − 1}, not all zero, such that AX^{r} + ∑^{p−1} ; k=0c_{k}(kX + Y)^{r} = 0, which

implies that

AX^{r} + c_{0}Y^{r} + ∑^{p−1} ; k=1kc_{k}X(kX + Y)^{r−1} + ∑^{p−1} ; k=1c_{k}k^{r−1}(X + k^{−1}Y)^{r−1}Y = 0.

But this shows that the standard spanning set {X^{r}, Y^{r}, X(kX + Y)^{r−1}, (X + lY)^{r−1}Y : k, l ∈ 𝔽_{p}} of X_{r−1} is linearly dependent, contradicting the hypothesis dim X_{r−1} = 2p + 2. □

For any r, let us set r′ := r − 1. The trick introduced in [GG15] of using the structure of X_{r′} ⊆ V_{r′} to study X_{r−1} ⊆ V_{r} via the map ϕ described below, turns out to be very useful in general.

Lemma 3.6. There exists an M-linear surjection ϕ : X_{r′} ⊗ V_{1} ↠ X_{r−1}.

Proof. The map ϕ_{r′,1} : V_{r′} ⊗ V_{1} ↠ V_{r} sending u ⊗ v ↦ uv, for u ∈ V_{r′} and v ∈ V_{1}, is M-linear by [G78, (5.1)]. Let ϕ be its restriction to the M-submodule X_{r′} ⊗V_{1} ⊆ V_{r′} ⊗V_{1}. The module X_{r′} ⊗V_{1} is generated by X^{r′} ⊗X and X^{r′} ⊗Y, which map to X^{r} and X^{r−1}Y ∈ X_{r−1} respectively. So the image of ϕ lands in X_{r−1} ⊆ V_{r}. The surjectivity follows as X^{r−1}Y generates X_{r−1}. □

Lemma 3.7. For p ≥ 3, if r ≡ 1 mod (p − 1), with Σ_{p}(r′) > p − 1, then

(i) X_{r′}^{∗∗} = X_{r′}^{∗} has dimension 1 over 𝔽_{p}. In fact, it is M-isomorphic to

D^{p−1}.
(ii) ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ V_{r}^{∗∗} and ϕ(X_{r′}^{∗} ⊗ V_{1}) ≅ V_{1} ⊗ D^{p−1}.

Proof. Consider F (X, Y) := X^{r′} + ∑_{k∈𝔽_{p}}(kX + Y)^{r′} ∈ X_{r′} ⊆ V_{r′}. It is easy to

see that

F (X, Y) ≡ − ∑_{\begin{matrix}0<j<r′ \\ j≡0 mod (p−1) \end{matrix}}\binom{r′}{j}X^{r′−j}Y^{j} mod p.

Using Lemmas 2.3 and 2.5 we check that F (X, Y) ∈ V_{r′}^{∗∗}, for p ≥ 3. Since Σ_{p}(r′) > p − 1 or equivalently Σ_{p}(r′) ≥ 2p − 2, using Lucas’ theorem one can show that at least one of the coefficients \binom{r′}{j} above is non-zero mod p. So we have 0 ≠ F (X, Y) ∈ X_{r′}^{∗∗} ⊆ X_{r′}^{∗}. Since r′ ≡ p − 1 mod (p − 1), [G78, (4.5)] gives the following short exact sequence of M-modules:

(3.3)  0 → X_{r′}^{∗} → X_{r′} → V_{p−1} → 0.

As dim X_{r′} ≤ p+1 and X_{r′}^{∗∗} ≠ 0, we must have dim X_{r′}^{∗∗} = dim X_{r′}^{∗} = 1. Hence X_{r′}^{∗∗} = X_{r′}^{∗} ≅ D^{n}, for some n ≥ 1. Checking the action of diagonal matrices on F (X, Y), we get n = p − 1. As X_{r′}^{∗∗} = X_{r′}^{∗}, each element of X_{r′}^{∗} is divisible by θ^{2}. Therefore it follows from the definition of the map ϕ that ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ V_{r}^{∗∗}. For any nonzero F ∈ X_{r′}^{∗}, note that ϕ(F ⊗ X) = F X ≠ 0. We know that X_{r′}^{∗} ⊗ V_{1} ≅ V_{1} ⊗ D^{p−1} is irreducible of dimension 2 and its image under ϕ is non-zero. Hence ϕ(X_{r′}^{∗} ⊗ V_{1}) ≅ V_{1} ⊗ D^{p−1} ⊆ X_{r−1}. □

Proposition 3.8. Let p ≥ 3, r > 2p and p ∤ r ≡ 1 mod (p − 1). If Σ = Σ_{p}(r − 1) > p − 1, then

(i) The M-module structures of X_{r−1} and X_{r} are given by the exact sequences

0 → V_{1} ⊗ D^{p−1} → X_{r−1} → V_{2p−1} → 0,
0 → V_{p−2} ⊗ D → X_{r} → V_{1} → 0.

(ii) X_{r}^{∗} ≅ V_{p−2} ⊗ D and X_{r−1}^{∗} ≅ V_{1} ⊗ D^{p−1} ⊕ V_{p−2} ⊗ D.

(iii) X_{r}^{∗∗} = 0 and X_{r−1}^{∗∗} ≅ V_{1} ⊗ D^{p−1}.

(iv) Q ≅ V_{1} as a Γ-module.

Proof. By Lemma 3.4, dim X_{r−1} = 2p + 2, so by Lemma 3.6, we must have dim X_{r′} ⊗ V_{1} ≥ 2p + 2. This forces X_{r′} to have its highest possible dimension, namely, p+1. Thus the M-map ϕ : X_{r′}⊗V_{1} ↠ X_{r−1} is actually an isomorphism. Tensoring the short exact sequence (3.3) by V_{1}, we get the exact sequence

0 → X_{r′}^{∗} ⊗ V_{1} → X_{r′} ⊗ V_{1} → V_{p−1} ⊗ V_{1} → 0.

The middle module is M-isomorphic to X_{r−1}, and the rightmost module is M-isomorphic to V_{2p−1}, by [G78, (5.3)]. Thus the exact sequence reduces to

(3.4)  0 → X_{r′}^{∗} ⊗ V_{1} → X_{r−1} → V_{2p−1} → 0,

where X_{r′}^{∗} ⊗ V_{1} ≅ V_{1} ⊗ D^{p−1}, by Lemma 3.7 (i). Since M-linear maps must take singular submodules to singular submodules, the above sequence gives rise to the following exact sequence

(3.5)  0 → V_{1} ⊗ D^{p−1} → X_{r−1}^{∗} → V_{2p−1}^{∗} ≅ V_{p−2} ⊗ D.

The rightmost module above is irreducible, so the map X_{r−1}^{∗} → V_{p−2} ⊗ D is either the zero map or it is a surjection. By Lemma 3.5, dim X_{r} = p + 1 and so by [G78, (4.5)], we have dim X_{r}^{∗} = p − 1. By Lemma 3.1 (i), we get X_{r}^{∗∗} = 0 and X_{r}^{∗} ≅ V_{p−2} ⊗ D must be a JH factor of X_{r−1}^{∗}. Therefore the rightmost map above must be surjective, as otherwise X_{r−1}^{∗} ≅ V_{1} ⊗ D^{p−1}. So we have

(3.6)  0 → X_{r′}^{∗} ⊗ V_{1} ≅ V_{1} ⊗ D^{p−1} → X_{r−1}^{∗} → V_{p−2} ⊗ D → 0.

Thus X_{r−1}^{∗} has two JH factors, of dimensions 2 and p−1 respectively. Moreover, since X_{r}^{∗} ≅ V_{p−2} ⊗ D is a submodule of X_{r−1}^{∗}, the sequence above must split, and we must have

X_{r−1}^{∗} = ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊕ X_{r}^{∗} ≅ V_{1} ⊗ D^{p−1} ⊕ V_{p−2} ⊗ D.

Knowing the structure of X_{r−1}^{∗} as above, next we want to see how the submodule X_{r−1}^{∗∗} sits inside it. By Lemma 3.7 (ii), we have ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ V_{r}^{∗∗}, on the other hand X_{r} ∩ V_{r}^{∗∗} = X_{r}^{∗∗} = 0. Therefore X_{r−1}^{∗∗} = X_{r−1}^{∗} ∩ V_{r}^{∗∗} = ϕ(X_{r′}^{∗} ⊗ V_{1}) ≅ V_{1} ⊗ D^{p−1} has dimension 2.

Now we count the dimension dim Q = 2p + 2 − dim X_{r−1} + dim X_{r−1}^{∗∗} = 2. The final statement Q ≅ V_{1} follows from Diagram (2.2) as in the proof of Proposition 3.3. □

Thus we know Q is isomorphic to V_{1} whenever r is prime to p. Next we treat the case p divides r. Since r ≡ 1 mod (p − 1), we see that r can be a pure p-power. We will show that Q has two JH factors as a Γ-module, irrespective of whether r is a p-power or not. The following result about dim X_{r−1} when p | r is stated without proof, as it follows from the more general Lemma 4.3 in Section 4.

Lemma 3.9. Let p ≥ 2 and r ≡ 1 mod (p − 1). If p | r but r is not a pure p-power, then dim X_{r−1} = 2p + 2.

Lemma 3.10. For p ≥ 2 and r = p^{n}, with n ≥ 2, we have dim X_{r−1} = p + 3.

Proof. We know that Γ = B ⊔ BwB, where B ⊆ Γ is the subgroup of uppertriangular matrices, and w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Using this decomposition and the fact that r = p^{n}, one can see that the 𝔽_{p}[Γ]-span of X^{r−1}Y ∈ V_{r} is generated by the set {X^{r}, X^{r−1}Y, Y^{r}, X(kX + Y)^{r−1} : k ∈ 𝔽_{p}} over 𝔽_{p}. We will show that this generating set is linearly independent. Suppose that

AX^{r} + BY^{r} + DX^{r−1}Y + ∑^{p−1} ; k=0c_{k}X(kX + Y)^{r−1} = 0,

where A, B, D, c_{k} ∈ 𝔽_{p}, for each k. Clearly, it is enough to show that c_{k} = 0, for each k ∈ 𝔽_{p}^{∗}. Since r − 1 = p^{n} − 1 for some n ≥ 2, Lucas’ theorem says that \binom{r−1}{i} ≢ 0 mod p, for 0 ≤ i ≤ r − 1. As r − p ≥ 2, equating the coefficients of

X^{i}Y^{r−i} on both sides, for 2 ≤ i ≤ p, we get that ∑^{p−1} ; k=1c_{k}k^{i−1} ≡ 0 mod p. The

non-vanishing of the Vandermonde determinant now shows that c_{k} = 0, for all k ∈ 𝔽_{p}^{∗}. □

Remark 3.11. Note that the proof does not work for r = p, since we need r − p ≥ 2. Also the lemma is trivially false for r = p, because then X_{r−1} ⊆ V_{r} must have dimension ≤ p + 1.

The next proposition describes the structure of Q for p | r. Note that if r > p is a multiple of p, then for r′ = r − 1, we have Σ_{p}(r′) > p − 1, so we can apply Lemma 3.7.

Proposition 3.12. For p ≥ 3, let r (> p) be a multiple of p such that r ≡ 1 mod (p − 1).

(i) If r = p^{n} with n ≥ 2, then X_{r}^{∗} = X_{r}^{∗∗} = 0 and X_{r−1}^{∗} = X_{r−1}^{∗∗} has dimension 2.

(ii) If r is not a pure p-power, then X_{r}^{∗} = X_{r}^{∗∗} ≅ V_{p−2} ⊗ D and X_{r−1}^{∗} =

X_{r−1}^{∗∗} has dimension p + 1. (iii) In either case, Q is a non-trivial extension of V_{1} by V_{p−2} ⊗ D, as a

Γ-module.

Proof. (i) By Lemma 3.7, dim X_{r′}^{∗} = 1 and dim X_{r′} = p + 1 by [G78, (4.5)]. By Lemma 3.10, dim X_{r−1} = p + 3. By Lemma 3.6, we get a surjection ϕ : X_{r′} ⊗V_{1} ↠ X_{r−1}, with a non-zero kernel of dimension 2(p+1)−(p+3) = p−1.

Note that W := \frac{X_{r−1}}{ϕ(X_{r′}^{∗} ⊗ V_{1})} is a quotient of (X_{r′}/X_{r′}^{∗}) ⊗ V_{1}, which is M-

isomorphic to V_{2p−1} by [G78, (4.5), (5.3)]. We have the exact sequence of M-modules

0 → X_{r′}^{∗} ⊗ V_{1} ⟶^{ϕ} X_{r−1} → W → 0.

Restricting it to the maximal singular submodules, we get the exact sequence

0 → X_{r′}^{∗} ⊗ V_{1} ⟶^{ϕ} X_{r−1}^{∗} → W^{∗},

where W^{∗} denotes the largest singular submodule of W. By Lemmas 3.10 and 3.7 (ii), we get dim W = (p + 3) − 2 = p + 1. Being a (p + 1)-dimensional quotient of V_{2p−1}, W must be M-isomorphic to V_{2p−1}/V_{2p−1}^{∗}. By [G78, (4.6)], W has a unique non-zero minimal submodule, namely,

W′ = (X_{2p−1} + V_{2p−1}^{∗}) /V_{2p−1}^{∗}.

Note that the singular matrix ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) acts trivially on X^{2p−1}, which is non-zero in W′. Thus the unique minimal submodule W′ is non-singular, so W^{∗} = 0, giving us an M-isomorphism X_{r′}^{∗} ⊗ V_{1} ⟶^{ϕ} X_{r−1}^{∗}. Now by Lemma 3.7 (ii), X_{r−1}^{∗} = ϕ(X_{r′}^{∗} ⊗ V_{1}) = X_{r−1}^{∗∗} has dimension 2.

(ii) If r = p^{n}u for some n ≥ 1 and p ∤ u ≥ 2p − 1, then dim X_{r−1} = 2p + 2, by Lemma 3.9. We have shown in the proof of Lemma 3.1 (ii) that X_{r}^{∗} = X_{r}^{∗∗} ≅ X_{u}^{∗}, which is isomorphic to V_{p−2} ⊗ D, as p ∤ u (cf. Propositions 3.3 and 3.8). We proceed exactly as in the proof of Proposition 3.8, to get that X_{r−1}^{∗} ≅ ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊕ X_{r}^{∗} has dimension p + 1. By Lemma 3.7, we know ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ V_{r}^{∗∗}. Thus both the summands of X_{r−1}^{∗} are contained in V_{r}^{∗∗}. Hence X_{r−1}^{∗∗} := X_{r−1}^{∗} ∩ V_{r}^{∗∗} = X_{r−1}^{∗}.

(iii) Using part (i), (ii) above and Lemmas 3.9, 3.10, we count that dim(X_{r−1}/X_{r−1}^{∗∗}) = p+1. Hence dim Q = 2p+2−dim X_{r−1}+dim X_{r−1}^{∗∗} = p+1. Since X_{r−1}^{∗} = X_{r−1}^{∗∗}, the natural map V_{r}^{∗}/V_{r}^{∗∗} → Q is injective, hence an isomorphism by dimension count. Now the Γ-module structure of Q follows from the short exact sequence (2.4). □

Note that in the course of studying the structure of Q, we have derived the complete structure of the M-submodule X_{r−1} ⊆ V_{r}, for r ≡ 1 mod (p − 1), summarized as follows:

Proposition 3.13. Let p ≥ 3, r > p, and r ≡ 1 mod (p − 1).

(i) If Σ_{p}(r − 1) = p − 1 (so p ∤ r), then X_{r−1} ≅ V_{2p−1} as an M-module. (ii) If Σ_{p}(r − 1) > p − 1 and r ≠ p^{n}, then we have a short exact sequence

of M-modules

0 → V_{1} ⊗ D^{p−1} → X_{r−1} → V_{2p−1} → 0.

(iii) If r = p^{n}, for some n ≥ 2 (so Σ_{p}(r − 1) > p − 1), then we have a short exact sequence of M-modules

0 → V_{1} ⊗ D^{p−1} → X_{r−1} → W → 0,

where W ≅ V_{2p−1}/V_{2p−1}^{∗} is a non-trivial extension of V_{p−2} ⊗ D by V_{1}.

4. Structure of X_{r−1}

In this section we study the M-submodule X_{r−1} of V_{r} generated by X^{r−1}Y, for r lying in any congruence class a modulo (p − 1). Recall that X_{r} is the M-submodule of V_{r} generated by X^{r}. For r ≤ p − 1, we have X_{r} = X_{r−1} = V_{r}, since V_{r} is irreducible. We begin with the following easily proved lemma showing that outside this small range of r, the module X_{r} is properly contained in X_{r−1}.

Lemma 4.1. For any p ≥ 2, if r ≥ p, then X_{r} ⊊ X_{r−1}.

The following lemma is valid for r lying in any congruence class a mod (p−1), with 1 ≤ a ≤ p − 1.

Lemma 4.2. Let p ≥ 2, r ≥ 2p + 1 and p ∤ r. If Σ := Σ_{p}(r − 1) ≥ p, then dim X_{r−1} = 2p + 2.

Proof. We claim that the spanning set {X^{r}, Y^{r}, X(kX +Y)^{r−1}, (X +lY)^{r−1}Y : k, l ∈ 𝔽_{p}} of X_{r−1} is linearly independent. Suppose there exist constants A, B, c_{k}, d_{l} ∈ 𝔽_{p}, for k, l = 0, 1, ⋯, p − 1, satisfying the equation

(4.1)  AX^{r} + BY^{r} + ∑^{p−1} ; k=0c_{k}X(kX + Y)^{r−1} + ∑^{p−1} ; l=0d_{l}(X + lY)^{r−1}Y = 0.

We want to show that A = B = c_{k} = d_{l} = 0, for all k, l ∈ 𝔽_{p}. It is enough to show that c_{k} = d_{l} = 0, for all k, l ≠ 0, since that implies that AX^{r} + BY^{r} + c_{0}XY^{r−1} + d_{0}X^{r−1}Y = 0, hence A = B = c_{0} = d_{0} = 0 too. As the matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) flips the coefficients c_{k}’s to d_{l}’s in (4.1), it is enough to show that d_{l} = 0, for each l = 1, 2, ⋯, p − 1. Let us define, for i, j ≥ 0,

C_{i} := ∑^{p−1} ; k=1c_{k}k^{i}, D_{j} := ∑^{p−1} ; l=1d_{l}l^{j}.

Note that C_{i}, D_{j} depend only on the congruence classes of i, j mod (p − 1). By the non-vanishing of the Vandermonde determinant, if D_{1} = D_{2} = ⋯ = D_{p−1} = 0, then (d_{1}, ⋯, d_{p−1}) = (0, ⋯, 0). Thus the proof reduces to showing that D_{t} = 0, for 1 ≤ t ≤ p − 1. Comparing the coefficients of X^{r−t−1}Y^{t+1} on both sides of (4.1), we get

(4.2)  \binom{r − 1}{t + 1}C_{r−2−t} + \binom{r − 1}{t}D_{t} = 0, for 1 ≤ t ≤ r − 3.

Let r − 1 = r_{m}p^{m} + ⋯ + r_{1}p + r_{0}, as in (3.1), with r_{i} being the rightmost non-zero coefficient. Note that 0 ≤ r_{0} < p − 1, as by hypothesis p ∤ r.

We will first consider the D_{t}’s with 1 ≤ t < r_{i}. Suppose r_{0} = 0, then for 1 ≤ t < r_{i}, choose t′ := tp^{i}. Clearly r − 3 ≥ t′ ≡ t mod (p − 1), so we apply it to equation (4.2), to get D_{t′} = D_{t} = 0, since \binom{r−1}{t′+1} ≡ 0 ≢ \binom{r−1}{t′} mod p, by Lucas’ theorem. If r_{0} ≠ 0, i.e., r_{i} = r_{0}, let i′ > 0 be the smallest positive integer such that r_{i′} ≠ 0. Then we choose t′ := t + p^{i′} − 1 ≡ t mod (p − 1) and check that t′ ≤ r − 3. Now equation (4.2) gives us the following two linear equations:

\binom{r − 1}{t + 1}C_{r−2−t} + \binom{r − 1}{t}D_{t} = 0,

\binom{r − 1}{t′ + 1}C_{r−2−t} + \binom{r − 1}{t′}D_{t} = 0.

The determinant of the corresponding matrix is congruent to

\binom{r_{0}}{t + 1}\binom{r_{i′}}{1}\binom{r_{0}}{t − 1} − \binom{r_{0}}{t}\binom{r_{i′}}{1}\binom{r_{0}}{t}

≡ −\binom{r_{i′}}{1}\binom{r_{0}}{t − 1}\binom{r_{0}}{t} · \frac{r_{0} + 1}{t(t + 1)} ≢ 0 mod p,

since we have 0 < r_{0} + 1, r_{i′} ≤ p − 1. This shows that D_{t} = C_{r−2−t} = 0. Next we deal with the D_{t}’s, for r_{i} ≤ t ≤ p − 1. As Σ = r_{i} + r_{i+1} + ⋯ + r_{m} ≥ p, we can write t as t = r_{i} + s_{i+1} + ⋯ + s_{m}, with 0 ≤ s_{j} ≤ r_{j}, for all j. Let us choose t′ := r_{i}p^{i} + s_{i+1}p^{i+1} + ⋯ + s_{m}p^{m}, clearly t′ ≡ t mod (p − 1).

We observe that since t = r_{i} + s_{i+1} + ⋯ + s_{m} ≤ p − 1 < p ≤ ∑^{m} ; j=ir_{j} = Σ,

there must exist at least one j > i, such that s_{j} < r_{j}. This implies that t′ ≤ r −1−p^{j} ≤ r −1−p^{i+1} ≤ r −3. By our choice of t′, we have t′ +1 ≡ r_{0} +1 mod p, with 0 ≤ r_{0} < r_{0} +1 ≤ p−1, as r_{0} < p−1. Hence \binom{r−1}{t′+1} ≡ 0 mod p, by Lucas’ theorem. Also note that \binom{r−1}{t′} ≡ \binom{r_{i}}{r_{i}}\binom{r_{i+1}}{s_{i+1}} ⋯ \binom{r_{m}}{s_{m}} ≢ 0 mod p. Now using (4.2) for t′, we get D_{t′} = D_{t} = 0. Thus we conclude D_{t} = 0, for all t = 1, 2, ⋯, p − 1. □

Lemma 4.3. Let p ≥ 2, r = p^{n}u with n ≥ 1 and p ∤ u, and say r ≡ a mod (p − 1), with 1 ≤ a ≤ p − 1. If Σ := Σ_{p}(u − 1) > a − 1, then dim X_{r−1} = 2p + 2.

Proof. We skip the details of this proof as the basic idea is the same as that of Lemma 4.2. □

Remark 4.4. The special cases a = 1 of the two lemmas above have been used in Section 3. Thus the proof of the structure of X_{r−1}, for a = 1, becomes complete now. Next we will study the module X_{r−1}, for r lying in higher congruence classes, i.e., for 2 ≤ a ≤ p − 1. The structure of X_{r−1} for r ≡ 1 mod (p−1) will be used as the first step of an inductive process. Note that the condition 2 ≤ a ≤ p − 1 implies that the prime p under consideration is odd.

In spite of the lemmas above, the module X_{r−1} often has small dimension, as we are going to show below. The next lemma is complementary to Lemma 4.2, for a ≥ 2. Note that the condition Σ_{p}(r − 1) = a − 1 forces r to be prime to p.

Lemma 4.5. Let p ≥ 3, r > p and let r ≡ a mod (p − 1), with 2 ≤ a ≤ p − 1. If Σ = Σ_{p}(r − 1) = a − 1, then dim X_{r−1} = 2a. In fact, X_{r} ≅ V_{a} and X_{r−1} ≅ V_{a−2} ⊗ D ⊕ V_{a} as M-modules.

Proof. Write r′ = r − 1 = r_{m}p^{m} + r_{m−1}p^{m−1} + ⋯ r_{1}p + r_{0}, where each r_{i} ∈ {0, 1, ⋯, p − 1} and r_{0} + ⋯ + r_{m} = a − 1, by hypothesis. For any α, β ∈ 𝔽_{p}, we have

(αX + βY)^{r′} ≡ (αX^{p^{m}} + βY^{p^{m}})^{r_{m}} ⋯ (αX^{p} + βY^{p})^{r_{1}} · (αX + βY)^{r_{0}} mod p.

Considering this as a polynomial in α and β, we get that

(αX + βY)^{r′} = ∑^{a−1} ; i=0α^{i}β^{a−1−i}F_{i}(X, Y),

where the polynomials F_{i} are independent of α and β. Hence X_{r′} is contained in the 𝔽_{p}-span of the polynomials F_{0}, F_{1}, …, F_{a−1}. Thus dim(X_{r′}) ≤ a. But by [G78, (4.5)], we know X_{r′}/X_{r′}^{∗} ≅ V_{a−1}, hence X_{r′} ≅ V_{a−1} and X_{r′}^{∗} = 0. By [G78, (5.2)] and Lemma 3.6, we have the surjection

(4.3)  X_{r′} ⊗ V_{1} ≅ V_{a−2} ⊗ D ⊕ V_{a} ⟶^{ϕ} X_{r−1}.

By [G78, (4.5)], we know that V_{a} is a quotient of X_{r}. In fact, by an argument similar to the one just given for X_{r′}, we have X_{r} ≅ V_{a} and X_{r}^{∗} = 0, since Σ_{p}(r) = a. Now by Lemma 4.1, the surjection in (4.3) has to be an isomorphism. □

Lemma 4.6. For any r > p, let r ≡ a mod (p − 1), for some 1 ≤ a ≤ p − 1. Then either X_{r}^{∗} = 0 or X_{r}^{∗} ≅ V_{p−a−1} ⊗ D^{a} as an M-module.

Proof. The statement for a = 1 is proved in Section 3 (cf. Propositions 3.3, 3.8, 3.12). Now we use the method of induction to prove it for 2 ≤ a ≤ p − 1. So assuming the statement for all congruence classes less than a, we want to prove it for r ≡ a mod (p − 1). Recall that for r = p^{n}u with p ∤ u, X_{u} ≅ X_{r}, where the M-linear isomorphism is simply given by F ↦ F^{p^{n}}, for any F ∈ X_{u}. So the largest singular submodules X_{u}^{∗} and X_{r}^{∗} are M-isomorphic. We also note that if u < p, then X_{u} = V_{u} and X_{r}^{∗} = X_{u}^{∗} = 0. Thus without loss of generality we may assume that u > p and p ∤ r, i.e., r = u > p. We denote r′ := r − 1, as usual. As p ∤ r and Σ ≡ a − 1 mod (p − 1), r satisfies the hypothesis of either Lemma 4.5 or Lemma 4.2. So dim X_{r−1} is either 2a or 2p + 2 respectively. In the first case we know X_{r} ≅ V_{a}, so X_{r}^{∗} = 0 by [G78, (4.5)]. In the second case, Lemma 4.2, Lemma 3.5 and [G78, (4.5)] together tell us that dim X_{r}^{∗} = p − a. By Lemma 3.6, we have the M-map ϕ : X_{r′} ⊗ V_{1} ↠ X_{r−1}. As dim X_{r−1} = 2p + 2 in this case, X_{r′} must have maximum possible dimension p + 1, and hence X_{r′}^{∗} ≠ 0. By the induction hypothesis, we get X_{r′}^{∗} ≅ V_{p−a} ⊗ D^{a−1}. Counting dimensions we conclude that the map ϕ above must be an isomorphism. The short exact sequence

0 ⟶ X_{r′}^{∗} ⊗ V_{1} ⟶ X_{r′} ⊗ V_{1} ⟶ V_{a−1} ⊗ V_{1} ⟶ 0

implies, by [G78, (5.2)], that we have

0 ⟶ V_{p−a−1} ⊗ D^{a} ⊕ V_{p−a+1} ⊗ D^{a−1} ⟶ X_{r−1} ⟶ V_{a−2} ⊗ D ⊕ V_{a} ⟶ 0,

giving the M-module structure of X_{r−1}. The obvious choice for the (p−a)-dimensional subspace X_{r}^{∗} ⊆ X_{r−1} is V_{p−a−1}⊗ D^{a}. In the particular case when 2a = p + 1, we eliminate the possibility of X_{r}^{∗} being isomorphic to V_{a−2} ⊗ D by using the fact that X_{r} is indecomposable as an M-module [G78, (4.6)], so X_{r} cannot be M-isomorphic to V_{a−2} ⊗ D ⊕ V_{a}. So X_{r}^{∗} ≅ V_{p−a−1} ⊗ D^{a}, as desired. □

Recall that in the case a = 1, we have X_{r}^{∗}/X_{r}^{∗∗} ≠ 0 if and only if p ∤ r, by Lemma 3.1. Our next lemma shows that, for all the higher congruence classes, X_{r}^{∗}/X_{r}^{∗∗} = 0 always.

Lemma 4.7. Let p ≥ 3, r ≥ 2p and r ≡ a mod (p − 1). If 2 ≤ a ≤ p − 1, then X_{r}^{∗} = X_{r}^{∗∗}.

Proof. For r = 2p, one has X_{r} ≅ V_{2} and X_{r}^{∗} = 0, so there is nothing to prove. So assume r > 2p. If X_{r}^{∗}/X_{r}^{∗∗} ≠ 0, then by Lemma 4.6 we must have X_{r}^{∗}/X_{r}^{∗∗} ≅ V_{p−a−1} ⊗ D^{a}. But the short exact sequences (2.5) and (2.6) given by Proposition 2.2 tell us that V_{p−a−1} ⊗ D^{a} is not a Γ-submodule of V_{r}^{∗}/V_{r}^{∗∗}. This is a contradiction. □

The next lemma is complementary to Lemma 4.3 for a ≥ 2 and is comparable to Lemma 3.10 for a = 1. It generates examples of X_{r−1} with relatively small dimension, under the hypothesis p | r.

Lemma 4.8. Let p ≥ 3, r > 2p and let p divide r ≡ a mod (p − 1), with 2 ≤ a ≤ p − 1. Write r = p^{n}u, where n ≥ 1 and p ∤ u. If Σ_{p}(u − 1) = a − 1, then dim X_{r−1} = a + p + 2.

Proof. Since p | r, we have p ∤ r′ := r − 1 ≡ p − 1 mod p. As r > p, we know that Σ_{p}(r′ − 1) is at least p − 1. Since Σ_{p}(r′ − 1) ≡ r′ − 1 ≡ a − 2 mod (p − 1), we have Σ_{p}(r′ − 1) ≥ p + a − 3. If a ≥ 3, then by Lemma 4.2 and Lemma 3.5 we get dim X_{r′} = p + 1. On the other hand if a = 2, then Propositions 3.3 and 3.8 show that dim X_{r′} = p+1. In any case, X_{r′}^{∗} ≠ 0, by [G78, (4.5)]. Therefore X_{r′}^{∗} ≅ V_{p−a} ⊗ D^{a−1}, by Lemma 4.6. Now using [G78, (4.5), (5.2)], we get

\xymatrix{0 \ar[r] & V_{p−a−1} ⊗ D^{a} ⊕ V_{p−a+1} ⊗ D^{a−1} \ar[r] & X_{r′} ⊗ V_{1} \ar@{->>}[d]_{ϕ} \ar[r] & V_{a−2} ⊗ D ⊕ V_{a} \ar[r] & 0. \\  &  & X_{r−1}}

Since r = p^{n}u, we know that X_{u} ≅ X_{r}. As Σ_{p}(u − 1) = a − 1, we have X_{u} ≅ V_{a}, by Lemma 4.5 if u > p, and by the fact that V_{u} = V_{a} is irreducible if u < p. So X_{r} ≅ V_{a} has dimension a + 1 < p + 1, and Lemma 3.5 implies that dim X_{r−1} < 2p + 2. Hence all of the four JH factors of X_{r′} ⊗ V_{1} given above cannot occur in the quotient X_{r−1}. We will show that the last three JH factors, i.e., V_{p−a+1} ⊗D^{a−1}, V_{a−2} ⊗D and V_{a}, always occur in X_{r−1}. Therefore V_{p−a−1} ⊗ D^{a} must die in X_{r−1}. Adding the dimensions of the JH factors we will get dim X_{r−1} = a + p + 2, as desired. For F (X, Y) ∈ V_{m}, let δ_{m}(F) := F_{X} ⊗ X + F_{Y} ⊗ Y ∈ V_{m−1} ⊗ V_{1}, where F_{X}, F_{Y} are the usual partial derivatives of F. It is shown on [G78, p. 449] that the injection V_{p−a+1} ⊗ D^{a−1} ↪ (V_{p−a} ⊗ D^{a−1}) ⊗ V_{1} can be given by the map \frac{1}{p−a+1} · δ_{p−a+1} (ϕ̄ in the notation of [G78]). So the monomial X^{p−a+1} ∈ V_{p−a+1} ⊗ D^{a−1} maps to the non-zero element X^{p−a} ⊗ X ∈ (V_{p−a} ⊗ D^{a−1}) ⊗ V_{1}. Let F ∈ X_{r′}^{∗} be the image of X^{p−a} under the isomorphism V_{p−a} ⊗D^{a−1} →^{∼} X_{r′}^{∗}. Then under the composition of maps

V_{p−a+1} ⊗ D^{a−1} ↪ (V_{p−a} ⊗ D^{a−1}) ⊗ V_{1} ⟶^{∼} X_{r′}^{∗} ⊗ V_{1} ⟶^{ϕ} X_{r−1},

the monomial X^{p−a+1} ↦ X^{p−a} ⊗ X ↦ F ⊗ X ↦ X · F ≠ 0 in X_{r−1}. This shows that V_{p−a+1} ⊗ D^{a−1} must be a JH factor of X_{r−1}. Note that by hypothesis r ≡ 0 ≢ a mod p. We refer to Lemma 6.1 in Section 6 to show that V_{a−2} ⊗D is a JH factor of X_{r−1}, if 3 ≤ a ≤ p−1. For a = 2, the hypothesis implies that r must be of the form r = p^{n}(p^{m} + 1) with m + n ≥ 2. An elementary calculation using Lucas’ theorem shows that in this case the M-linear map X_{r−1} ↠ V_{0} ⊗ D, sending X^{r−1}Y ↦ 1, is well-defined. Thus V_{a−2} ⊗ D is a JH factor of X_{r−1} for all a ≥ 2. Finally, V_{a} is always a JH factor of X_{r−1}, by [G78, (4.5)]. □

We write r = p^{n}u, where p ∤ u and n ≥ 0 is an integer. Note that if p ∤ r, then u simply equals r, and n = 0. Clearly u ≡ a mod (p − 1) as well, so the sum of the p-adic digits of u − 1 lies in the congruence class a − 1 mod (p − 1). We denote this sum by Σ := Σ_{p}(u − 1). Then the M-module structure of X_{r−1} proved in this section can be summarized as follows:

Proposition 4.9. Let p ≥ 3, r > 2p, and r ≡ a mod (p − 1), with 2 ≤ a ≤ p − 1. Then with the notation above

(i) If Σ = a − 1 and p ∤ r (i.e., n = 0), then dim X_{r−1} = 2a and as an

M-module

X_{r−1} ≅ V_{a−2} ⊗ D ⊕ V_{a}.

(ii) If Σ = a − 1 and p | r (i.e., n > 0), then dim X_{r−1} = a + p + 2, and we have the following exact sequence of M-modules

0 ⟶ V_{p−a+1} ⊗ D^{a−1} ⟶ X_{r−1} ⟶ V_{a−2} ⊗ D ⊕ V_{a} → 0.

(iii) If Σ > a − 1, or equivalently Σ ≥ p + a − 2, then dim X_{r−1} = 2p + 2, and we have the following exact sequence of M-modules

0 → V_{p−a−1} ⊗ D^{a} ⊕ V_{p−a+1} ⊗ D^{a−1} → X_{r−1} → V_{a−2} ⊗ D ⊕ V_{a} → 0.

5. The case r ≡ 2 mod (p − 1)

With the notation of Section 2.3, let a = 2. In particular, p will denote an odd prime throughout this section. By Proposition 2.2, we get V_{r}^{∗}/V_{r}^{∗∗} ≅ V_{p−1} ⊗D ⊕V_{0} ⊗D as a Γ-module. For r′ := r −1, we know by [G78, (4.5)] that X_{r′}/X_{r′}^{∗} ≅ V_{1}. As usual by Lemma 3.6 and [G78, (5.2)], we get the following commutative diagram of M-modules:

\xymatrix{0 \ar[r] & X_{r′}^{∗} ⊗ V_{1} \ar@{->>}[d] \ar[r] & X_{r′} ⊗ V_{1} \ar@{->>}[d]_{ϕ} \ar[r] & V_{0} ⊗ D ⊕ V_{2} \ar@{->>}[d] \ar[r] & 0 \\ 0 \ar[r] & ϕ(X_{r′}^{∗} ⊗ V_{1}) \ar[r] & X_{r−1} \ar[r] & \frac{X_{r−1}}{ϕ(X_{r′}^{∗} ⊗ V_{1})} \ar[r] & 0.} (5.1)

Lemma 5.1. Let p ≥ 3, r > 2p and r ≡ 2 mod (p − 1). If p ∤ r, then there is an M-linear surjection

X_{r−1}^{∗}/X_{r−1}^{∗∗} ↠ V_{0} ⊗ D.

Proof. We claim that the following composition of maps is non-zero, hence surjective:

\frac{X_{r−1}^{∗}}{X_{r−1}^{∗∗}} ↪ \frac{V_{r}^{∗}}{V_{r}^{∗∗}} ≅ \frac{V_{r−p−1}}{V_{r−p−1}^{∗}} ⊗ D ⟶^{ψ^{−1}⊗ id} \frac{V_{2p−2}}{V_{2p−2}^{∗}} ⊗ D ↠ V_{0} ⊗ D,

where ψ is the M-isomorphism in [G78, (4.2)] and the rightmost surjection is induced from the Γ-linear map in [B03b, 5.3(ii)]. Indeed, one checks that under the above composition of maps, F (X, Y) = X^{r−1}Y − XY^{r−1} ∈ X_{r−1}^{∗} maps to \frac{r−2p}{p−1} ∈ V_{0}⊗D, which is non-zero, as p ∤ r. Note that the composition is automatically M-linear, as both its domain and range are singular modules. □

Lemma 5.2. Let p ≥ 3, r > 2p, and r ≡ 2 mod (p − 1). If p ∤ r′ := r − 1, then ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ X_{r−1}^{∗}, but ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊈ X_{r−1}^{∗∗}.

Proof. The inclusion is obvious from the definition of the map ϕ. For the rest,

note that F (X, Y) := ∑^{p−1} ; k=0(kX + Y)^{r′} ∈ X_{r′}^{∗}, as shown in the proof of Lemma

3.1 (i). The coefficient of X^{r−1}Y in ϕ(F ⊗ X) = F (X, Y) · X is the same as the coefficient of X^{r′−1}Y in F, which is congruent to −r′ mod p. By hypothesis p ∤ r′, so θ^{2} cannot divide ϕ(F ⊗ X). Therefore ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊈ X_{r−1}^{∗∗}. □

Lemmas 5.1 and 5.2 will be used to study the Γ-module Q. If p ∤ r(r − 1), then we can use both the lemmas simultaneously to prove the following simple structure of Q.

Proposition 5.3. Let p ≥ 3, r > 2p and r ≡ 2 mod (p − 1). If p ∤ r(r − 1), then Q ≅ V_{p−3} ⊗ D^{2} as a Γ-module.

Proof. Since p ∤ r(r − 1), we have Σ = Σ_{p}(r − 1) > a − 1 = 1 with the notation of Proposition 4.9, as otherwise Σ = 1, which forces r − 1 to be a p-power. So by Proposition 4.9 (iii), dim X_{r−1} = 2p + 2, and we have

(5.2)  0 ⟶ V_{p−3} ⊗ D^{2} ⊕ V_{p−1} ⊗ D ⟶ X_{r−1} ⟶ V_{0} ⊗ D ⊕ V_{2} ⟶ 0,

noting that here V_{p−3} ⊗ D^{2} ⊕ V_{p−1} ⊗ D = ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ X_{r−1}. Restricting this short exact sequence of M-modules to the largest singular submodules, we get the structure of X_{r−1}^{∗}:

(5.3)  0 ⟶ V_{p−3} ⊗ D^{2} ⊕ V_{p−1} ⊗ D ⟶ X_{r−1}^{∗} ⟶ V_{0} ⊗ D ⟶ 0,

where the last surjection follows from the fact that V_{0} ⊗ D is a JH factor of X_{r−1}^{∗}, by Lemma 5.1. Next we want to compute the JH factors of X_{r−1}^{∗∗}. By Lemma 3.5 and [G78, (4.5)], we have dim X_{r}^{∗} = p − 2. By Lemmas 4.7 and 4.6, we get that X_{r}^{∗∗} = X_{r}^{∗} ≅ V_{p−3} ⊗ D^{2}, hence this submodule of ϕ(X_{r′}^{∗} ⊗ V_{1}) (⊆ X_{r−1}^{∗}) is in fact contained in X_{r−1}^{∗∗}. By Lemma 5.2, the other JH factor V_{p−1} ⊗D of ϕ(X_{r′}^{∗} ⊗V_{1}) cannot occur in X_{r−1}^{∗∗}. Since V_{0} ⊗ D occurs only once in X_{r−1}, Lemma 5.1 tells that V_{0} ⊗ D cannot be a JH factor of X_{r−1}^{∗∗}. Thus X_{r−1}^{∗∗} ≅ V_{p−3} ⊗ D^{2}. So now we know all the JH factors of X_{r−1}, X_{r−1}^{∗} and X_{r−1}^{∗∗}. As X_{r−1}^{∗}/X_{r−1}^{∗∗} has two JH factors V_{p−1} ⊗ D and V_{0} ⊗ D, we have X_{r−1}^{∗}/X_{r−1}^{∗∗} = V_{r}^{∗}/V_{r}^{∗∗}, and the left module in the bottom row of Diagram (2.2) vanishes. On the other hand X_{r−1}/X_{r−1}^{∗} has only one JH factor V_{2}, so the short exact sequence (2.3) of Γ-modules implies that the rightmost module in the bottom row of Diagram (2.2) is V_{p−3} ⊗ D^{2}. Therefore Q is Γ-isomorphic to V_{p−3} ⊗ D^{2}. □

Next we will treat the case p | r(r − 1). Note that if p | (r − 1), then p ∤ r and we can still use Lemma 5.1. Similarly if p | r, then p ∤ (r − 1) and we can use Lemma 5.2.

Proposition 5.4. Let p ≥ 3, r > 2p and r ≡ 2 mod (p − 1). If p | r − 1, then Q ≅ V_{p−1} ⊗ D ⊕ V_{p−3} ⊗ D^{2} as a Γ-module.

Proof. If r−1 is a pure p-power, then by Proposition 4.9 (i), X_{r−1} ≅ V_{0}⊗D⊕V_{2} as an M-module. Hence X_{r−1}^{∗} ≅ V_{0} ⊗ D, being the largest singular submodule of X_{r−1}. By Lemma 5.1, X_{r−1}^{∗∗} must be zero. In other words, X_{r−1}/X_{r−1}^{∗} ≅ V_{2} and X_{r−1}^{∗}/X_{r−1}^{∗∗} ≅ V_{0} ⊗ D. If r − 1 is not a pure p-power, then Σ_{p}(r − 1) ≥ p. By Proposition 4.9 (iii), dim X_{r−1} = 2p + 2 and we have the exact sequence of M-modules

0 ⟶ V_{p−3} ⊗ D^{2} ⊕ V_{p−1} ⊗ D ⟶ X_{r−1} ⟶ V_{0} ⊗ D ⊕ V_{2} ⟶ 0.

Note that V_{p−3} ⊗ D^{2} ⊕ V_{p−1} ⊗ D = ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ X_{r−1}, with the notation of Diagram (5.1). Similarly as in the proof of Proposition 5.3, we use Lemma 5.1 to obtain

0 ⟶ ϕ(X_{r′}^{∗} ⊗ V_{1}) ⟶ X_{r−1}^{∗} ⟶ V_{0} ⊗ D ⟶ 0.

As p | r′, by Lemma 3.1 (ii) we have X_{r′}^{∗} = X_{r′}^{∗∗}, therefore ϕ(X_{r′}^{∗} ⊗ V_{1}) = ϕ(X_{r′}^{∗∗} ⊗ V_{1}) must be contained inside X_{r−1}^{∗∗}. On the other hand V_{0} ⊗ D occurs only once in X_{r−1}, so Lemma 5.1 implies that V_{0} ⊗ D cannot be a JH factor of X_{r−1}^{∗∗}. Thus X_{r−1}^{∗∗} = ϕ(X_{r′}^{∗} ⊗ V_{1}) ≅ V_{p−3} ⊗ D^{2} ⊕ V_{p−1} ⊗ D. Thus again we get X_{r−1}/X_{r−1}^{∗} ≅ V_{2} and X_{r−1}^{∗}/X_{r−1}^{∗∗} ≅ V_{0} ⊗ D.

Therefore by the exact sequences (2.3) and (2.5), the bottom row of Diagram (2.2) reduces to

0 ⟶ V_{p−1} ⊗ D ⟶ Q ⟶ V_{p−3} ⊗ D^{2} ⟶ 0.

Now the result follows as V_{p−1} ⊗ D is an injective Γ-module. □

Proposition 5.5. Let p ≥ 3, r > 2p and r ≡ 2 mod (p − 1). If p | r, then the Γ-module structure of Q is given by

0 ⟶ V_{0} ⊗ D ⟶ Q ⟶ V_{p−3} ⊗ D^{2} ⟶ 0.

Proof. Let us write r = p^{n}u, n ≥ 1 and p ∤ u. Looking at Diagram (5.1) and using Proposition 4.9, we get the exact sequence of M-modules

0 ⟶ W ⟶ X_{r−1} ⟶ V_{0} ⊗ D ⊕ V_{2} ⟶ 0,

where W = ϕ(X_{r′}^{∗} ⊗ V_{1}) = \begin{cases}V_{p−1} ⊗ D, & if Σ_{p}(u − 1) = 1, \\ V_{p−1} ⊗ D ⊕ V_{p−3} ⊗ D^{2}, & if Σ_{p}(u − 1) ≥ p. \end{cases}

Restricting the above sequence to the largest singular submodules, we get

(5.4)  0 ⟶ W ⟶ X_{r−1}^{∗} ⟶ V_{0} ⊗ D ⟶ 0,

where the last map has to be a surjection, as otherwise V_{0} ⊗ D would be a JH factor of V_{r}/V_{r}^{∗}, which is not true by the sequence (2.3). Now we claim that the restriction of the map X_{r−1}^{∗} ↠ V_{0} ⊗ D in (5.4) to the submodule X_{r−1}^{∗∗} is also non-zero, and hence surjective. \underline{Proof of claim}: If not, then the above map factors via η_{1} : X_{r−1}^{∗}/X_{r−1}^{∗∗} ↠ V_{0} ⊗ D ⊆ V_{1} ⊗ V_{1}. We can check that the elements X^{r′}, Y^{r′} ∈ X_{r′} map to X, Y ∈ V_{1} respectively under the map X_{r′} ↠ X_{r′}/X_{r′}^{∗} ≅ V_{1}. Hence, for

F = F (X, Y) = X^{r−1}Y − Y^{r−1}X = ϕ(X^{r′} ⊗ Y − Y^{r′} ⊗ X) ∈ ϕ(X_{r′} ⊗ V_{1}),

we get η_{1}(F) = X ⊗ Y − Y ⊗ X ∈ V_{1} ⊗ V_{1}. Note that the non-zero element X ⊗ Y − Y ⊗ X of V_{1} ⊗ V_{1} ≅ V_{0} ⊗ D ⊕ V_{2} actually belongs to V_{0} ⊗ D, as it projects to XY − Y X = 0 ∈ V_{2}. So η_{1}(F) ≠ 0. Now we consider the composition of maps η_{2} : X_{r−1}^{∗}/X_{r−1}^{∗∗} ↪ V_{r}^{∗}/V_{r}^{∗∗} ↠ V_{0} ⊗ D, as described in the proof of Lemma 5.1. If η_{2} is the zero map, then the short exact sequence (2.4) implies that X_{r−1}^{∗}/X_{r−1}^{∗∗} ⊆ V_{p−1} ⊗ D, contradicting the existence of the map η_{1} : X_{r−1}^{∗}/X_{r−1}^{∗∗} ↠ V_{0} ⊗ D. So η_{2} is a non-zero map. But the calculation in the proof of Lemma 5.1 shows that η_{2}(F) = 0, since p | r. Thus we end up with two different surjections η_{1}, η_{2} : X_{r−1}^{∗}/X_{r−1}^{∗∗} ↠ V_{0} ⊗ D, one containing F in its kernel and the other not. This forces V_{0} ⊗ D to be a repeated JH factor of X_{r−1}^{∗}/X_{r−1}^{∗∗}, contradicting the fact that it occurs only once in X_{r−1}^{∗}. Thus the claim is proved. If Σ_{p}(u − 1) ≥ p, then the extra JH factor V_{p−3} ⊗ D^{2} of W is actually the submodule X_{r}^{∗} (cf. Lemma 4.6), which equals X_{r}^{∗∗}, by Lemma 4.7. So this JH factor is in fact contained in X_{r−1}^{∗∗}. By Lemma 5.2, we know that W = ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊈ X_{r−1}^{∗∗}. Therefore

W_{1} := W ∩ X_{r−1}^{∗∗} = \begin{cases}0, & if Σ_{p}(u − 1) = 1, \\ V_{p−3} ⊗ D^{2}, & if Σ_{p}(u − 1) ≥ p. \end{cases}

By the claim above, the structure of X_{r−1}^{∗∗} is now given by

0 → W_{1} → X_{r−1}^{∗∗} → V_{0} ⊗ D → 0.

Hence we always have X_{r−1}/X_{r−1}^{∗} ≅ V_{2}, and X_{r−1}^{∗}/X_{r−1}^{∗∗} ≅ W/W_{1} ≅ V_{p−1}⊗D. Next we use the exact sequences (2.3) and (2.5) to see that the bottom row of Diagram (2.2) reduces to

0 ⟶ V_{0} ⊗ D ⟶ Q ⟶ V_{p−3} ⊗ D^{2} ⟶ 0.
□

Remark 5.6. The short exact sequences in Proposition 2.2 split only when a = 2, the case we are dealing with. Note that all three possible non-zero submodules of V_{r}^{∗}/V_{r}^{∗∗} ≅ V_{0} ⊗ D ⊕ V_{p−1} ⊗ D occur as X_{r−1}^{∗}/X_{r−1}^{∗∗} in the three distinct cases p ∤ r(r − 1), p | (r − 1) and p | r.

6. The case r ≡ a mod (p − 1), with 3 ≤ a ≤ p − 1

In this section we describe the Γ-module structure of Q, for r ≡ a mod (p−1), with 3 ≤ a ≤ p − 1. Note that this bound on a forces p to be at least 5.

Lemma 6.1. Let p ≥ 5, r > 2p, r ≡ a mod (p − 1), with 3 ≤ a ≤ p − 1. If r ≢ a mod p, then X_{r−1}^{∗}/X_{r−1}^{∗∗} contains V_{a−2} ⊗ D as a Γ-submodule.

Proof. By the non-split short exact sequence (2.6), it is enough to show the submodule X_{r−1}^{∗}/X_{r−1}^{∗∗} of V_{r}^{∗}/V_{r}^{∗∗} is non-zero. Using Lemma 2.3 (i) and Lemma 2.6, one may check that the polynomial

F (X, Y) = (a − 1)X^{r−1}Y + ∑^{p−1} ; k=0k^{p+1−a}(kX + Y)^{r−1}X ∈ X_{r−1}^{∗}.

But F ∉ V_{r}^{∗∗}, as the coefficient of X^{r−1}Y in F (X, Y) is (a−1)−\binom{r−1}{r−2} = a−r ≢ 0 mod p, by hypothesis. So F maps to a non-zero element in X_{r−1}^{∗}/X_{r−1}^{∗∗}. □

Lemma 6.2. Let p ≥ 5 and r ≡ a mod (p − 1), with 3 ≤ a ≤ p − 1. If r ≡ a mod p, then we have X_{r−1} ⊆ X_{r} + V_{r}^{∗∗} and X_{r−1}^{∗}/X_{r−1}^{∗∗} = 0.

Proof. The lemma is trivial for r = a as V_{a} is irreducible. Thus for each a, r = p^{2} − p + a is the first non-trivial integer satisfying the hypotheses. We begin by expanding the following element of X_{r}:

∑_{k∈𝔽_{p}}k^{p−2}(kX + Y)^{r} ≡ −rXY^{r−1} − ∑_{\begin{matrix}0<j<r−1 \\ j≡a−1 mod (p−1) \end{matrix}}\binom{r}{j} · X^{r−j}Y^{j} mod p.

Using Lemma 2.3 (ii) and Lemma 2.6, one checks that

F (X, Y) := ∑_{\begin{matrix}0<j<r−1 \\ j≡a−1 mod (p−1) \end{matrix}}\binom{r}{j} · X^{r−j}Y^{j}

is in fact contained in V_{r}^{∗∗}, implying that the monomial −rXY^{r−1} is contained in X_{r} + V_{r}^{∗∗}. Since r ≡ a ≢ 0 mod p, we have X_{r−1} ⊆ X_{r} + V_{r}^{∗∗}. Hence X_{r−1}^{∗} ⊆ X_{r−1} ∩ (X_{r}^{∗} + V_{r}^{∗∗}), which equals X_{r−1} ∩ (X_{r}^{∗∗} + V_{r}^{∗∗}), by Lemma 4.7. But X_{r}^{∗∗} ⊆ V_{r}^{∗∗}, so we have X_{r−1}^{∗} ⊆ X_{r−1} ∩V_{r}^{∗∗} = X_{r−1}^{∗∗}. The reverse inclusion is obvious and therefore X_{r−1}^{∗} = X_{r−1}^{∗∗}. □

Remark 6.3. It was proved in [GG15, Prop. 4] that X_{r−1} ⊄ X_{r} + V_{r}^{∗∗}, for p ≤ r ≤ p^{2} − p + 2 and all primes p. The lemma above shows that the upper bound is sharp, at least if p ≥ 5, since the smallest non-trivial r covered by the lemma is p^{2} − p + 3. As we show just below, the fact that X_{r−1} ⊂ X_{r} + V_{r}^{∗∗} causes Q to have 3 JH factors, leading to additional complications.

Proposition 6.4. Let p ≥ 5, r > 2p and r ≡ a mod (p−1), with 3 ≤ a ≤ p−1. The Γ-module structure of Q is as follows.

(i) If r ≢ a mod p, then

0 ⟶ V_{p−a+1} ⊗ D^{a−1} ⟶ Q ⟶ V_{p−a−1} ⊗ D^{a} ⟶ 0,

and moreover the exact sequence above is Γ-split. (ii) If r ≡ a mod p, then

0 ⟶ V_{r}^{∗}/V_{r}^{∗∗} ⟶ Q ⟶ V_{p−a−1} ⊗ D^{a} ⟶ 0,

where V_{r}^{∗}/V_{r}^{∗∗} is the non-trivial extension of V_{p−a+1} ⊗D^{a−1} by V_{a−2} ⊗ D.

Proof. Let r′ := r − 1. As explained in Section 4 (cf. Proposition 4.9), we have

0 ⟶ ϕ(X_{r′}^{∗} ⊗ V_{1}) ⟶ X_{r−1} ⟶ V_{a−2} ⊗ D ⊕ V_{a} → 0.

Restricting the above M-linear maps to the largest singular submodules, we get

0 ⟶ ϕ(X_{r′}^{∗} ⊗ V_{1}) ⟶ X_{r−1}^{∗} ⟶ V_{a−2} ⊗ D → 0.

Indeed, the short exact sequence (2.3) shows that V_{a−2}⊗D is not a JH factor of V_{r}/V_{r}^{∗}, so the surjection X_{r−1} ↠ V_{a−2} ⊗ D cannot factor through X_{r−1}/X_{r−1}^{∗}, hence the rightmost map above is surjective. As 2 ≤ a − 1 ≤ p − 2, by Lemma 4.7 we have X_{r′}^{∗} = X_{r′}^{∗∗}. Hence by the definition of the ϕ map, we get ϕ(X_{r′}^{∗} ⊗ V_{1}) ⊆ X_{r−1}^{∗∗}. If r ≢ a mod p, then Lemma 6.1 implies that X_{r−1}^{∗}/X_{r−1}^{∗∗} must have exactly one JH factor, namely V_{a−2} ⊗ D. So we have X_{r−1}/X_{r−1}^{∗} ≅ V_{a} and X_{r−1}^{∗}/X_{r−1}^{∗∗} ≅ V_{a−2} ⊗ D. Now using the short exact sequences (2.3) and (2.6), the bottom row of Diagram (2.2) reduces to

0 ⟶ V_{p−a+1} ⊗ D^{a−1} ⟶ Q ⟶ V_{p−a−1} ⊗ D^{a} ⟶ 0.

That the sequence splits follows from [BP12, Cor. 5.6 (i)], which implies that there exists no non-trivial extension between the Γ-modules V_{p−a−1} ⊗ D^{a} and V_{p−a+1} ⊗ D^{a−1}. On the other hand if r ≡ a mod p, then X_{r−1}/X_{r−1}^{∗} ≅ V_{a} as before, but X_{r−1}^{∗}/X_{r−1}^{∗∗} = 0, by Lemma 6.2. Now using the short exact sequence (2.3), the bottom row of Diagram (2.2) reduces to

0 ⟶ V_{r}^{∗}/V_{r}^{∗∗} ⟶ Q ⟶ V_{p−a−1} ⊗ D^{a} ⟶ 0,

where the structure of V_{r}^{∗}/V_{r}^{∗∗} is given by the exact sequence (2.6). □

7. Combinatorial lemmas

In this section we prove some technical lemmas which are used repeatedly in the next two sections. Only the first lemma is proved in detail as the techniques used to prove the others are similar.

Lemma 7.1. Let r ≡ a mod (p − 1), with 2 ≤ a ≤ p − 1. Then one can choose integers α_{j} ∈ ℤ, for all j ≡ a mod (p − 1), with 0 < j < r, such that

(i) \binom{r}{j} ≡ α_{j} mod p, for all j as above,

(ii) ∑_{j}α_{j} ≡ 0 mod p^{3},

(iii) ∑_{j≥1}j α_{j} ≡ 0 mod p^{2},

(iv) ∑_{j≥2}\binom{j}{2} α_{j} ≡ \begin{cases}\binom{r}{2} mod p, & if a = 2, \\ 0 mod p, & if 3 ≤ a ≤ p − 1. \end{cases}

Proof. For r ≤ ap, note that Σ_{p}(r) = a and one can check using Lucas’ theorem that \binom{r}{j} ≡ 0 mod p for all the j’s listed above. In this case we simply choose α_{j} = 0, for all j. So now assume r > ap, hence j = a, ap are both contained in the list of j’s above. Let a′ be a fixed integer such that a′a ≡ 1 mod p^{2}, and then let us choose the α_{j}, for all 0 < j < r, with j ≡ a mod (p − 1), as follows:

(7.1)  α_{j} = \binom{r}{j}, for j ≠ a, ap,
α_{a} = − ∑_{\begin{matrix}a < j < r \\ j≡a mod (p−1) \end{matrix}} a′j \binom{r}{j}, (7.2)

(7.3)  α_{ap} = − ∑_{j≠a, ap}\binom{r}{j} − α_{a}.

We will show that these α_{j} satisfy the properties (i), (ii), (iii) and (iv).

(i) Note that j \binom{r}{j} = r \binom{r−1}{j−1}, for any j ≥ 1. Using Lemma 2.5, for r − 1 and r respectively, we obtain

α_{a} = − ∑_{\begin{matrix}a<j<r \\ j≡a mod (p−1) \end{matrix}} a′r \binom{r − 1}{j − 1} ≡^{(2.5)} a′r \binom{r − 1}{a − 1} = a′a \binom{r}{a} ≡ \binom{r}{a} mod p,

α_{ap} ≡^{(2.5)} \binom{r}{a} + \binom{r}{ap} − α_{a} ≡ \binom{r}{ap} mod p.

For j ≠ a, ap, property (i) is trivially satisfied. (ii) By our choice of α_{j}, in fact ∑_{j}α_{j} = 0.

(iii) Since a′a ≡ 1 mod p^{2}, using equation (7.2) we get a · α_{a} ≡

− ∑_{\begin{matrix}a<j<r, \\ j≡a mod (p−1) \end{matrix}}j \binom{r}{j} mod p^{2}. Since α_{ap} ≡ \binom{r}{ap} mod p, we have

ap · α_{ap} ≡ ap\binom{r}{ap} mod p^{2}. Using these two congruences we conclude that ∑_{j} jα_{j} ≡ 0 mod p^{2}, as desired.

(iv) We use property (i) and Lemma 2.5 for r − 2 to get

∑_{\begin{matrix}0<j<r \\ j≡a mod (p−1) \end{matrix}}\binom{j}{2}α_{j} ≡ ∑_{\begin{matrix}0<j<r \\ j≡a mod (p−1) \end{matrix}}\binom{j}{2}\binom{r}{j}

= ∑_{\begin{matrix}0<j<r \\ j≡a mod (p−1) \end{matrix}}\binom{r}{2}\binom{r − 2}{j − 2} ≡ \begin{cases}\binom{r}{2} mod p, & if a = 2, \\ 0 mod p, & if 3 ≤ a ≤ p − 1. \end{cases}

□

Lemma 7.2. Let r ≡ b mod (p − 1), and 3 ≤ b ≤ p. If p | r − b, then one can choose integers β_{j}, for all j ≡ b − 1 mod (p − 1), with b − 1 ≤ j < r − 1, satisfying:

(1) β_{j} ≡ \binom{r}{j} mod p, for all j as above,
(2) ∑_{j≥n} \binom{j}{n} β_{j} ≡ 0 mod p^{3−n}, for n = 0, 1 and 2.

Proof. As p | b−r, we have r ≡ b mod (p^{2}−p), so we may assume r ≥ p^{2}−p+b. Thus we have j = b − 1, (b − 1)p are two of the j’s in the expression for T_{r} in Lemma 2.6. Let us choose

(7.4)  β_{j} = \binom{r}{j}, for all j ≠ b − 1, (b − 1)p,
(7.5)  β_{b−1} = − ∑_{\begin{matrix}b−1 <j < r−1 \\ j≡b−1 mod (p−1) \end{matrix}} b′j\binom{r}{j},

(7.6)  β_{(b−1)p} = − ∑_{\begin{matrix}j≠b−1 \\ j≠(b−1)p \end{matrix}} \binom{r}{j} − β_{b−1},

where b′ is any integer satisfying (b−1)b′ ≡ 1 mod p^{2}. One can now check that the integers β_{j} satisfy the required properties. The proof uses the congruence in Lemma 2.6 and is similar to that of Lemma 7.1, so we leave it as an exercise. □

Lemma 7.3. Let p ≥ 3, r ≡ 1 mod (p − 1), i.e., b = p. Suppose that p^{2} | r − p. Then

(i) One can choose α_{j} ∈ ℤ, for all j ≡ 1 mod (p − 1), with p ≤ j < r,

satisfying: (1) α_{j} ≡ \binom{r}{j} mod p^{2}, for all j as above,

(2) ∑_{j≥n} \binom{j}{n}α_{j} ≡ 0 mod p^{4−n}, for n = 0, 1 and 2,

(3) ∑_{j≥3}\binom{j}{3}α_{j} ≡ \begin{cases}0 mod p, & if p ≥ 5 \\ 1 mod p, & if p = 3. \end{cases}

(ii) One can choose γ_{j} ∈ ℤ, for all j ≡ 0 mod (p−1), with p−1 ≤ j < r−1,

satisfying: (1) γ_{j} ≡ \binom{r}{j} mod p^{2}, for all j as above,

(2) ∑_{j≥n} \binom{j}{n}γ_{j} ≡ 0 mod p^{4−n}, for n = 0, 1 and 2,

(3) ∑_{j≥3}\binom{j}{3}γ_{j} ≡ \begin{cases}0 mod p, & if p ≥ 5 \\ −1 mod p, & if p = 3. \end{cases}

Proof. Similar to that of Lemma 7.1 and 7.2, and uses the congruences given in Lemma 2.7. □

Remark 7.4. The integers α_{j}, β_{j}, γ_{j} in the lemmas above are not unique, but their existence will be crucial for us. We will use these integers in §8 and §9 to construct functions to eliminate JH factors of Q, and to compute the reduction V̄_{k,a_{p}}, which is the main goal of this paper.

8. Elimination of JH factors

For the rest of this paper, we will work under the assumption 1 < v(a_{p}) < 2. Let us recall Proposition 3.3 in [BG09]: If Θ̄_{k,a_{p}} is a quotient of ind_{KZ}^{G}(V_{s}⊗D^{n}) for some 0 ≤ s ≤ p − 1, then V̄_{k,a_{p}} ≅ ind(ω_{2}^{s+1+(p+1)n}), unless s = p − 2, where one has the additional possibility that V̄_{k,a_{p}} is reducible and isomorphic to ω^{n} ⊕ ω^{n} on I_{p}, the inertia subgroup at p. Using this result one can specify the shape of V̄_{k,a_{p}} when Q is irreducible as a Γ-module, up to the fact that V̄_{k,a_{p}} may be occasionally reducible, as mentioned above. For example, we have

Theorem 8.1. Let p ≥ 3 and r > 2p.

(i) If r ≡ 1 mod (p − 1) and p ∤ r, then V̄_{k,a_{p}} ≅ ind(ω_{2}^{2}). If p = 3, then V̄_{k,a_{p}} can also possibly be reducible and trivial on I_{p}. (ii) If r ≡ 2 mod (p − 1) and p ∤ r(r − 1), then V̄_{k,a_{p}} ≅ ind(ω_{2}^{3}).

Proof. As 1 < v(a_{p}) < 2, there exists a surjection ind_{KZ}^{G}Q ↠ Θ̄_{k,a_{p}}. In part (i), we have Q ≅ V_{1} by Propositions 3.3 and 3.8. In part (ii), Q ≅ V_{p−3} ⊗ D^{2} by Proposition 5.3. Since Q is irreducible in both cases, we apply [BG09, Prop. 3.3] to determine V̄_{k,a_{p}}. □

Remark 8.2. In fact, Theorem 9.1 in the next section will imply that the reducible possibility in part (i) above never occurs. Note that for p = 3, the condition p ∤ r implies that \binom{r−1}{2} ≡ 0 mod p and therefore the hypothesis of Theorem 9.1 is automatically satisfied.

As we have already seen, Q is usually not irreducible. It can have two and sometimes even three JH factors as a Γ-module depending on the congruence class of r modulo both p−1 and p. In these cases we will use the explicit formula for the Hecke operator T acting on the space ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2}, to eliminate one or two JH factors of Q, so that we can use [BG09, Prop. 3.3]. To work explicitly with the Hecke operator T, we need to recall some wellknown formulas involving T from [B03b]. For m = 0, set I_{0} = {0}, and for m > 0, let

I_{m} = {[λ_{0}] + [λ_{1}]p + ⋯ + [λ_{m−1}]p^{m−1} : λ_{i} ∈ 𝔽_{p}} ⊂ ℤ_{p},

where the square brackets denote Teichmüller representatives. For m ≥ 1, there is a truncation map [ ]_{m−1} : I_{m} → I_{m−1} given by taking the first m − 1 terms in the p-adic expansion above; for m = 1, [ ]_{m−1} is the 0-map. Let

α = ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ). For m ≥ 0 and λ ∈ I_{m}, let

g_{m,λ}^{0} = (\begin{matrix}p^{m} & λ \\ 0 & 1 \end{matrix})  and g_{m,λ}^{1} = (\begin{matrix}1 & 0 \\ pλ & p^{m+1} \end{matrix}) ,

noting that g_{0,0}^{0} = Id is the identity matrix and g_{0,0}^{1} = α in G. Recall the decomposition

G = ∐_{\begin{matrix}m≥0, λ∈I_{m}, \\ i∈\{0,1\} \end{matrix}} KZ(g_{m,λ}^{i})^{−1}.

Thus, a general element in ind_{KZ}^{G}V, for a KZ-module V, is a finite sum of elementary functions of the form [g, v], with g = g_{m,λ}^{0} or g_{m,λ}^{1}, for some λ ∈ I_{m} and v ∈ V. For a ℤ_{p}-algebra R, let v = ∑_{i=0}^{r} c_{i}X^{r−i}Y^{i} ∈ V = Sym^{r}R^{2} ⊗ D^{s}.

Expanding the formula (2.1) for the Hecke operator T one may write T = T^{+} + T^{−}, where

T^{+}([g_{n,μ}^{0}, v]) = ∑_{λ∈I_{1}} [g_{n+1,μ+p^{n}λ}^{0}, ∑_{j=0}^{r} (p^{j} ∑_{i=j}^{r} c_{i}\binom{i}{j}(−λ)^{i−j}) X^{r−j}Y^{j}] ,

T^{−}([g_{n,μ}^{0}, v]) = [g_{n−1,[μ]_{n−1}}^{0}, ∑_{j=0}^{r} (∑_{i=j}^{r} p^{r−i}c_{i}\binom{i}{j} ( \frac{μ − [μ]_{n−1}}{p^{n−1}} )^{i−j}) X^{r−j}Y^{j}]
(n > 0),
T^{−}([g_{n,μ}^{0}, v]) = [α, ∑_{j=0}^{r} p^{r−j}c_{j}X^{r−j}Y^{j}] (n = 0).

The formulas for T^{+} and T^{−} will be used to calculate the (T − a_{p})-image of the functions f ∈ ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2}. Though T is a G-linear operator, note that T^{+} and T^{−} are not G-linear. Formulas similar to those above describe how T acts on functions of the form [g_{n,μ}^{1}, v] but we will not use these functions in this article. An integral function, i.e., an element of ind_{KZ}^{G}Sym^{r}ℤ̄_{p}^{2} will be said to “die mod p”, if it maps to zero in ind_{KZ}^{G}Sym^{r}𝔽̄_{p}^{2} under the standard reduction map.

Theorem 8.3. Let p ≥ 5, r > 2p, r ≡ a mod (p − 1), with 3 ≤ a ≤ p − 1. If r ≢ a mod p, then there exists a surjection

ind_{KZ}^{G}(V_{p−a+1} ⊗ D^{a−1}) ↠ Θ̄_{k,a_{p}}.

As a consequence, V̄_{k,a_{p}} ≅ ind(ω_{2}^{a+p}), if a > 3. For a = 3, we have the additional possibility that V̄_{k,a_{p}} is reducible and its restriction to I_{p} is ω^{2} ⊕ ω^{2}.

Proof. By Proposition 6.4 (i), we have Q ≅ J_{1} ⊕ J_{2} as a Γ-module, where J_{1} = V_{p−a+1} ⊗ D^{a−1} and J_{2} = V_{p−a−1} ⊗ D^{a}. Let F_{1} be the image of ind_{KZ}^{G}J_{1} in Θ̄_{k,a_{p}} under the surjection ind_{KZ}^{G}Q ↠ Θ̄_{k,a_{p}}, and let F_{2} denote the quotient Θ̄_{k,a_{p}}/F_{1}. Then we have the commutative diagram:

\xymatrix{0 \ar[r] & ind_{KZ}^{G}J_{1} \ar@{->>}[d] \ar[r] & ind_{KZ}^{G}Q \ar@{->>}[d] \ar[r] & ind_{KZ}^{G}J_{2} \ar@{->>}[d] \ar[r] & 0 \\ 0 \ar[r] & F_{1} \ar[r] & Θ̄_{k,a_{p}} \ar[r] & F_{2} \ar[r] & 0.}

We will construct a function in X_{k,a_{p}} = ker(ind_{KZ}^{G}V_{r} ↠ Θ̄_{k,a_{p}}), which maps to a function of the form [g, v] ∈ ind_{KZ}^{G}J_{2} under the surjection in the top row above, for some g ∈ G and 0 ≠ v ∈ J_{2}. Since the G-span of [g, v] is all of ind_{KZ}^{G}J_{2}, we get F_{2} = 0 and Θ̄_{k,a_{p}} ≅ F_{1} is a quotient of ind_{KZ}^{G}J_{1}. The final conclusion follows by applying [BG09, Prop. 3.3].

Consider the function f ∈ ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2} defined by f := f_{2} + f_{1} + f_{0} with

f_{2} = ∑_{λ∈𝔽_{p}} [ g_{2,p[λ]}^{0}, \frac{1}{p} · (Y^{r} − X^{p−1}Y^{r−p+1}) ] ,

f_{1} = [ g_{1,0}^{0}, \frac{(p − 1)}{pa_{p}} · ∑_{\begin{matrix}0<j<r \\ j≡a mod (p−1) \end{matrix}} α_{j} · X^{r−j}Y^{j} ] ,

f_{0} = \begin{cases}0, & if a < p − 1, \\ [Id, \frac{(1−p)}{p} · (X^{r} − X^{r−p+1}Y^{p−1})] , & if a = p − 1, \end{cases}

where the α_{j} ∈ ℤ are integers satisfying the four properties stated in Lemma 7.1. Applying the explicit formulas for T^{+} and T^{−}, using Lemma 7.1 and the facts r > 2p, 3 ≤ p − 1 and 1 < v(a_{p}) < 2, we get that the functions T^{−}f_{0}, T^{−}f_{1}, T^{+}f_{1}, a_{p}f_{0}, a_{p}f_{2} are all integral and die mod p. We compute that T^{+}f_{2} ∈ ind_{KZ}^{G}⟨X^{r−1}Y ⟩ℤ̄_{p} + p · ind_{KZ}^{G}Sym^{r}ℤ̄_{p}^{2}, hence it maps to 0 ∈ ind_{KZ}^{G}Q. Next we use the formulas for T^{−}, T^{+} and the identity (2.7) to compute that

T^{−}f_{2} − a_{p}f_{1} + T^{+}f_{0} ≡ [g_{1,0}^{0}, F (X, Y)] mod p,

where

F (X, Y) = ∑_{\begin{matrix}0<j<r \\ j≡a mod (p−1) \end{matrix}} \frac{(p − 1)}{p} (\binom{r}{j} − α_{j}) · X^{r−j}Y^{j} + Y^{r}.

Note that F (X, Y) is integral, as α_{j} ≡ \binom{r}{j} mod p, for each j, by Lemma 7.1. All the information above together implies that (T − a_{p})f ∈ ind_{KZ}^{G}Sym^{r}ℤ̄_{p}^{2}, so its reduction lies in X_{k,a_{p}}. Moreover, the reduction \overline{(T − a_{p})f} maps to [g_{1,0}^{0}, , Pr_{2}(F)] ∈ ind_{KZ}^{G}J_{2}, where Pr_{2} : Q ↠ J_{2} is the projection map. It is

clear from Diagram (2.2) that Pr_{2} is induced by the map V_{r} ↠ \frac{V_{r}}{V_{r}^{∗}} ↠ J_{2}.

Noting that the monomial Y^{r} maps to 0 in Q, we have

Pr_{2}(F) = −Pr_{2} (∑_{j} \frac{1}{p} (\binom{r}{j} − α_{j}) · X^{r−j}Y^{j})

= −∑_{j} \frac{1}{p} (\binom{r}{j} − α_{j}) · Pr_{2}(X^{r−a}Y^{a}),

since all the mixed monomials in F (X, Y) are congruent to scalar multiples of

X^{r−a}Y^{a} modulo V_{r}^{∗}. Under the composition V_{r}/V_{r}^{∗} ⟶^{∼} V_{a+p−1}/V_{a+p−1}^{∗} ↠

V_{p−a−1} ⊗ D^{a} = J_{2}, where the first isomorphism is ψ^{−1} of [G78, (4.2)] and the next surjection is induced from [B03b, Lem. 5.3], we have

X^{r−a}Y^{a} mod V_{r}^{∗} ↦ X^{p−1}Y^{a} mod V_{a+p−1}^{∗} ↦ X^{p−1−a} ≠ 0.

Therefore Pr_{2}(F) = c · Pr_{2}(X^{r−a}Y^{a}) = c · X^{p−1−a}, where c is the mod p

reduction of the sum −∑_{j} \frac{1}{p} ·(\binom{r}{j} − α_{j}) ∈ ℤ. By property (ii) of the integers α_{j}

stated in Lemma 7.1, c is the reduction of −\frac{1}{p} ·∑_{j}\binom{r}{j} ∈ ℤ, which is congruent to

\frac{r − a}{a} mod p, by Lemma 2.5. By hypothesis, r ≢ a mod p, so we get c ∈ 𝔽̄_{p}^{∗}.

Thus the reduction \overline{(T − a_{p})f} maps to [g, v] ∈ ind_{KZ}^{G}J_{2}, where g = g_{1,0}^{0} ∈ G and v = c · X^{p−1−a} is a non-zero element in J_{2}, and we are done. □

The following theorem is complementary to Theorem 8.1 (ii).

Theorem 8.4. Let p ≥ 3, r > 2p, r ≡ 2 mod (p − 1). If p | r(r − 1), then V̄_{k,a_{p}} ≅ ind(ω_{2}^{2+p}).

Proof. If p | r(r − 1), then Q has two JH factors by Propositions 5.4 and 5.5. We will eliminate the JH factor J_{2} = V_{p−3} ⊗ D^{2}. We take the function f = f_{0} + f_{1} + f_{2} as in the proof of Theorem 8.3, for a = 2. Note that property (iv) in Lemma 7.1 implies that ∑_{j}\binom{j}{2}α_{j} ≡ \binom{r}{2} ≡ 0 mod p, by hypothesis.

Now by the same argument as in Theorem 8.3, we eliminate the JH factor J_{2}, except for the following subtlety: to show T^{−}f_{1} ≡ 0 mod p, we used the bound 3 ≤ p−1 in Theorem 8.3. This cannot be used in the present case as a = 2 and so p = 3 is allowed. But p | r(r − 1) implies that α_{r−p+1} ≡ \binom{r}{r−p+1} ≡ 0 mod p by Lucas’ theorem. This ensures that T^{−}f_{1} dies mod p even in this case. Therefore if p | r − 1, then we have a surjection ind_{KZ}^{G}(V_{p−1} ⊗ D) ↠ Θ̄_{k,a_{p}} by Proposition 5.4, and if p | r, then we have ind_{KZ}^{G}(V_{0} ⊗ D) ↠ Θ̄_{k,a_{p}} by Proposition 5.5. Now we use [BG09, Prop. 3.3] to draw the final conclusion. □

Next we treat some cases where Q has three JH factors and those coming from V_{r}^{∗}/V_{r}^{∗∗} are to be eliminated. We begin by stating the following easy lemma. Note that we already have a complete solution to the problem in the case b = 2 by Theorem 8.1 (ii) and Theorem 8.4. So we assume b ≥ 3 from now on.

Lemma 8.5. Let p ≥ 3, r > 2p, r ≡ b mod (p − 1), with 3 ≤ b ≤ p. Then we have the non-split short exact sequence of Γ-modules

0 → J_{0} := V_{b−2} ⊗ D → V_{r}^{∗}/V_{r}^{∗∗} → J_{1} := V_{p−b+1} ⊗ D^{b−1} → 0, where

(i) The monomials Y^{b−2}, X^{b−2} ∈ J_{0} map to θY^{r−p−1} and θX^{r−p−1} re-

spectively in V_{r}^{∗}/V_{r}^{∗∗}.

(ii) The polynomials θY^{r−p−1}, θX^{r−p−1} ∈ V_{r}^{∗}/V_{r}^{∗∗} map to 0 ∈ J_{1} and

θX^{r−p−b+1}Y^{b−2} maps to X^{p−b+1} ∈ J_{1}.

Proof. The exact sequence is given by Proposition 2.2. By [G78, (4.1), (4.2)],

we have an isomorphism V_{r}^{∗}/V_{r}^{∗∗} ⟶^{ψ^{−1}⊗id} (V_{p+b−3}/V_{p+b−3}^{∗}) ⊗ D. Then one computes the images of the polynomials mentioned above under the Γ-maps V_{b−2} ⊗D ↪ (V_{p+b−3}/V_{p+b−3}^{∗})⊗D and (V_{p+b−3}/V_{p+b−3}^{∗})⊗D ↠ V_{p−b+1} ⊗D^{b−1} respectively, using the explicit formulas from [B03b, Lem. 5.3]. □

The next two theorems are complementary to Theorem 8.3 above.

Theorem 8.6. Let p ≥ 5, r > 2p and r ≡ b mod (p − 1), with 4 ≤ b ≤ p − 1. If r ≡ b mod p, then V̄_{k,a_{p}} ≅ ind(ω_{2}^{b+1}).

Proof. By Proposition 6.4 (ii), Q contains V_{r}^{∗}/V_{r}^{∗∗} as a submodule, which is an extension of J_{1} by J_{0}, with the notation of Lemma 8.5. Let F_{0,1} denote the image of ind_{KZ}^{G}(V_{r}^{∗}/V_{r}^{∗∗}) inside Θ̄_{k,a_{p}}, and let F_{2} := Θ̄_{k,a_{p}}/F_{0,1}. Then we have the following commutative diagram of G-maps

\xymatrix{0 \ar[r] & ind_{KZ}^{G}(V_{r}^{∗}/V_{r}^{∗∗}) \ar@{->>}[d] \ar[r] & ind_{KZ}^{G}Q \ar@{->>}[d] \ar[r] & ind_{KZ}^{G}J_{2} \ar@{->>}[d] \ar[r] & 0 \\ 0 \ar[r] & F_{0,1} \ar[r] & Θ̄_{k,a_{p}} \ar[r] & F_{2} \ar[r] & 0,}

where J_{2} = V_{p−b−1} ⊗ D^{b}. We will show that F_{0,1} = 0, under the hypothesis r ≡ b mod p. Consider f = f_{0} + f_{1} ∈ ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2}, given by

f_{1} = ∑_{λ∈𝔽_{p}^{∗}} [ g_{1,[λ]}^{0}, \frac{p}{a_{p}} [λ]^{p−2} · (Y^{r} − X^{r−b}Y^{b})]

+ [ g_{1,0}^{0}, \frac{r(1 − p)}{a_{p}} · (XY^{r−1} − X^{r−b+1}Y^{b−1})] ,

f_{0} = [Id, ∑_{\begin{matrix}0<j<r−1 \\ j≡b−1 mod (p−1) \end{matrix}} \frac{p(p − 1)}{a_{p}^{2}} · β_{j} X^{r−j}Y^{j} ] ,

where the β_{j} are the integers from Lemma 7.2. Using b > 2 and the fact that p | r − b, we check that T^{+}f_{1} ≡ 0 mod p. Similarly, T^{−}f_{0} ≡ 0 mod p, since v(a_{p}^{2}/p) < 3 ≤ p. We use the fact b > 3, together with the properties satisfied by the integers β_{j} in Lemma 7.2 to conclude that T^{+}f_{0} ≡ 0 mod p as well. Next we compute that

T^{−}f_{1} − a_{p}f_{0} ≡ [Id, ∑_{\begin{matrix}0<j<r−1 \\ j≡b−1 mod (p−1) \end{matrix}} \frac{(p − 1)p}{a_{p}} (\binom{r}{j} − β_{j}) · X^{r−j}Y^{j}] ,

which again dies mod p, since β_{j} ≡ \binom{r}{j} mod p for each j, by Lemma 7.2. Finally we get (T − a_{p})f is integral and (T − a_{p})f ≡ −a_{p}f_{1} mod p. As r ≡ b mod p, we have (T − a_{p})f

≡ − [g_{1,0}^{0}, r(1 − p) · (XY^{r−1} − X^{r−b+1}Y^{b−1})]
≡ − [g_{1,0}^{0}, −b · θ (\frac{r − p − b + 1}{p − 1} · X^{r−p−b+1}Y^{b−2} + Y^{r−p−1}) mod V_{r}^{∗∗}]
≡ [g_{1,0}^{0}, −b · θ (X^{r−p−b+1}Y^{b−2} − Y^{r−p−1})] mod p.

Let v be the image of −b · θ (X^{r−p−b+1}Y^{b−2} − Y^{r−p−1}) in V_{r}^{∗}/V_{r}^{∗∗}. Then the reduction \overline{(T − a_{p})f} maps to [g_{1,0}^{0}, v] ∈ ind_{KZ}^{G}(V_{r}^{∗}/V_{r}^{∗∗}) ⊆ ind_{KZ}^{G}Q. By Lemma 8.5, v maps to the non-zero element −b·X^{p−b+1} ∈ J_{1} = V_{p−b+1}⊗D^{b−1}. As the short exact sequence (2.6) is non-split, v generates the whole module V_{r}^{∗}/V_{r}^{∗∗} over Γ, so the element [g_{1,0}^{0}, v] generates ind_{KZ}^{G}(V_{r}^{∗}/V_{r}^{∗∗}) over G. Thus F_{0,1} = 0 and hence Θ̄_{k,a_{p}} ≅ F_{2} is a quotient of ind_{KZ}^{G}(V_{p−b−1} ⊗ D^{b}). Finally we apply [BG09, Prop. 3.3] to get the structure of V̄_{k,a_{p}}. □

However, for b = 3 ≤ p−1, we do not have a complete solution to the problem of computing V̄_{k,a_{p}} when r ≡ b mod p(p − 1). But we have the following theorem which is applicable whenever v(a_{p}) ≠ \frac{3}{2}. It is also applicable if v(a_{p}) = \frac{3}{2},

unless the unit \frac{a_{p}^{2}}{p^{3}} reduces to 1 in 𝔽̄_{p}.

Theorem 8.7. Let p ≥ 5, r > 2p and r ≡ 3 mod p(p − 1). If v(a_{p}) = \frac{3}{2}, then assume that v(a_{p}^{2} − p^{3}) = 3. Then V̄_{k,a_{p}} ≅ ind(ω_{2}^{4}).

Proof. If v(a_{p}) ≤ 3/2, then we consider f = f_{0} + f_{1}, where f_{0} are f_{1} are as in the proof of Theorem 8.6 with b = 3. The formula for the Hecke operator shows that (T − a_{p})f is still integral. As b = 3, now T^{+}f_{0} does not necessarily die mod p. In fact we have

T^{+}f_{0} ≡ [g_{1,0}^{0}, \frac{p^{3}(p − 1)}{a_{p}^{2}} \binom{2}{2}β_{2} · X^{r−2}Y^{2}] mod p,

which is integral because v(a_{p}^{2}) ≤ 3, and we have

(T − a_{p})f ≡ T^{+}f_{0} − a_{p}f_{1}

≡ [g_{1,0}^{0}, \frac{p^{3}(p − 1)}{a_{p}^{2}} β_{2} · X^{r−2}Y^{2} − r(1 − p) (XY^{r−1} − X^{r−2}Y^{2})] mod p.

Note that r ≡ 3 mod p and also β_{2} ≡ \binom{r}{2} ≡ \binom{3}{2} = 3 mod p by Lucas’ theorem. As XY^{r−1} vanishes in Q, the reduction \overline{(T − a_{p})f} maps to the image of [g_{1,0}^{0}, 3 (1 − p^{3}/a_{p}^{2}) (X^{r−2}Y^{2} − XY^{r−1})] in ind_{KZ}^{G}Q. The hypothesis implies that 1 − p^{3}/a_{p}^{2} is a p-adic unit. So, the module ind_{KZ}^{G}(V_{r}^{∗}/V_{r}^{∗∗}) maps to 0 in Θ̄_{k,a_{p}}, as explained in the proof of Theorem 8.6.

If v(a_{p}) > 3/2, then we consider the new function f′ = \frac{a_{p}^{2}}{p^{3}} · f = f′_{0} + f′_{1}, with

f′_{1} = ∑_{λ∈𝔽_{p}^{∗}} [ g_{1,[λ]}^{0}, \frac{a_{p}}{p^{2}} [λ]^{p−2} · (Y^{r} − X^{r−3}Y^{3})]

+ [ g_{1,0}^{0}, \frac{(1 − p)ra_{p}}{p^{3}} · (XY^{r−1} − X^{r−2}Y^{2})] ,

f′_{0} = [Id, ∑_{\begin{matrix}0<j<r−1 \\ j≡2 mod (p−1) \end{matrix}} \frac{(p − 1)}{p^{2}} · β_{j} X^{r−j}Y^{j} ] ,

where the β_{j} are the integers from Lemma 7.2. The computations are very similar to the previous case, except that now we have a_{p}f′_{1} ≡ 0 mod p, since v(a_{p}^{2}/p^{3}) > 0. Finally we get (T − a_{p})f′ ≡ T^{+}f′_{0} ≡ [g_{1,0}^{0}, (p − 1)β_{2} · X^{r−2}Y^{2}] mod p. So the reduction \overline{(T − a_{p})f′} maps to the image of [g_{1,0}^{0}, 3(XY^{r−1} − X^{r−2}Y^{2})] in ind_{KZ}^{G}Q. The rest of the proof follows as in the previous case. □

Remark 8.8. Note that when b = 3 and p | r − b, the hypothesis (⋆) in Theorem 1.1 is equivalent to the condition v(a_{p}^{2} − p^{3}) = 3 above. If v(a_{p}^{2} − p^{3}) > 3, so necessarily v(a_{p}) = \frac{3}{2}, we can only show that V̄_{k,a_{p}} is either ind(ω_{2}^{4}) or ind(ω_{2}^{3+p}), or it is reducible of the form ω^{2} ⊕ ω^{2} or ω^{3} ⊕ ω on I_{p}.

The following theorem is complementary to Theorem 8.1 (i). It treats the case p | r ≡ 1 mod p−1, where Q has two JH factors. With the notation of Lemma 7.2, a = 1 is equivalent to b = p, and so the condition p | r can also be stated as r ≡ b mod p.

Theorem 8.9. For p ≥ 3, let p < r ≡ 1 mod (p − 1) and suppose p | r. If p = 3 and v(a_{p}) = \frac{3}{2}, then further assume that v(a_{p}^{2} − p^{3}) = 3. Then we have

(i) If p^{2} ∤ r − p, then there is a surjection ind_{KZ}^{G}V_{1} ↠ Θ̄_{k,a_{p}}. As a consequence, V̄_{k,a_{p}} ≅ ind(ω_{2}^{2}) unless p = 3, in which case V̄_{k,a_{3}} may be reducible, and trivial on I_{3}.

(ii) If p^{2} | r − p, then there is a surjection ind_{KZ}^{G}(V_{p−2} ⊗ D) ↠ Θ̄_{k,a_{p}}. As a consequence, either V̄_{k,a_{p}} ≅ ind(ω_{2}^{2}) or V̄_{k,a_{p}} is reducible, with the shape ω ⊕ ω on I_{p}.

Proof. By Proposition 3.12 (iii), we have Q ≅ V_{r}^{∗}/V_{r}^{∗∗} is an extension of J_{1} = V_{1} by J_{0} = V_{p−2} ⊗ D. Let F_{0} ⊆ Θ̄_{k,a_{p}} be the image of ind_{KZ}^{G}J_{0} under the map ind_{KZ}^{G}Q ↠ Θ̄_{k,a_{p}}. Then F_{1} := Θ̄_{k,a_{p}}/F_{0} is a quotient of ind_{KZ}^{G}J_{1} and we have the following commutative diagram:

\xymatrix{0 \ar[r] & ind_{KZ}^{G}J_{0} \ar@{->>}[d] \ar[r] & ind_{KZ}^{G}Q \ar@{->>}[d] \ar[r] & ind_{KZ}^{G}J_{1} \ar@{->>}[d] \ar[r] & 0 \\ 0 \ar[r] & F_{0} \ar[r] & Θ̄_{k,a_{p}} \ar[r] & F_{1} \ar[r] & 0.}

We show F_{0} = 0 if p^{2} ∤ r − p and F_{1} = 0 if p^{2} | r − p. Then (i) and (ii) will follow as usual.

(i) Consider f = f_{2} + f_{1} + f_{0} ∈ ind_{KZ}^{G}Sym^{r}ℚ̄_{p}^{2}, given by

f_{2} = ∑_{λ∈𝔽_{p}} [ g_{2,p[λ]}^{0}, \frac{[λ]^{p−2}}{p} · (Y^{r} − X^{r−p}Y^{p}) ] ,

f_{1} = [ g_{1,0}^{0}, \frac{p − 1}{pa_{p}} · ∑_{\begin{matrix}0<j<r−1 \\ j≡0 mod (p−1) \end{matrix}} β_{j} · X^{r−j}Y^{j} ] ,

f_{0} = [Id, \frac{1 − p}{p} · (X^{r} − X^{p}Y^{r−p})] ,

where the β_{j} are the integers from Lemma 7.2. Using the formula for the Hecke operator, one checks that T^{+}f_{2}, T^{−}f_{1}, −a_{p}f_{2}, T^{−}f_{0}, −a_{p}f_{0} are all integral and die mod p. Also T^{+}f_{1} ≡ 0 mod p by Lemma 7.2, i.e., by the properties satisfied by the integers β_{j}. Moreover,

T^{−}f_{2} + T^{+}f_{0} − a_{p}f_{1} ≡

[ g_{1,0}^{0}, \frac{(p − 1)}{p} · ( ∑_{\begin{matrix}0<j<r−1 \\ j≡0 mod (p−1) \end{matrix}} (\binom{r}{j} − β_{j}) · X^{r−j}Y^{j} + rXY^{r−1}) ] mod p.

Note that the function above is integral because each β_{j} ≡ \binom{r}{j} mod p by Lemma 7.2, and p | r by hypothesis. Now modifying the polynomial above by a suitable XY^{r−1}-term, we can see that \overline{(T − a_{p})f} has the same image as

[g_{1,0}^{0}, (p − 1) (F (X, Y) + \frac{(p − r)θY^{r−p−1}}{p} )] in ind_{KZ}^{G}Q, where

F (X, Y) = ∑_{\begin{matrix}0<j<r−1 \\ j≡0 mod (p−1) \end{matrix}} \frac{1}{p} (\binom{r}{j} − β_{j}) · X^{r−j}Y^{j} − \frac{(p − r)}{p} · X^{p}Y^{r−p}.

Using Lemmas 7.2, 2.7 and 2.6 we see that F (X, Y) ∈ V_{r}^{∗∗}, by Lemma 2.3.

Hence \overline{(T − a_{p})f} maps to the image of \frac{(r − p)}{p} · [g_{1,0}^{0}, θY^{r−p−1}] in ind_{KZ}^{G}Q.

By hypothesis, p^{2} ∤ r − p. Thus c = \overline{(r − p)/p} is a non-zero element in 𝔽̄_{p}. By Lemma 8.5 (i), the element c · [g_{1,0}^{0}, Y^{p−2}] ∈ ind_{KZ}^{G}J_{0} maps to 0 ∈ F_{0} ⊆ Θ̄_{k,a_{p}}.

Since c · [g_{1,0}^{0}, Y^{p−2}] generates all of ind_{KZ}^{G}J_{0} as a G-module, we have F_{0} = 0 and Θ̄_{k,a_{p}} ≅ F_{1} is a quotient of ind_{KZ}^{G}J_{1}.

(ii) If p^{2} | r − p, we consider the function f = f_{2} + f_{1} + f_{0} ∈ ind_{KZ}^{G}Sym ℚ̄_{p}^{2}, given by

f_{2} = ∑_{λ∈𝔽_{p}^{∗}} [ g_{2,p[λ]}^{0}, \frac{[λ]^{p−2}}{a_{p}} · (Y^{r} − X^{r−p}Y^{p}) ]

+ [g_{2,0}^{0}, \frac{(1 − p)r}{pa_{p}} · (XY^{r−1} − X^{r−p+1}Y^{p−1})] ,

f_{1} = [ g_{1,0}^{0}, \frac{p − 1}{a_{p}^{2}} · ∑_{\begin{matrix}0<j<r−1 \\ j≡0 mod (p−1) \end{matrix}} γ_{j} · X^{r−j}Y^{j} ] ,

f_{0} = [Id, \frac{1 − p}{a_{p}} · (X^{r} − X^{p}Y^{r−p})] ,

where the γ_{j} are integers from Lemma 7.3 (ii). Using Lemma 7.3 we check that if either p ≥ 5 or if p = 3 and v(a_{p}) < 3/2, then T^{+}f_{2}, T^{+}f_{0} + T^{−}f_{2} − a_{p}f_{1}, T^{−}f_{1}, T^{+}f_{1}, T^{−}f_{0} are all integral and die mod p. That leaves us with

(T − a_{p})f ≡ −a_{p}f_{0} − a_{p}f_{2} ≡ ∑_{λ∈𝔽_{p}^{∗}} [ g_{2,p[λ]}^{0}, −[λ]^{p−2} · (Y^{r} − X^{r−p}Y^{p}) ]

+ [g_{2,0}^{0}, \frac{(p − 1)r}{p} (XY^{r−1} − X^{r−p+1}Y^{p−1})]

+ [Id, (p − 1) · (X^{r} − X^{p}Y^{r−p})] mod p.

Hence the image of the reduction \overline{(T − a_{p})f} in ind_{KZ}^{G}Q is the same as that of

∑_{λ∈𝔽_{p}^{∗}}[ g_{2,p[λ]}^{0}, −[λ]^{p−2} · (X^{r−1}Y − X^{r−p}Y^{p}) ]

+[ g_{2,0}^{0}, \frac{(p−1)r}{p} (XY^{r−1} − X^{r−p+1}Y^{p−1})] + [Id, (p − 1) · (XY^{r−1} − X^{p}Y^{r−p})].

As we have r/p ≡ 1 mod p by hypothesis, the function above is congruent to

∑_{λ∈𝔽_{p}^{∗}}[g_{2,p[λ]}^{0}, −[λ]^{p−2}θX^{r−p−1}]+[g_{2,0}^{0}, X^{r−p+1}Y^{p−1}−XY^{r−1}]+[Id, θY^{r−p−1}] mod p.

Since X^{r−p+1}Y^{p−1} − XY^{r−1} = θ · (X^{r−2p+1}Y^{p−2} + ⋯ + Y^{r−p−1}), we have

X^{r−p+1}Y^{p−1} − XY^{r−1} ≡ θ · ( \frac{(r − 2p + 1)}{p − 1} · X^{r−2p+1}Y^{p−2} + Y^{r−p−1}) mod V_{r}^{∗∗}.

Applying Lemma 8.5 (ii) we get that \overline{(T − a_{p})f} maps to [g_{2,0}^{0}, −X] ∈ ind_{KZ}^{G}J_{1} under the map ind_{KZ}^{G}Q ↠ ind_{KZ}^{G} J_{1}. As [g_{2,0}^{0}, −X] generates all of ind_{KZ}^{G}J_{1} as a G-module, we get F_{1} = 0 and so Θ̄_{k,a_{p}} ≅ F_{0} is a quotient of ind_{KZ}^{G}J_{0}.

If p = 3 and v(a_{p}) ≥ 3/2, then note that T^{−}f_{1} and T^{+}f_{1} do not die mod p any more, and (T − a_{p})f is not necessarily integral. In this case we consider the modified new function f′ := (a_{p}^{2}/p^{3}) · f, with f as above. Then one checks (T − a_{p})f′ is integral and maps to c · [g_{2,0}^{0}, X] ∈ ind_{KZ}^{G}J_{1}, where c = \overline{1 − a_{p}^{2}/p^{3}}. By the extra hypothesis in the case v(a_{p}) = 3/2, c is always a non-zero element in 𝔽̄_{p}, and thus the JH factor J_{1} is killed again. □

Remark 8.10. In the next section we will show that in part (i) above the reducible case does not occur when p = 3, and that in part (ii) V̄_{k,a_{p}} is always reducible.

9. Separating out reducible and irreducible cases

If Θ̄_{k,a_{p}} is a quotient of ind_{KZ}^{G}(V_{p−2} ⊗ D^{n}), then [BG09, Prop. 3.3] fails to determine V̄_{k,a_{p}}. In this case, either V̄_{k,a_{p}} ≅ ind(ω_{2}^{p−1+n(p+1)}) is irreducible or it is reducible with V̄_{k,a_{p}}|_{I_{p}} ≅ ω^{n} ⊕ ω^{n}. We have faced this problem in the following cases, cf. Theorems 8.1 (i), 8.3 and 8.9.

(1) If b = 3 and p^{1+v(b)} ∤ r − b, then we have ind_{KZ}^{G} (V_{p−2} ⊗ D^{2}) ↠ Θ̄_{k,a_{p}}, hence

V̄_{k,a_{p}}|_{I_{p}} ≅ \begin{cases}ind(ω_{2}^{p+3}), or \\ ω^{2} ⊕ ω^{2}. \end{cases}

(2) If b = p and p^{2} | r − b, then under the extra hypothesis ‘v(a_{p}) = 3/2 ⇒ v(a_{p}^{2}−p^{3}) = 3, when p = 3’, we have ind_{KZ}^{G} (V_{p−2} ⊗ D) ↠ Θ̄_{k,a_{p}}, hence

V̄_{k,a_{p}}|_{I_{p}} ≅ \begin{cases}ind(ω_{2}^{2}), or \\ ω ⊕ ω. \end{cases}

In this section we mostly separate out the reducible and irreducible possibilities above. We will show that V̄_{k,a_{p}} is ‘almost always’ irreducible in the first case whereas it is always reducible in the second case above. In the first case, we work under the mild hypothesis (⋆) in Theorem 1.1. Note that (⋆) holds trivially if p | \binom{r−1}{2}. In particular, it holds for the smallest new weight treated in this paper, namely, k = 2p + 3.

Theorem 9.1. Let p ≥ 3, r > 2p, r ≡ 3 mod (p − 1) and p^{1+v(3)} ∤ r − 3. If v(a_{p}) = \frac{3}{2}, then further assume that v (a_{p}^{2} − \binom{r−1}{2}(r − 2)p^{3}) = 3. Then V̄_{k,a_{p}} ≅ ind(ω_{2}^{p+3}) is irreducible.

Proof. Assuming the hypothesis, we will show that the G-map ind_{KZ}^{G}J_{1} ↠ Θ̄_{k,a_{p}} given by Propositions 3.3, 3.8 and Theorem 8.9 (i) for p = 3, and by Theorem 8.3 for p ≥ 5, factors through the cokernel of the Hecke operator T acting on ind_{KZ}^{G}J_{1}, where J_{1} = V_{p−2} ⊗ D^{2}.

If v(a_{p}^{2}) ≤ v (\binom{r−1}{2}) + 3, we consider the function f = f_{2} + f_{1} + f_{0}, where

f_{2} = ∑_{λ∈𝔽_{p}} [g_{2,p[λ]}^{0}, \frac{1}{a_{p}} · θ(X^{r−p−2}Y − Y^{r−p−1})] ,

f_{1} = [g_{1,0}^{0}, ∑_{\begin{matrix}0<j<r−1 \\ j≡2 mod (p−1) \end{matrix}} \frac{(p − 1)p}{a_{p}^{2}} · α_{j}X^{r−j}Y^{j}] ,

f_{0} = \begin{cases}0, & if p ≥ 5 \\ [Id, \frac{(1−p)p}{a_{p}} · (X^{r} − X^{r−p+1}Y^{p−1})] , & if p = 3, \end{cases}

where the α_{j} are integers from Lemma 7.1, applied with r−1 ≡ 2 mod (p−1), instead of r. It is easy to see using the formula for the Hecke operator that T^{+}f_{2}, T^{−}f_{1} are integral and die mod p. Moreover,

T^{−}f_{2} − a_{p}f_{1} + T^{+}f_{0} ≡ [g_{1,0}^{0}, ∑_{\begin{matrix}0<j<r−1 \\ j≡2 mod (p−1) \end{matrix}} \frac{(p − 1)p}{a_{p}} · (\binom{r − 1}{j} − α_{j}) X^{r−j}Y^{j} ]

dies mod p, since v(a_{p}) < 2 and each α_{j} ≡ \binom{r−1}{j} mod p, by Lemma 7.1. Also, T^{−}f_{0} and a_{p}f_{0} die mod p, which is relevant only when p = 3. We use the four properties of the α_{j} in Lemma 7.1, and the fact that v(a_{p}^{2}) ≤ v (\binom{r−1}{2}) + 3 to conclude that T^{+}f_{1} is also integral and that T^{+}f_{1} ≡

∑_{λ∈𝔽_{p}} [g_{2,p[λ]}^{0}, \frac{p^{3}(p − 1)}{a_{p}^{2}} \binom{r−1}{2} · X^{r−2}Y^{2}] mod p. Thus (T −a_{p})f is integral and

is congruent to

∑_{λ∈𝔽_{p}} [g_{2,p[λ]}^{0}, −θ(X^{r−p−2}Y − Y^{r−p−1}) + \frac{p^{3}(p − 1)}{a_{p}^{2}} \binom{r − 1}{2} · X^{r−2}Y^{2}] mod p.

The image of X^{r−2}Y^{2} in Q is the same as that of X^{r−2}Y^{2} − XY^{r−1}, which is

θ · (X^{r−p−2}Y + ⋯ + Y^{r−p−1}) ≡ θ · ( \frac{r − p − 2}{p − 1} · X^{r−p−2}Y + Y^{r−p−1}) mod V_{r}^{∗∗}.

Hence \overline{(T − a_{p})f} maps to

∑_{λ∈𝔽_{p}} [g_{2,p[λ]}^{0}, −X^{p−2} + \frac{p^{3}}{a_{p}^{2}} \binom{r − 1}{2}(r − 2) · X^{p−2}] ∈ ind_{KZ}^{G}(V_{p−2} ⊗ D^{2}),

by Lemma 8.5. This equals c · T ([g_{1,0}^{0}, X^{p−2}]), with c = \overline{\frac{p^{3}}{a_{p}^{2}} \binom{r−1}{2}(r − 2) − 1}.

Using the hypothesis one checks that the constant c ∈ 𝔽̄_{p} is non-zero, hence the map ind_{KZ}^{G} J_{1} ↠ Θ̄_{k,a_{p}} factors through π(p − 2, 0, ω^{2}). Therefore the reducible

case cannot occur and V̄_{k,a_{p}} ≅ ind (ω_{2}^{p+3}).

Now let v(a_{p}^{2}) > v (\binom{r−1}{2}) + 3. As v(a_{p}^{2}) < 4, this forces p ∤ \binom{r−1}{2} and so v(a_{p}^{2}) > 3. Note that in this case (T − a_{p})f is not integral for the f above. However, we can use the following modified function f′ = f′_{2} + f′_{1} + f′_{0} :

f′_{2} = \frac{a_{p}^{2}}{p^{3}} · f_{2} = ∑_{λ∈𝔽_{p}} [g_{2,p[λ]}^{0}, \frac{a_{p} · θ(X^{r−p−2}Y − Y^{r−p−1})}{p^{3}}] ,

f′_{1} = \frac{a_{p}^{2}}{p^{3}} · f_{1} = [g_{1,0}^{0}, ∑_{\begin{matrix}0<j<r−1 \\ j≡2 mod (p−1) \end{matrix}} \frac{(p − 1)}{p^{2}} · α_{j}X^{r−j}Y^{j}] ,

f′_{0} = \frac{a_{p}^{2}}{p^{3}} · f_{0} = \begin{cases}0, & if p ≥ 5 \\ [Id, \frac{(1−p)a_{p}}{p^{2}} · (X^{r} − X^{r−p+1}Y^{p−1})] , & if p = 3, \end{cases}

with the same α_{j} as before. Now a_{p}f′_{2}, a_{p}f′_{0} dies mod p, as v(a_{p}^{2}) > 3. Also T^{+}f′_{2}, T^{−}f′_{1}, T^{−}f′_{0} and T^{−}f′_{2} − a_{p}f′_{1} + T^{+}f′_{0} die mod p as before, and hence

(T − a_{p})f′ ≡ T^{+}f′_{1} ≡ ∑_{λ∈𝔽_{p}} [g_{2,p[λ]}^{0}, (p − 1)\binom{r − 1}{2}X^{r−2}Y^{2}] mod p.

This maps to ∑_{λ∈𝔽_{p}} [g_{2,p[λ]}^{0}, \overline{(r − 2)\binom{r−1}{2}} · X^{p−2}] = \overline{(r − 2)\binom{r−1}{2}}· T ([g_{1,0}^{0}, X^{p−2}])

under the map ind_{KZ}^{G}V_{r} ↠ ind_{KZ}^{G}Q ↠ ind_{KZ}^{G}J_{1}, as shown above. Since (r − 2)\binom{r−1}{2} is a p-adic unit, the map ind_{KZ}^{G}J_{1} ↠ Θ̄_{k,a_{p}} factors through the image of T, and we have V̄_{k,a_{p}} ≅ ind(ω_{2}^{p+3}). □

Surprisingly, if b = p and p^{2} | r − p, then V̄_{k,a_{p}} is always reducible, at least if p ≥ 5. This is the first time in this paper that we have obtained a family of examples where V̄_{k,a_{p}} is reducible, for slopes in the range 1 < v(a_{p}) < 2. The following theorem describes the action of both inertia and Frobenius elements.

Theorem 9.2. Let p ≥ 3, r > 2p and r ≡ 1 mod (p − 1), i.e., b = p. If p = 3 and v(a_{p}) = \frac{3}{2}, then further assume that v(a_{p}^{2} − p^{3}) = 3. If p^{2} | r − p, then V̄_{k,a_{p}} is reducible and

V̄_{k,a_{p}} ≅ unr (\sqrt{−1}) ω ⊕ unr (−\sqrt{−1}) ω.

Proof. We claim that the map ind_{KZ}^{G}J_{0} ↠ Θ̄_{k,a_{p}} given by Theorem 8.9 (ii), where J_{0} = V_{p−2} ⊗ D, factors through

\frac{ind_{KZ}^{G}(V_{p−2} ⊗ D)}{(T^{2} + 1)} ≅ π (p − 2, \sqrt{−1}, ω) ⊕ π (p − 2, −\sqrt{−1}, ω) .

Once this claim is proved, the result follows as we know that Θ̄_{k,a_{p}} lies in the image of the mod p Local Langlands Correspondence.

\underline{Proof of the claim}: We consider f = f_{2} + f_{1} + f_{0} ∈ ind_{KZ}^{G} Sym^{r}ℚ̄_{p}^{2}, given by

f_{2} = ∑_{\begin{matrix}λ∈𝔽_{p}, \\ μ∈𝔽_{p}^{∗} \end{matrix}} [ g_{2, p[μ]+[λ]}^{0} , \frac{1}{a_{p}} · (Y^{r} − X^{r−p}Y^{p}) ]

+ ∑_{λ∈𝔽_{p}} [g_{2,[λ]}^{0} , \frac{(1 − p)}{a_{p}} · (Y^{r} − X^{r−p}Y^{p})] ,

f_{1} = ∑_{λ∈𝔽_{p}} [ g_{1,[λ]}^{0}, \frac{(p − 1)}{a_{p}^{2}} · ∑_{\begin{matrix}1<j<r \\ j≡1 mod (p−1) \end{matrix}} α_{j} · X^{r−j}Y^{j} ] ,

f_{0} = [Id, \frac{r}{pa_{p}} · (X^{r−1}Y − X^{r−p}Y^{p})] ,

where the α_{j} are integers from Lemma 7.3 (i). If either p ≥ 5 or p = 3 and v(a_{p}) < 3/2, then we use the fact that p^{2} | r − p and the properties satisfied by the α_{j} to conclude that all of T^{+}f_{2}, T^{+}f_{1}, T^{−}f_{1}, T^{−}f_{0}, T^{−}f_{2} − a_{p}f_{1} + T^{+}f_{0} are integral and die mod p. Thus (T − a_{p})f is integral, and is congruent to −a_{p}f_{2} − a_{p}f_{0} mod p, which equals

∑_{λ, μ∈𝔽_{p}} [ g_{2, p[μ]+[λ]}^{0} , X^{r−p}Y^{p} − Y^{r} ] − [Id, \frac{r}{p} · (X^{r−1}Y − X^{r−p}Y^{p})] .

Since Y^{r}, X^{r−1}Y map to 0 in Q, and as r/p ≡ 1 mod p by hypothesis, the image of the integral function above in ind_{KZ}^{G}Q is the same as that of

− ∑_{λ, μ∈𝔽_{p}} [ g_{2, p[μ]+[λ]}^{0}, θX^{r−p−1}] − [Id, θX^{r−p−1}] , which, by the formula for T^{2}

and by Lemma 8.5, is the image of

(T^{2} + 1)[Id, −X^{p−2}] = ∑_{λ, μ∈𝔽_{p}} [ g_{2, p[μ]+[λ]}^{0}, −X^{p−2}] + [Id, −X^{p−2}] ∈ ind_{KZ}^{G}J_{0}

in ind_{KZ}^{G}Q. As [Id, −X^{p−2}] generates ind_{KZ}^{G}J_{0}, the image (T^{2} + 1)(ind_{KZ}^{G}J_{0}) must map to 0 under the G-map ind_{KZ}^{G}J_{0} ↠ Θ̄_{k,a_{p}}. When p = 3 and v(a_{p}) ≥ 3/2, then (T − a_{p})f is not necessarily integral for the function f above. However, if we consider f′ := (a_{p}^{2}/p^{3}) · f, then (T − a_{p})f′ is integral with reduction equal to the image of c·(T^{2}+1)[Id, X] ∈ ind_{KZ}^{G}J_{0} inside ind_{KZ}^{G}Q, for c = \overline{1 − a_{p}^{2}/p^{3}} ∈ 𝔽̄_{p}. By the extra hypothesis when v(a_{p}) = 3/2, we see c is non-zero, and the result follows as before. □

Errata to [GG15]

• p. 256, l. 21: α = ( \begin{matrix}0 & 1 \\ p & 0 \end{matrix} ) should be α = ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ).

• p. 267, l. 13: “hat” should be “that”.

• p. 273, l. 15: Sym ℚ̄_{p}^{2} should be Sym^{2p}ℚ̄_{p}^{2}.

• p. 274, l. 15 and l. 18: “3 ≤ a ≤ p − 3” should be “3 ≤ a ≤ p − 1”.

• p. 284: When a = p − 1, part (2) of Lemma 28 is not true, since d_{0} = −\frac{p−1}{p} is not integral and so w is non-integral. This can be corrected by modifying the function f_{2}. Set f = f_{0} + f_{2}, with f_{0} = [Id, \frac{1}{p}(X^{r−p+1}Y^{p−1} − X^{r})]. One checks that T^{−}f_{0} is integral and vanishes mod p and that T^{+}f_{0} + T^{−}f_{2} = [g_{1,0}^{0}, w′], with w′ integral. Thus, (T − a_{p})f is integral and the proof of Theorem 27 proceeds as before with w replaced by w′.

Acknowledgements

The first author wishes to thank her advisor, the second author, for his constant encouragement, and also the School of Mathematics, TIFR, for its support. Part of this work was supported by the National Science Foundation under Grant No. 0932078 000 while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2014 special semester.

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Shalini Bhattacharya School of Mathematics Tata Institute of Fundamental Research 1, Homi Bhabha Road Mumbai-400005 India shalini@math.tifr.res.in Eknath Ghate School of Mathematics Tata Institute of Fundamental Research 1, Homi Bhabha Road Mumbai-400005 India eghate@math.tifr.res.in 988

Documenta Mathematica 20 (2015)