Stable reduction of modular curves

Irene I. Bouw  Stefan Wewers

Abstract

We determine the stable reduction at p of all three point covers of the projective line with Galois group SL_{2}(p). As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves X_{0}(p) and X_{1}(p). Our method does not use the fact that modular curves are moduli spaces. Instead, we rely on results of Raynaud and the authors which describe the stable reduction of three point covers whose Galois group is strictly divisible by p.

Introduction

Modular curves are quotients of the upper half plane by congruence subgroups of SL_{2}(ℤ). The most prominent examples are X_{0}(N), X_{1}(N) and X(N), for N ≥ 1. Modular curves are also moduli spaces for elliptic curves endowed with a level structure. Therefore, they are defined over small number fields and have a rich arithmetic structure. Deligne and Rapoport [4] determine the reduction behavior of X_{0}(p) and X_{1}(p) at the prime p. Using this result, they prove a conjecture of Shimura saying that the quotient of the Jacobian of X_{1}(p) by the Jacobian of X_{0}(p) acquires good reduction over ℚ(ζ_{p}). Both the reduction result and its corollary have been generalized by Katz and Mazur [9] to arbitrary level N and various level structures.

The basic method employed in [4] and [9] is to generalize the moduli problem defining the modular curve in question, in such a way that it makes sense in arbitrary characteristic. By general results on representability of moduli problems one then obtains a model of the modular curve over the ring of integers of a subfield of ℚ(ζ_{N}). This model has bad reduction at all primes dividing N, and it is far from being semistable, in general. In spite of the very general results of [9], it remains an unsolved problem to describe the stable reduction of X_{0}(N) at p if p^{3} divides N (see [5] for the case p^{2}||N).

In this note we suggest a different approach to study the reduction of modular curves. Our starting point is the observation that the j-invariant

X(N) ⟶ X(1) ≅ ℙ^{1}

presents the modular curve X(N) as a Galois cover of the projective line (with Galois group PSL_{2}(ℤ/N)) which is branched only at the three rational points 0, 1728 and ∞. Let us call a Galois covers of the projective line branched only at three points a three point cover. In [11] Raynaud studies the stable reduction of three point covers under the assumption that the residue characteristic p strictly divides the order of the Galois group. His results have been sharpened in [15] and [16]. In the present paper we determine the stable reduction of all three point covers with Galois group PSL_{2}(p), using the results of [16]. As a corollary we obtain a new proof of the results of Deligne and Rapoport on the reduction of X_{0}(p) and X_{1}(p). Somewhat surprisingly, our proof does not use the modular interpretation of these curves. On the other hand, the determination of the stable reduction of some of the PSL_{2}(p)-covers which are not modular curves does use the fact that they are moduli spaces of a certain kind. Here we build on the results of [2].

What is so special about the group PSL_{2}(p)? From our point of view there are two main aspects. The first is rigidity. Let G be a finite group and C = (C_{1}, C_{2}, C_{3}) a triple of conjugacy classes of G. Suppose that there exists a triple g = (g_{1}, g_{2}, g_{3}) of generators of G with g_{i} ∈ C_{i} and g_{1}g_{2}g_{3} = 1. The triple g corresponds to a G-cover Y → ℙ^{1} with three branch points. If G equals PSL_{2}(p), such a triple g is unique, up to uniform conjugation in PGL_{2}(p). Therefore there exists at most one three point cover with a given branch cycle description C, up to isomorphism.

We use rigidity as follows. By a result of [16] we can construct three point G-covers Y → ℙ^{1} with bad reduction at p by lifting a certain type of ‘stable G-cover’ Ȳ → X̄ from characteristic p to characteristic zero. Here Ȳ → X̄ is a finite map between semistable curves in characteristic p together with some extra structure (g, ω) which we call the special deformation datum. In the case of the modular curve X(p) (where G = PSL_{2}(p)) one can construct Ȳ → X̄ very explicitly; the special deformation datum (g, ω) corresponds to a solution of a hypergeometric differential equation. (A similar phenomenon occurs in Ihara’s work on congruence relations [7].) The PSL_{2}(p)-cover Y → ℙ^{1} resulting from the lifting process has branch cycle description (3A, 2A, pA). By rigidity, Y → ℙ^{1} is isomorphic to X(p) → X(1). In particular, the stable reduction of X(p) is isomorphic to Ȳ.

The other nice thing about PSL_{2}(p) is that p strictly divides its order. The results of [11] and [16] which describe the stable reduction of a given G-cover require that p strictly divides the order of G, whereas the Sylow p-subgroup of PSL_{2}(ℤ/p^{n}) is rather big for n > 1. This is the main obstruction for extending the method of the present paper to modular curves of higher p-power level. There are partial results of the authors generalizing some of the results of [11] and [16] to groups G with a cyclic or an elementary abelian Sylow p-subgroup (unpublished). It seems hopeless to obtain general results beyond these cases. But maybe a combination of the methods presented in the present paper with the modular approach might shed some light on the stable reduction of modular curves of higher p-power level.

The organization of this paper is as follows. In Section 1 we define special deformation data and explain how to associate a special deformation datum to a three point cover with bad reduction. We recall a result from [16] which essentially says that we can reverse this process. In Section 2 we introduce hypergeometric deformation data. These are special deformation data satisfying an addition condition (Definition 2.1). We classify all hypergeometric deformation data by showing that they correspond to the solution in characteristic p of some hypergeometric differential equation. In Sections 3 and 4 we use these results to give a new proof of the stable reduction of the modular curves X(p) and X_{0}(p). In Section 5 we generalize these results to all three point covers with Galois group SL_{2}(p).

1 The special deformation datum

Let k be an algebraically closed field of characteristic p > 2. Let H be a finite group of order prime to p. Fix a character χ : H → 𝔽_{p}^{×}. Let g : Z_{k} → ℙ_{k}^{1} be an H-Galois cover and ω be a meromorphic differential on Z_{k}. We assume that ω is logarithmic (i.e. can be written as ω = du/u) and

β^{∗}ω = χ(β) · ω, for all β ∈ H.

Let ξ ∈ Z_{k} be a closed point and τ its image in ℙ_{k}^{1}. Denote by H_{ξ} the stabilizer of ξ in H. Define

m_{τ} := |H_{ξ}|,  h_{τ} := ord_{ξ}(ω) + 1,  σ_{τ} = h_{τ} /m_{τ} .  (1)

Since ω is logarithmic, we have h_{τ} ≥ 0. We say that τ is a critical point of the differential ω if (m_{ξ}, h_{ξ}) ≠ (1, 1). Let (τ_{i})_{i∈𝔹} be the critical points of ω, indexed by a finite set 𝔹. For i ∈ 𝔹, we write m_{i}, h_{i}, σ_{i} instead of m_{τ_{i}}, h_{τ_{i}}, σ_{τ_{i}}. For every i, choose a point ξ_{i} ∈ Z_{k} above τ_{i} and write H(ξ_{i}) ⊂ H for its stabilizer. Define 𝔹_{wild} := { i ∈ 𝔹 | h_{i} = 0 }.

Definition 1.1 A special deformation datum of type (H, χ) is a pair (g, ω), where g : Z_{k} → ℙ_{k}^{1} is an H-Galois cover and ω is a logarithmic differential on Z_{k} such that the following holds.

(i) We have

β^{∗}ω = χ(β) · ω,  for all β ∈ H.  (2)

(ii) For every i ∈ 𝔹 − 𝔹_{wild}, we have that 0 < σ_{i} ≤ 2.

(iii) Define 𝔹_{prim} = {i ∈ 𝔹 | 0 < σ_{i} ≤ 1}. Then |𝔹_{prim} ∪ 𝔹_{wild}| = 3.

Write 𝔹_{0} = 𝔹_{wild} ∪ 𝔹_{prim} and 𝔹_{new} = 𝔹 − 𝔹_{0}. See [16] for more details and an explanation of the terminology. If σ_{i} ≠ 1, 2 for all i, the above definition coincides with [16, Definition 2.7].

Definition 1.2 Let G be a finite group. A G-tail cover is a (not necessarily connected) G-Galois cover f_{k} : Y_{k} → ℙ_{k}^{1} such that f_{k} is wildly branched at ∞ of order pn with n prime to p and tamely branched at no more than one other point. We say that f is a primitive tail cover if it is branched at two points. Otherwise, we call f a new tail cover.

If the group G is understood, we talk about tail covers instead of G-tail covers. To a G-tail cover f_{k} : Y_{k} → ℙ_{k}^{1} we associate its ramification invariant σ(f) = h/n, where h is the conductor of f_{k} at ∞ and n the order of the prime-to-p ramification, as in Definition 1.2. The ramification invariant is the jump in the filtration of higher ramification groups in the upper numbering.

Three point covers with bad reduction give rise to a special deformation datum and a set of tail covers. Essentially, the stable reduction of the cover is determined by these data. Proposition 1.3 states that given a special deformation datum and a set of tail covers satisfying some compatibility conditions, there exists a three point cover in characteristic zero which gives rise to the given datum.

To be more precise, let R be a complete discrete valuation ring with fraction field K of characteristic zero and residue field an algebraically closed field k of characteristic p. Let G be a finite group and let f : Y → X = ℙ_{K}^{1} be a G-Galois cover branched at three points x_{1}, x_{2}, x_{3}. We assume that the points x_{1}, x_{2}, x_{3} specialize to pairwise distinct points on the special fiber ℙ_{k}^{1}. Denote the ramification points of f by y_{1}, …, y_{s}. We consider the y_{i} as markings on Y. After replacing K by a finite extension, there exists a unique extension (Y_{R}; y_{i}) of (Y; y_{i}) to a stably marked curve over R. The action of G extends to Y_{R}; write X_{R} for the quotient of Y_{R} by G. The map f_{R} : Y_{R} → X_{R} is called the stable model of f; its special fiber f̄: Ȳ → X̄ is called the stable reduction of f, [16, Definition 1.1].

We say that f has good reduction if f̄ is separable. This is equivalent to X̄ being smooth. If f does not have good reduction, we say it has bad reduction.

Suppose that f has bad reduction and that p strictly divides the order of G. Then X̄ is a ‘comb’, see [16, Theorem 2.6] and Figure 1. The central component X̄_{0} ⊂ X̄ is canonically isomorphic to ℙ_{k}^{1}. Choose a component Ȳ_{0} of Ȳ above X̄_{0}. Let G_{0} ⊂ G denote the decomposition group of the component Ȳ_{0} and I_{0} ◁ G_{0} the inertia group. Then I_{0} ≅ ℤ/p. Therefore, the restriction of f̄ to Ȳ_{0} factors as Ȳ_{0} → Z̄_{0} → X̄_{0}, with g : Z̄_{0} → X̄_{0} a separable Galois cover of prime-to-p order and Ȳ_{0} → Z̄_{0} purely inseparable of degree p. The map Ȳ_{0} → Z̄_{0} is generically endowed with the structure of a μ_{p}-torsor. This structure is encoded in a logarithmic differential ω. Define H := Gal(Z̄_{0}/X̄_{0}); then G_{0} is an extension of H by I_{0}. The action of H on I_{0} by conjugation gives rise to a character χ : H → 𝔽_{p}^{×}. It is proved in [16, Proposition 2.15] that (g, ω) is a special deformation datum of type (H, χ).

The irreducible components of X̄ different from the central component X̄_{0} are called the tails of X̄. The tails are parameterized by i ∈ 𝔹 − 𝔹_{wild}, [16, Proposition1.7]. For i ∈ 𝔹 − 𝔹_{wild}, write X̄_{i} for the corresponding tail of X̄ and write f̄_{i} : Ȳ_{i} → X̄_{i} for the restriction of f̄ to X̄_{i}. The map f̄_{i} is a primitive tail cover if i ∈ 𝔹_{prim} and a new tail cover if i ∈ 𝔹_{new}, [16, Lemma 1.4]. We say that X̄_{i} is a primitive (resp. new) tail of X̄ if i ∈ 𝔹_{prim} (resp. i ∈ 𝔹_{new}).

The three branch points specialize as follows. If p divides the ramification index of x_{j}, then x_{j} specializes to τ_{i} ∈ X̄_{0} for a unique i ∈ 𝔹_{wild}. Otherwise, x_{j} specializes to one of the primitive tails

[figure: ⋯; ⋯; ⋮; ⋮; ⋮; Ȳ; X̄; _{𝔹_{prim}}; _{𝔹_{new}}; τ_{1}; x̄_{2}; x̄_{3}; X̄_{0}]

Figure 1: The stable reduction of f : Y → ℙ^{1}

X̄_{i}, for i ∈ 𝔹_{prim}. From now on, we will identify the set 𝔹_{0} = 𝔹_{prim} ∪ 𝔹_{wild} with {1, 2, 3} such that i ∈ 𝔹_{wild} if and only if p divides the order of the ramification index at x_{i}. For i ∈ 𝔹_{prim} we denote by x̄_{i} ∈ X̄_{i} the specialization of x_{i}.

The special deformation datum (g, ω) and the tail covers f̄_{i} satisfy the following compatibility condition. For every i ∈ 𝔹 − 𝔹_{wild}, there exists a point ξ_{i} ∈ Z̄_{0} above τ_{i} and a point η_{i} ∈ Ȳ_{i} which is wildly ramified in f̄_{i} such that

I(η_{i}) ⊂ G_{0},  I(η_{i}) ∩ H = H(ξ_{i}).  (3)

Here I(η_{i}) ⊂ G (resp. H(ξ_{i}) ⊂ H) denotes the stabilizer of η_{i} (resp. of ξ_{i}). Furthermore, the ramification invariant σ(f̄_{i}) of the tail cover f̄_{i} is equal to σ_{i} as in Definition 1.1.

Proposition 1.3 Let G be a finite group. Fix a subgroup G_{0} ≃ ℤ/p ⋊_{χ} H of G, where H has order prime-to-p and acts on ℤ/p via a character χ : H → 𝔽_{p}^{×}. Let (g, ω) be a special deformation datum of type (H, χ). Let 𝔹, τ_{i} and σ_{i} = h_{i}/m_{i} be as in Definition 1.1.

For i ∈ 𝔹_{prim} (resp. i ∈ 𝔹_{new}), suppose we are given a primitive (resp. new) G-tail cover f̄_{i} : Ȳ_{i} → X̄_{i} = ℙ_{k}^{1} with ramification invariant σ_{i}. Suppose furthermore that there exists a point ξ_{i} ∈ Z_{k} above τ_{i} ∈ ℙ_{k}^{1} and a point η_{i} ∈ Ȳ_{i} which is wildly ramified in f̄_{i} such that (3) holds.

Let K_{0} be the fraction field of W (k) (the Witt vectors over k) and let K/K_{0} be the (unique) tame extension of degree

N := (p − 1) · lcm_{i∉𝔹_{wild}}(h_{i}).  (4)

There exists a G-Galois cover f : Y → X = ℙ_{K}^{1} over K, branched at 0, 1 and ∞, which gives rise to the special deformation datum (g, ω) and the tail covers f̄_{i}.

Proof. The datum (g, ω, f̄_{i}) together with the choice of the points ξ_{i} and η_{i} is called a special G-deformation datum in [16, Definition 2.13]. Hence the proposition follows from [16, Corollary 4.6]. □

In this paper, the group G is either SL_{2}(p) or PSL_{2}(p). Therefore, the group H is cyclic and H(ξ_{i}) only depends on i and not on the choice of ξ_{i}.

The G-cover f : Y → ℙ_{K}^{1} in Proposition 1.3 which gives rise to the datum (g, ω, f̄_{i}) is not unique. In [16, Theorem 4.5] one finds an explicit parameterization of the the set of isomorphism classes of all such G-covers.

2 Hypergeometric deformation data

In this section we classify a certain type of special deformation data which we call hypergeometric. A hypergeometric (special) deformation datum corresponds to a polynomial solution of a hypergeometric differential equation. For this reason it is possible to find all such deformation data. The definition of hypergeometricity looks very restrictive, but we show in Section 5 that the special deformation datum corresponding to a three point cover with Galois group PSL_{2}(p) or SL_{2}(p) is hypergeometric. In particular, this holds for the PSL_{2}(p)-cover X(p) → X(1) ≃ ℙ_{j}^{1}.

Definition 2.1 A hypergeometric deformation datum is a special deformation datum (g, ω) with σ_{i} = (p + 1)/(p − 1) for all i ∈ 𝔹_{new}.

Let (g, ω) be a hypergeometric deformation datum. Recall that 𝔹_{0} = {1, 2, 3}, by assumption. In this section we assume x_{1} = 0, x_{2} = 1, x_{3} = ∞. In particular, τ_{i} ≠ ∞ for all i ∈ 𝔹_{new}. For i ∈ 𝔹, we define integers 0 ≤ a_{i} < p − 1 such that

a_{i} ≡ (p − 1)σ_{i} mod p − 1.

In particular, a_{i} = 2 for i ∈ 𝔹_{new}.

To g : Z_{k} → ℙ_{k}^{1} we associate the (not necessarily connected) 𝔽_{p}^{×}-Galois cover

g′ : Z′_{k} := Ind_{Im(χ)}^{𝔽_{p}^{×}}(Z_{k}/Ker χ) → ℙ_{k}^{1}.

It follows easily from Definition 2.1 and the expansion of ω into local coordinates that Z′_{k} is the complete nonsingular curve defined by

z^{p−1} = x^{a_{1}}(x − 1)^{a_{2}} ∏_{i∈𝔹_{new}}(x − τ_{i})^{2},  (5)

where β^{∗}z = χ(β) · z. We call (a_{1}, a_{2}, a_{3}) the signature of the hypergeometric deformation datum (g, ω).

We may identify the differential ω on Z_{k} with a differential on Z′_{k} which we also denote by ω. Definition 1.1(i) and (1) imply that

ω = ϵ\frac{z dx}{x(x − 1)},  (6)

for some ϵ ∈ k^{×}. The condition that ω is logarithmic imposes a strong condition on u := ∏_{i∈𝔹_{new}}(x − τ_{i}). Proposition 2.2 below states that the differential (6) is logarithmic if and only if u is the solution to a certain hypergeometric differential equation.

It follows from (1) and the Riemann–Roch Theorem applied to ω that

|𝔹_{new}| = (p − 1 − a_{1} − a_{2} − a_{3})/2.  (7)

In particular, a_{1}+a_{2}+a_{3} is an even integer between 0 and p−1. Note that if a_{1}+a_{2}+a_{3} = p−1 then deg(u) = |𝔹_{new}| = 0. Therefore, we exclude this case in Proposition 2.2.(i). However, Proposition 2.2.(ii) applies also if a_{1} + a_{2} + a_{3} = p − 1.

Proposition 2.2 (i) Let (g, ω) be a hypergeometric deformation datum. Let (a_{1}, a_{2}, a_{3}) be its signature, and suppose that a_{1} + a_{2} + a_{3} < p − 1. Then u(x) := ∏_{i∈𝔹_{new}}(x − τ_{i}) is a solution to the hypergeometric differential equation

x(x − 1)u′′ + [(A + B + 1)x − C]u′ + ABu = 0,

where A = (1 + a_{1} + a_{2} + a_{3})/2, B = (1 + a_{1} + a_{2} − a_{3})/2 and C = 1 + a_{1}. The degree of u in x is d := (p − 1 − a_{1} − a_{2} − a_{3})/2.

(ii) Let 0 ≤ a_{1}, a_{2}, a_{3} < p−1 be integers with a_{1} +a_{2} +a_{3} ≤ p−1. We assume that a_{1} +a_{2} +a_{3} is even. Let H = ℤ/m, where m = (p − 1)/2 if a_{1}, a_{2}, a_{3} are all even and m = p − 1 otherwise. Choose an injective character χ : H → 𝔽_{p}^{×}. There exists a hypergeometric deformation datum (g, ω) of signature (a_{1}, a_{2}, a_{3}) and type (H, χ). The deformation datum (g, ω) is uniquely determined by (a_{1}, a_{2}, a_{3}), up to multiplying the differential ω by an element of 𝔽_{p}^{×}.

Proof. The following proof is inspired by [1, Lemma 3]. A similar argument can be found in [7, Theorem 5]. Write

Q = x^{1+a_{1}}(x − 1)^{1+a_{2}}, F = ϵ\frac{z}{x(x − 1)} = \frac{ϵz^{p}}{Qu^{2}} .

It is well known that the fact that ω = F dx is logarithmic is equivalent to D^{p−1}F = −F^{p}, where D := d/dx. Since D^{p−1}F = ϵz^{p}D^{p−1}[1/(Qu^{2})], we find

D^{p−1} \frac{1}{Qu^{2}} = −ϵ^{p−1} \frac{1}{x^{p}(x − 1)^{p}} .  (8)

Choose i ∈ 𝔹_{new} and write

\frac{1}{Qu^{2}} = ∑_{n≥−2} a_{n}(x − τ_{i})^{n}.

Then

D^{p−1} \frac{1}{Qu^{2}} = − [\frac{a_{−1}}{(x − τ_{i})^{p}} + a_{p−1} + ⋯] = −ϵ^{p−1} \frac{1}{x^{p}(x − 1)^{p}} .

We conclude that a_{−1} = 0.

Write

Qu^{2} = [Q(τ_{i}) + Q′(τ_{i})(x − τ_{i}) ⋯][u′(τ_{i})(x − τ_{i}) + \frac{1}{2}u′′(τ_{i})(x − τ_{i})^{2} + ⋯]^{2}
= (x − τ_{i})^{2} [ Q(τ_{i})u′(τ_{i}) + ( Q′(τ_{i})u′(τ_{i}) + Q(τ_{i})u′′(τ_{i}) ) (x − τ_{i}) + ⋯].

We see that −a_{−1} = u′(τ_{i})[ Q′(τ_{i})u′(τ_{i})+Q(τ_{i})u′′(τ_{i}) ]. Since u′(τ_{i}) ≠ 0, we have that Q′(τ_{i})u′(τ_{i})+ Q(τ_{i})u′′(τ_{i}) = 0 for all i ∈ 𝔹_{new}.

Define G = Q′u′ + Qu′′. This is a polynomial of degree less than or equal to e := deg(Q) + deg(u) − 2 = |𝔹_{new}| + a_{1} + a_{2}. The coefficient of x^{e} in G is g_{e} = deg(u)(deg(Q) + deg(u) − 1). Since deg(Q) ≥ 2 and deg(u) ≠ 0, the coefficient g_{e} is nonzero. The polynomial G is obviously divisible by x^{a_{1}}(x − 1)^{a_{2}}, and G(τ_{i}) = 0, for all i ∈ 𝔹_{new}. Therefore G = g_{e}x^{a_{1}}(x − 1)^{a_{2}}u. Dividing by x^{a_{1}}(x − 1)^{a_{2}}, we find that u is a solution to the hypergeometric differential equation

P_{0}u′′ + P_{1}u′ + P_{2}u = 0,  (9)

where P_{0} = x(x−1), P_{1} = P_{0}Q′/Q = (2+a_{1}+a_{2})x−(1+a_{1}), and P_{2} = −g_{e} = ((1+a_{1}+a_{2})^{2}−a_{3}^{2})/4. Part (i) of the proposition follows. The proof also shows that

ϵ^{p−1} = (−1)^{a_{1}} (\begin{matrix}p − 1 − a_{2} \\ a_{1} \end{matrix}) .  (10)

To prove (ii), let 0 ≤ a_{1}, a_{2}, a_{3} < p − 1 be integers such that a_{1} + a_{2} + a_{3} is even and less than or equal to p − 1. Define d = (p − 1 − a_{1} − a_{2} − a_{3})/2. We claim that the differential equation (9) has a unique polynomial solution u = ∑_{i} u_{i}x^{i} ∈ k[x] of degree d with u_{d} = 1.

Suppose that u = ∑_{i∈ℤ} u_{i}x^{i} is a solution of (9). Put A_{i} = (i + 1)(i + a_{1} + 1) and B_{i} = (i + (1 + a_{1} + a_{2} + a_{3})/2)(i + (1 + a_{1} + a_{2} − a_{3})/2). The differential equation gives a recursion

A_{i}u_{i+1} = B_{i}u_{i}  (11)

for the coefficients of u. One checks that A_{i} and B_{i} are nonzero for all 0 ≤ i < d and that A_{−1} = B_{d} = 0 and A_{d} · B_{−1} ≠ 0. Therefore there is a unique polynomial solution u of degree d with u_{d} = 1. This implies the claim.

Let m be as in the statement of the proposition. Let Z′_{k} be the complete nonsingular curve defined by

z^{p−1} = u^{2}x^{a_{1}}(x − 1)^{a_{2}}

and let Z_{k} be a connected component of Z′_{k}. Write g : Z_{k} → ℙ_{k}^{1} for the m-cyclic cover defined by (x, z) ↦ x. We may identify H with Gal(Z_{k}/ℙ_{k}^{1}) such that H acts on z via some chosen character χ : H → 𝔽_{p}^{×}. Let

ω = \frac{ϵ z dx}{x(x − 1)}.

Since u is the solution to a hypergeometric differential equation, it has at most simple zeros outside 0 and 1. We know that u(0) = u_{0} = ∏_{i=0}^{d−1}(B_{i}/A_{i}) ≠ 0. Therefore u does not have a zero at x = 0. We claim that u does not have a zero at 1, also. To see this, let t = 1 − x be a new coordinate on ℙ_{k}^{1} and write ũ(t) = u(1 − t). The coefficients of ũ satisfy a recursion similar to (11), but with a_{1} and a_{2} interchanged. The same argument as above applied to ũ shows now that ũ(0) = u(1) ≠ 0. We conclude that u has exactly d zeros different from 0 and 1. Analogous to the proof of (i), one shows that for ϵ as in (10) the differential ω is logarithmic. It is clear that (g, ω) is a special deformation datum. The uniqueness statement is obvious. □

3 The reduction of X(p)

In the rest of the paper we suppose that p ≥ 5. Choose a primitive (p − 1)th root of unity ζ ∈ 𝔽_{p} and a primitive (p + 1)th root of unity ζ̃ ∈ 𝔽_{p^{2}}. Define

C(l) = {A ∈ SL_{2}(p) | tr(A) = ζ^{l} + ζ^{−l}}, for 0 < l ≤ (p − 1)/2

and

C̃(l) = {A ∈ SL_{2}(p) | tr(A) = ζ̃^{l} + ζ̃^{−l}}, for 0 < l < (p + 1)/2.

These are the conjugacy classes of SL_{2}(p) of nontrivial elements of order prime to p. We write pA and pB for the two conjugacy classes of elements of order p. Note that the elements of order p have trace 2.

Let C = (C_{1}, C_{2}, C_{3}) be a triple of conjugacy classes of SL_{2}(p). In the rest of this section, we suppose that the C_{i} do not contain ±I. We write Ni_{3}^{in}(C) for the set of isomorphism classes of SL_{2}(p)-covers Y → ℙ^{1} branched at 0, 1, ∞ with class vector C. This means that the canonical generator of some point of Y above x_{i} ∈ {0, 1, ∞} with respect to the chosen roots of unity is contained in the conjugacy class C_{i}. In this paper, we call two G-covers f_{i} : Y_{i} → X isomorphic if there exists a G-equivariant automorphism ϕ : Y_{1} → Y_{2} such that f_{1} = f_{2} ∘ ϕ.

Proposition 3.1 (Linear rigidity) Let C = (C_{1}, C_{2}, C_{3}) be a triple of conjugacy classes of SL_{2}(p).

(i) Suppose that the elements of C_{i} have prime-to-p order. Then # Ni_{3}^{in}(C) ∈ {0, 2}.

(ii) Otherwise, # Ni_{3}^{in}(C) ∈ {0, 1}.

Proof. See [13]. The difference between the two cases comes from the fact that the outer automorphism group of G has order two and interchanges the two conjugacy classes of order p but fixes all other conjugacy classes. □

In the following we denote by 2A (resp. 3A) the unique conjugacy classes in PSL_{2}(p) of elements of order 2 (resp. 3). Note that elements of 2A (resp. elements of 3A) lift to elements of order 4 (resp. of order 3 or 6) in SL_{2}(p). Furthermore, we denote by pA and pB the images in PSL_{2}(p) of the conjugacy classes of SL_{2}(p) with the same name.

Let X(N) be the modular curve parameterizing (generalized) elliptic curves with full level Nstructure [4]. Consider the cover X(2p) → X(2) ≃ ℙ_{λ}^{1}. This is a Galois cover with Galois group PSL_{2}(p) which is branched at 0, 1, ∞ of order p and unbranched elsewhere. One easily checks as in [14, Lemma 3.27] that there are no PSL_{2}(p)-covers with class vector (pA, pA, pB). After renaming the conjugacy classes, the class vector of X(2p) → X(2) is therefore C = (pA, pA, pA).

Proposition 3.1 implies that the triple C = (pA, pA, pA) is rigid, i.e. there is a unique PSL_{2}(p)- cover of ℙ^{1} whose class vector is C, up to automorphism. Since PSL_{2}(p) has trivial center, it follows that the PSL_{2}(p)-cover X(2p) → X(2) has a unique model X(2p)_{K} → X(2)_{K} = ℙ_{K}^{1} over any field K of characteristic zero containing \sqrt{p^{∗}}, [12, Chapter 7]. Let K_{0} be the fraction field of the ring of Witt vectors over k = 𝔽̄_{p}, and let K/K_{0} be a sufficiently large finite extension. We define the stable reduction of X(2p) → X(2) at p as the stable reduction of X(2p)_{K} → ℙ_{K}^{1}.

The cover X(2p) → X(2) may be considered as a variant with ordered branch points of the cover X(p) → X(1) ≃ ℙ_{j}^{1}. We find it easier to first compute the stable reduction of the former and then deduce the stable reduction of the latter.

Proposition 3.2 Let Ȳ → X̄ be the stable reduction of X(2p) → X(2) at p. Then X̄ has (p−1)/2 new tails with ramification invariant σ = (p + 1)/(p − 1). The points τ_{i} ∈ ℙ_{k}^{1}, i ∈ 𝔹_{new} (where the new tails occur) are precisely the zeros of the polynomial

u = ∑_{j=0}^{(p−1)/2} \binom{(p − 1)/2}{j}^{2}x^{j}.  (12)

Moreover, the stable reduction occurs over the (unique) tame extension K/K_{0} of degree (p^{2} − 1)/2.

The polynomial u is known as the Hasse invariant. It has the property that the elliptic curve given by y^{2} = x(x − 1)(x − λ) is supersingular in characteristic p if and only if λ is a zero of u. It is a solution to the hypergeometric differential equation with parameters (1/2, 1/2, 1). The relation between this differential equation and the reduction of elliptic curves is well understood, see for example [8].

Proof. Let a_{1} = a_{2} = a_{3} = 0 and m = (p − 1)/2. Choose an injective character χ : ℤ/m → 𝔽_{p}^{×}. Let (g, ω) be the special deformation datum of signature (0, 0, 0) constructed in Proposition 2.2.(ii). Recall that we associate to the triple (a_{1}, a_{2}, a_{3}) = (0, 0, 0) the hypergeometric differential equation with parameters A = B = 1/2 and C = 1. Using the recursion (11) one checks that the polynomial u defined by (12) is a solution in characteristic p to this differential equation. This differential equation has a unique monic solution of degree (p−1)/2. Therefore, u is the same as in Proposition 2.2 (ii).

Let Y′_{k} → ℙ_{k}^{1} be a connected and new PSL_{2}(p)-tail cover with ramification invariant σ = (p + 1)/(p − 1). The existence of such a tail cover is proved in Lemma 3.3 below. The inertia group of a point above ∞ has order p(p − 1)/2. For every i ∈ 𝔹_{new}, we let f̄_{i} : Ȳ_{i} → X̄_{i} be a copy of Y′_{k} → ℙ_{k}^{1}. Let η_{i} ∈ Ȳ_{i} be the point whose inertia group consists of the upper triangular matrices in PSL_{2}(p). Then the condition of Proposition 1.3 is satisfied. Therefore, there exists a PSL_{2}(p)-cover f_{K} : Y_{K} → ℙ_{K}^{1} branched at three points of order p whose stable reduction gives rise to the special deformation datum (g, ω) and the tail covers f̄_{i}. This cover is defined and has stable reduction over the tame extension K/K_{0} of degree (p^{2} − 1)/2. It follows from rigidity that f_{K} is isomorphic to X(2p)_{K} → X(2)_{K}. This proves the proposition. □

Lemma 3.3 There exist a connected new SL_{2}(p)-tail cover h : Y_{k} → X_{k} = ℙ_{k}^{1}, with inertia group of order p(p − 1) and ramification invariant σ = (p + 1)/(p − 1).

Proof. Let Y_{k} be the complete nonsingular curve defined by

y^{p+1} = x^{p} − x.

The group SL_{2}(p) acts on Y_{k} as follows. For

A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(p), define Ax = \frac{ax + b}{cx + d}, Ay = \frac{y}{cx + d}.

It is easy to check that the quotient map h : Y_{k} → X_{k} ≃ ℙ_{k}^{1} is branched at exactly one point which we may assume to be ∞. The inertia group of some point above ∞ has order p(p − 1) and σ = (p + 1)/(p − 1). □

Let X(p) → X(1) ≃ ℙ_{j}^{1} be the projection of the modular curve X(p) to the j-line. This is a PSL_{2}(p)-cover with branch cycle description (3A, 2A, pA) , which we may assume to be branched at x_{1} = 0, x_{2} = 1728, x_{3} = ∞. By Proposition 3.1, it has a unique model X(p)_{K} → ℙ_{K}^{1} over any field K of characteristic 0 containing \sqrt{p^{∗}}. As before, we take K/K_{0} a sufficiently large finite extension and define the stable reduction of X(p) → X(1) at p as the stable reduction of the model X(p)_{K} → ℙ_{K}^{1}.

Corollary 3.4 Define α = ⌊p/12⌋. Let Ȳ′ → X̄′ be the stable reduction of X(p) → X(1). Then X̄′ has two primitive tails and α new tails. For i = 1, 2, denote by X̄_{i} the primitive tail to which x_{i} specializes. Then

σ_{i} = \{ \begin{matrix}\frac{p + 1}{p − 1} & if i ∈ 𝔹_{new}, \\ \frac{⌈(p − 1)/3⌉}{p − 1} & if i = 1, \\ \frac{2⌈(p − 1)/4⌉}{p − 1} & if i = 2. \end{matrix}

Moreover, the stable reduction occurs over the tame extension K/K_{0} of degree (p^{2} − 1)/2.

Proof. Consider the commutative diagram

X(2p)_{K} ⟶ X(p)_{K}
↓  ↓
X(2)_{K} ⟶ X(1)_{K}.

The horizontal arrows are Galois covers whose Galois group is the symmetric group S_{3} on three letters. Let Y_{R} be the stable model of X(2p)_{K}. The action of S_{3} extends to Y_{R} and Y_{R}/S_{3} := Y_{R}^{∗} is a semistable curve, [10, Appendice].

Since S_{3} acts on Y_{R} it follows that the set of new tails of X̄_{0} is stable under the action of S_{3}. We claim that the action of S_{3} on the set of new tails of X̄ has α orbits of length six, one orbit of length three if p ≡ 3 mod 4 and one orbit of length two if p ≡ 2 mod 3. Since X̄_{0} has genus zero, an element of order three in S_{3} has two fixed points in X̄_{0} − {0, 1, ∞}. An element of order two in S_{3} fixes one of the points 0, 1, ∞ and has therefore one fixed point in X̄_{0} − {0, 1, ∞}. The claim now follows by distinguishing the four possibilities for p mod 12.

The curve X_{R}^{∗} := Y_{R}^{∗}/G is almost equal to X′_{R}. The special fiber X̄^{∗} has a tail at zero (resp. 1728) if and only if p ≡ −1 mod 3 (resp. p ≡ −1 mod 4), i.e. exactly when zero (resp. 1728) is supersingular. To obtain X′_{R} one has to perform a suitable blow up centered at the point j = 0 (resp. j = 1728) on X̄^{∗} if p ≡ 1 mod 3 (resp. p ≡ 1 mod 4.) The corollary follows, by analyzing the ramification of Ȳ_{0}^{∗} → X̄_{0}^{∗}. For future reference, we note that a connected component of Ȳ′_{1} has decomposition group PSL_{2}(p) if p ≡ −1 mod 3 and the nonabelian group of order 3p if p ≡ 1 mod 3. A connected component of Ȳ′_{2} has decomposition group PSL_{2}(p) if p ≡ −1 mod 4 and the dihedral group of order 2p if p ≡ 1 mod 3. □

The hypergeometric differential equation corresponding to X(p) → X(1) is

x(x − 1728)u′′ + [(2 + a_{1} + a_{2})x − 1728(1 + a_{1})]u′ + (1 + a_{1} + a_{2})^{2}u/4,

where a_{1} = ⌈(p − 1)/3⌉ and a_{2} = 2⌈(p − 1)/4⌉. The polynomial

u = (1728)^{α} ∑_{n=0}^{α} \binom{a_{1} + α}{α − n}\binom{α}{n} (\frac{x}{1728})^{n} ∈ k[x]

is a solution to this differential equation. Recall that j ∈ k is supersingular if and only if the corresponding elliptic curve over k is supersingular. It follows that j ≠ 0, 1728 is supersingular if and only if u(j) = 0.

4 The reduction of X_{0}(p)

Let X_{0}(p) denote the modular curve parameterizing (generalized) elliptic curves with a Γ_{0}(p)- structure [4]. We may and will identify X_{0}(p) with the quotient curve X(p)/G_{0}, where G_{0} ⊂ G := PSL_{2}(p) is the standard Borel group, modulo ±I. The G-cover X(p) → X(1) = ℙ_{j}^{1} gives rise to a non-Galois cover X_{0}(p) → ℙ_{j}^{1} which we call the j-invariant. We claim that the curve X_{0}(p) has a unique ℚ-model X_{0}(p)_{ℚ} such that the j-invariant descends to a cover X_{0}(p)_{ℚ} → ℙ_{ℚ}^{1}. (Of course, the existence of such a ℚ-model is also a consequence of the modular interpretation of X_{0}(p).)

To prove the claim, let G ↪ S_{p+1} denote the standard permutation representation of G coming from the natural action on ℙ^{1}(𝔽_{p}). Note that G_{0} is precisely the stabilizer of ∞. It is well known that the normalizer of G in S_{p+1} is equal to PGL_{2}(p) = Aut(G). In particular, the centralizer of G in S_{p+1} is trivial. Let C = (3A, 2A, pA) denote the triple of conjugacy classes of G corresponding to the G-cover X(p) → X(1), as in the previous section. Let C′ denote the image of C in PGL_{2}(p). Then C′ is rigid and rational. Hence the claim follows from standard Galois theory.

For any field K of characteristic zero we denote by X_{0}(p)_{K} the K-model of X_{0}(p) obtained from X_{0}(p)_{ℚ} by base change. In [4, VI.6.16] Deligne and Rapoport prove the following result.

Theorem 4.1 Let X_{0}(p)_{ℤ_{p}} denote the normalization of ℙ_{ℤ_{p}}^{1} inside X_{0}(p)_{ℚ_{p}}. Set k := 𝔽̄_{p}.

(i) The ℤ_{p}-curve X_{0}(p)_{ℤ_{p}} is semistable (and stable if g(X_{0}(p)) > 1).

(ii) The geometric special fiber X_{0}(p)_{k} := X_{0}(p)_{ℤ_{p}} ⊗k is the union of two smooth curves W̄′_{0}, W̄′′_{0} of genus 0. The induced map W̄′_{0} → ℙ_{k}^{1} (resp. W̄′′_{0} → ℙ_{k}^{1}) is an isomorphism (resp. purely inseparable of degree p). The components W̄′_{0} and W̄′′_{0} intersect precisely in the supersingular points, i.e. the points of X_{0}(p)_{k} with supersingular j-invariant.

(iii) Let x ∈ X_{0}(p)_{k} be a supersingular point, with j-invariant j_{x} ∈ k. If j_{x} ≡ 0 then X_{0}(p)_{ℤ_{p}} has a singularity of type A_{3} at x. If j_{x} ≡ 1728 then x presents a singularity of type A_{2}. Otherwise, x is a regular point of X_{0}(p)_{ℤ_{p}}.

We give a proof of this theorem using the results of the previous section.

Proof. Let R_{0} := W (k) denote the ring of Witt vectors and K_{0} the fraction field of R_{0}. Note that

X_{0}(p)_{R_{0}} := X_{0}(p)_{ℤ_{p}} ⊗_{ℤ_{p}} R_{0}

is equal to the normalization of ℙ_{R_{0}}^{1} in the function field of X_{0}(p)_{K_{0}}. Hence it suffices to prove the theorem with the ring ℤ_{p} replaced by R_{0}.

Let K/K_{0} be the tame extension of degree (p^{2} − 1)/2. By Corollary 3.4 the PSL_{2}(p)-cover Y_{K} := X(p)_{K} → X_{K} := ℙ_{K}^{1} extends to a stable model Y_{R} → X_{R} over the ring of integers R of K. As before, we denote by Ȳ → X̄ the special fiber of Y_{R} → X_{R}.

Set W_{R} := Y_{R}/G_{0}. By [10, Appendice], W_{R} is a semistable curve over R with generic fiber W_{K} = X_{0}(p)_{K}. By construction, we have a finite map W_{R} → X_{R}. Let W̄ denote the special fiber of W_{R}. The formation of the quotient Y_{R}/G_{0} does not commute with base change, in general. However, the canonical map Ȳ /G_{0} → W̄ is a homeomorphism. Let V be an irreducible component of Ȳ and U its image in W̄. Let D(V) ⊂ G (resp. I(V) ⊂ G) denote the decomposition group (resp. the inertia group) of V. Then the natural map V/(D(V) ∩ G_{0}) → U is purely inseparable of degree |I(V) ∩ G_{0}|. Using Corollary 3.4 and the description of the stable reduction in Section 1, we see that W̄ has four different types of components (see Figure 2 for the case p = 29). In the description below we call an irreducible component of W̄ horizontal if it is mapped to the central component X̄_{0} of X̄ and vertical otherwies, cf. Figure 1.

(a) A horizontal component W̄′_{0} which is the image of the (unique) horizontal component Ȳ_{0} ⊂ Ȳ with decomposition group G_{0}. The natural map W̄′_{0} → X̄_{0} = ℙ_{k}^{1} is an isomorphism.

(b) Another horizontal component W̄′′_{0} which is the image of the set of horizontal components of Ȳ different from Ȳ_{0}. Note that if Ȳ′′ ≠ Ȳ_{0} is such a horizontal component, with decomposition group D′′, then D′′ ∩ G_{0} is a cyclic group of order (p − 1)/2 (a split torus). Therefore, the natural map W̄′′_{0} → X̄_{0} is purely inseparable of degree p.

(c) For i ∈ 𝔹_{new} ∪ 𝔹_{prim} we let W̄_{i} be the image of the tail cover Ȳ_{i}. Suppose that i is supersingular, i.e. that the tail X̄_{i} intersects X̄_{0} = ℙ_{k}^{1} in a supersingular point. Then Ȳ_{i} and — hence W̄_{i} as well — is connected. The curve W̄_{i} intersects each of the horizontal components W̄_{0} and W̄′_{0} in a unique point.

(d) With notation as in (c), suppose that i is not supersingular (in particular, i ∈ 𝔹_{prim}). Then Ȳ_{i} is not connected, and the decomposition group of any connected component is isomorphic to ℤ/p ⋊ ℤ/2 or ℤ/p ⋊ ℤ/3 (depending on whether x_{i} ≡ 1728 or x_{i} ≡ 0). It follows that W̄_{i} has exactly two connected components W̄′_{i} and W̄′′_{i}. The curve W̄′_{i} (resp. W̄′′_{i}) intersects W̄′_{0} (resp. W̄′′_{0}) in a unique point. (See the end of the proof of Corollary 3.4.)

In the modular picture, the component W̄′_{0} (resp. W̄′′_{0}) corresponds generically to ordinary elliptic curves together with a subgroup scheme isomorphic to μ_{p} (resp. to ℤ/p), see [4, VI.6]. The vertical components have no easy modular interpretation. Note that every component of W̄ has genus 0.

W̄′_{0} W̄′′_{0}  ^{j≡1728}  ^{j^{2}+2j−8≡0}  ^{j≡0}

Figure 2: The curve W̄ for p = 29

Let W′_{R} be the curve obtained from W_{R} by contracting all vertical components of W̄. Since these vertical components have genus zero and self-intersection number −1 or −2, the curve W′_{R} is semistable. In particular, W′_{R} is a normal scheme. Recall that the curve obtained from X_{R} by contracting the tails X̄_{i} is equal to ℙ_{R}^{1}. Therefore the finite map W_{R} → X_{R} gives rise to a finite map W′_{R} → ℙ_{R}^{1}. We conclude that W′_{R} is the normalization of ℙ_{R}^{1} inside the function field of W_{K} = X_{0}(p)_{K}.

Let Γ := Gal(K/K_{0}) ≅ ℤ/N, and recall that N = (p^{2} − 1)/2 is prime to p. Since W_{K} = X_{0}(p)_{K_{0}} ⊗_{K_{0}} K, the group Γ acts naturally on W_{K}. Since W′_{R} is the normalization of ℙ_{R}^{1} in W_{K} this action extends to the R-model W′_{R} and

X_{0}(p)_{R_{0}} = W′_{R}/Γ.  (13)

Since the order of Γ is prime to p, formation of the quotient commutes with base change. In particular, we have

X_{0}(p)_{k} = W̄′/Γ.  (14)

It is clear that the action of Γ commutes with the map W̄′ → ℙ_{k}^{1} coming from the j-invariant. Since this map is an isomorphism on one and purely inseparable of degree p on the other component, Γ acts trivially on W̄′. Now Parts (i) and (ii) of the theorem follow from (14) and the description of the components of W̄ given above.

It remains to prove Part (iii) of the theorem. Let x ∈ X_{0}(p)_{k} = W̄′ be a supersingular point, with j-invariant j_{x} ∈ k. Let v denote the valuation on K, normalized such that v(p) = 1. Since x is an ordinary double point of W̄′, the local ring of X_{0}(p)_{R_{0}} at x is of the form R_{0}[[u, v | uv = π]], with π ∈ R_{0}. We define the thickness of X_{0}(p)_{R_{0}} at x as the rational number e(x) := v(π). We have to show that e(x) is equal to 3 if j_{x} ≡ 1728, equal to 2 if j_{x} ≡ 0 and equal to 1 otherwise.

We assume that j_{x} ≢ 0, 1728. The other cases follow in the same manner. Let x′ ∈ W̄′ be the unique point lying above x. Also, let x′′ and x′′′ be the two double points on W̄ lying above x′. We assume that x′′ ∈ W̄′_{0} and x′′′ ∈ W̄′′_{0}. It is easy to see that

e(x) = e(x′) = e(x′′) + e(x′′′).  (15)

Let y′′, y′′′ ∈ Ȳ be points above x′′, x′′′. Recall that W_{R} = Y_{R}/G_{0}. By [10, Appendice], we have e(x′′) = |Stab_{G_{0}}(y′′)| · e(y′′) and e(x′′′) = |Stab_{G_{0}}(y′′′)| · e(y′′′). Using (15) and our knowledge of the G-action on Ȳ, we see that

e(x) = \frac{p(p − 1)}{2} · e(y′′) + \frac{p − 1}{2} · e(y′′′).  (16)

It is proved in [16] that the thickness of any double point y ∈ Ȳ is equal to (h_{i}(p − 1))^{−1} if y lies on the tail Ȳ_{i} and h_{i} denotes the conductor of the tail cover Ȳ_{i} → X̄_{i} at ∞. Since we assumed that j_{x} ≢ 0, 1728 we have y′′, y′′′ ∈ Ȳ_{i} for some i ∈ 𝔹_{new}. By Corollary 3.4 we have h_{i} = (p + 1)/2 for i ∈ 𝔹_{new}. With (16) we conclude that

e(x) = \frac{p}{p + 1} + \frac{1}{p + 1} = 1.

This finishes the proof of the theorem. □

One can analyze the natural R_{0}-model of X_{1}(p) in an entirely analogous manner. This would give a new proof of the result of [4] that J_{1}(p)/J_{0}(p) has good reduction over ℚ(ζ_{p}). One can also derive this result directly from our knowledge of the stable reduction of X(p).

5 Three point covers with Galois group SL_{2}(p)

In this section we compute the stable reduction of all SL_{2}(p)-covers of ℙ^{1} branched at three points. In particular, we obtain an explicit formula for the number of such covers with good reduction (Theorem 5.6). To prove this we use the results of the previous sections. In addition, we use the fact that some SL_{2}(p)-covers have a modular interpretation as a Hurwitz space.

Let us give an outline of the proof of Theorem 5.6. We first show that the SL_{2}(p)-covers of ℙ^{1} branched at three points of order p, p, n are essentially Hurwitz spaces. These covers have bad reduction to characteristic p; their stable reduction is described in [2]. By considering the (unique) primitive tail of the stable reduction, we obtain one primitive tail cover for every noncentral conjugacy class of SL_{2}(p) whose order is prime to p. Proposition 5.5 states that every primitive SL_{2}(p)-tail cover with group SL_{2}(p) arises in this way. Then we show (Proposition 5.3) that the only new tail cover that occurs is the cover constructed in Lemma 3.3. This implies that the special deformation datum of an SL_{2}(p)-cover branched at three points is hypergeometric (Definition 2.1). Therefore, Proposition 1.3 allows to construct all SL_{2}(p)-covers with bad reduction at p, by lifting. This also gives a formula for the number of covers with good reduction. Passing to the quotient modulo −I, one obtains similar result for three point covers with Galois group PSL_{2}(p).

In this section, R is a complete mixed characteristic discrete valuation ring which contains \sqrt{p^{∗}} and whose residue field is an algebraically closed field k of characteristic p. We let K denote the fraction field of R.

Lemma 5.1 Let D ∈ {C(l), C̃(l)}. Let f : Y → ℙ_{K}^{1} be an SL_{2}(p)-Galois cover with class vector C_{1} = (pA, pA, D) or C_{2} = (pA, pB, D). Let Ȳ → X̄ be the stable reduction of f. The curve X̄ has exactly one primitive tail, with ramification invariant

σ = \{ \begin{matrix}\frac{p − 1 − 2l}{p − 1} & if D = C(l), \\ \frac{p + 1 − 2l}{p − 1} & if D = C̃(l). \end{matrix}

The inertia group of the primitive tail has order p(p − 1)/ gcd(p − 1, l) if D = C(l) and p(p − 1)/ gcd(p − 1, l − 1) otherwise. The new tails have ramification invariant σ = (p + 1)/(p − 1) and an inertia group of order p(p − 1).

For SL_{2}(p)-covers with ramification of order n, one can prove a strengthening of Proposition 3.1, cf. [14, Lemma 3.29]. Either there exists a cover with class vector C_{1} or there exists a cover with class vector C_{2}. In both cases, there is a unique such cover. The outer automorphism of SL_{2}(p) conjugates the cover with class vector C_{1} (resp. C_{2}) to a cover with class vector (pB, pB, D) (resp. (pB, pA, D)). Therefore, in both cases, there are exactly two covers with ramification (p, p, D), up to isomorphism.

Proof. The case D = C(l) is treated in [2, Section 8]. The idea of the proof is as follows. Define m = (p − 1)/ gcd(p − 1, l). For λ ∈ ℙ^{1} − {0, 1, ∞}, let W_{λ} be a connected component of the nonsingular projective curve corresponding to

w^{p−1} = x^{l}(x − 1)^{l}(x − λ)^{p−1−l}.

This defines a family of m-cyclic covers g_{λ} : W_{λ} → ℙ^{1}, branched at four points. Fix an injective character χ : ℤ/m → 𝔽_{p}^{×} and let N = ℤ/p ⋊_{χ} ℤ/m. Define the Hurwitz space ℍ parameterizing N-Galois covers branched at 0, 1, λ, ∞ of order prime to p which factor via g_{λ}. Define ℍ → ℙ_{λ}^{1} by mapping an N-cover to the branch point λ and let f : ℍ̃ → ℙ_{λ}^{1} its Galois closure. The Galois group of f is either SL_{2}(p) or PSL_{2}(p). If it is SL_{2}(p), the class vector of f is (D_{1}, D_{2}, D), with D_{1}, D_{2} ∈ {pA, pB}. Otherwise, there exists a unique lift of f to a cover with such a class vector. Therefore, f is isomorphic to the cover of the statement of the proposition. In [2], the stable reduction of f is computed, by using the moduli interpretation of ℍ.

The proof for D = C̃(l) is analogous. (Take m|(p + 1) and N ≃ (ℤ/p)^{2} ≃ ℤ/m.) □

The following lemma is due to R. Pries (unpublished).

Lemma 5.2 Let G be a finite group. Let φ : W → ℙ_{k}^{1} be a new G-tail cover with ramification invariant σ and inertia group I of order pn. Suppose 1 < σ < 2. Then there exists a G-cover f : Y → ℙ_{K}^{1} with signature (p, p, n) such that φ occurs as the unique new tail of the stable reduction of f.

If σ = 2 then φ can be lifted to a cover in characteristic zero which is branched at three points of order p, [3, Section 3.2].

Proof. Write n = m_{1}m_{2}, where m_{2} is the order of the prime-to-p part of the center of I. Then m_{1} divides p − 1 and σ − 1 = a/m_{1} for some 0 < a < m_{1}. Write α = (p − 1)/m_{1}. Choose λ ∈ k − {0, 1} such that (1 − λ)^{αa} = 1. Note that σ ≠ p/(p − 1), since conductors are prime to p. Therefore, αa ≠ 1 and it is possible to choose λ as asserted.

Let g_{0} : Z̄_{0} → X̄_{0} be the m_{1}-cyclic cover where Z̄_{0} is defined by z^{m_{1}} = x^{−a}(x − λ)^{a}. Let g : Z̄ → ℙ_{k}^{1} be the n-cyclic cover branched only at 0 and λ which has g_{0} as a quotient. Define the differential

ω = \frac{ϵ z dx}{x(x − 1)}

on Z̄, with ϵ as in (10). A straightforward computation, as in the proof of Proposition 2.2, shows that (g, ω) is a special deformation datum. There is one primitive critical point at zero and one new critical point at λ.

Define a primitive I-tail cover ψ′ : Ȳ′ → ℙ_{k}^{1} which is totally ramified at ∞ with ramification invariant σ = (m_{1} − a)/m_{1} and branched of order n at 0. Let ψ := Ind_{I}^{G} ψ′. One checks that the datum (g, ω, ψ, φ) satisfies the conditions of Proposition 1.3. We conclude that there exists a G-Galois cover f : Y → ℙ_{K}^{1} which is branched at three points with ramification of order (p, p, n) such that its stable reduction gives rise to (g, ω, ψ, φ). This proves the lemma. □

It follows from rigidity that the special deformation datum associated to an SL_{2}(p)-cover with bad reduction is uniquely determined by the invariants h_{i} and m_{i}. On the other hand, the special deformation datum (g, ω) in the proof of Lemma 5.2 depends also on the choice of λ ∈ k − {0, 1} with (1 − λ)^{αa} = 1. It follows that we have αa = 2 in case G = SL_{2}(p). This statement is also a consequence of the next proposition.

Proposition 5.3 Suppose that G is either SL_{2}(p) or PSL_{2}(p). Let φ : W → ℙ_{k}^{1} be a new G-tail cover with ramification invariant σ. Suppose that 1 < σ ≤ 2. Then σ = (p + 1)/(p − 1). The order of the inertia group is p(p − 1) if G = SL_{2}(p) and p(p − 1)/2 if G = PSL_{2}(p). Furthermore, the cover φ is unique, up to isomorphism.

Proof. If there exists a new SL_{2}(p)-tail cover φ with σ ≠ 2, Lemma 5.2 implies that φ occurs as the restriction to a new tail of the stable reduction of a SL_{2}(p)-cover with signature (p, p, n) for some n prime-to-p. Therefore Lemma 5.1 implies that σ = (p + 1)/(p − 1).

Similarly, one shows that every SL_{2}(p)-cover of ℙ_{k}^{1} branched at one point, with σ = 2, can be lifted to a cover of ℙ_{K}^{1} which is branched at three points of order p. The proposition follows in this case from Proposition 3.2 and Proposition 3.1, cf. [3, Corollary 3.2.2].

Since every PSL_{2}(p)-cover of ℙ_{K}^{1} with ramification of order p, p, n′ can be lifted to an SL_{2}(p)- cover with ramification of order p, p, n (where n is n′ or 2n′), the proposition also follows for G = PSL_{2}(p). □

Corollary 5.4 Let G be SL_{2}(p) or PSL_{2}(p). Let f : Y → ℙ_{K}^{1} be a G-Galois cover branched at three points. Suppose that f has bad reduction. Then the special deformation datum corresponding to f is hypergeometric.

Proof. f : Y → ℙ_{K}^{1} be a G-Galois cover branched at three points. Suppose that f has bad reduction. Proposition 5.3 implies that σ_{i} = (p + 1)/(p − 1) for i ∈ 𝔹_{new}. Therefore the corollary follows from Definition 2.1. □

Proposition 5.5 (i) Let φ : W → ℙ_{k}^{1} be a (possibly nonconnected) primitive SL_{2}(p)-tail cover defined over k, with ramification invariant σ. Suppose that 0 < σ < 1. Denote by D the conjugacy class in SL_{2}(p) of the canonical generator of a point of W above 0, with respect to the fixed roots of unity. Then

σ = \{ \begin{matrix}\frac{p − 1 − 2l}{p − 1} & if D = C(l), \\ \frac{p + 1 − 2l}{p − 1} & if D = C̃(l). \end{matrix}

(ii) The decomposition group of an irreducible component of W is SL_{2}(p) if D = C̃(l) and is a semidirect product of order p(p − 1)/gcd(p − 1, l) if C = C(l).

Proof. Write σ = a/(p − 1). The normalizer N_{G}(P) of a Sylow p-subgroup P in SL_{2}(p) has order p(p − 1). Since the center of N_{G}(P) has order two, it follows that a is even. Therefore (p − 1 − a)/2 is a positive integer.

Proposition 2.2 (ii) shows that there exists a hypergeometric deformation datum (g′, ω′) of signature (0, 0, a), where g′ : Z′_{k} → ℙ_{k}^{1} is a (p − 1)/2-cyclic cover, given by

z′^{(p−1)/2} = ∏_{i∈𝔹_{new}}(x − τ_{i}).

We define g : Z_{k} → ℙ_{k}^{1} to be the (p − 1)-cyclic cover given by

z^{p−1} = ∏_{i∈𝔹_{new}}(x − τ_{i}).

Let ω be the pull back of ω′ to Z_{k}. Then (g, ω) is a special deformation datum. Note that |𝔹_{prim}| = 1 and |𝔹_{new}| = (p − 1 − a)/2, with a = p − 1 − 2l if D = C(l) and a = p + 1 − 2l if D = C̃(l). The primitive critical point is at ∞, where the cover g is branched of order (p − 1)/(p − 1, l) if D = C(l) and of order (p − 1)/(p − 1, l − 1) otherwise.

Let h be the SL_{2}(p)-tail cover constructed in Lemma 3.3. For every i ∈ 𝔹_{new}, let f̄_{i} be a copy of h. Proposition 1.3 shows that there exists an SL_{2}(p)-cover f : Y → ℙ_{K}^{1} whose stable reduction gives rise to the special deformation datum (g, ω), the new tail covers f̄_{i} and the primitive tail cover φ. By construction, the class vector of f is (D_{1}, D_{2}, D) with D_{1}, D_{2} ∈ {pA, pB}. Therefore f is isomorphic to a cover described in Lemma 5.1. Part (i) of the proposition follows from this.

Let N denote the decomposition group of an irreducible component of W. Suppose D = C̃(l), for some l < (p − 1)/2. Then N contains an element of order p and an element of order (p + 1)/ gcd(p + 1, l) > 2. Therefore, N = SL_{2}(p), [6, Section II.8].

Next suppose that D = C(l). Put n_{l} = (p − 1)/ gcd(p − 1, l). It is easy to construct a primitive tail cover with Galois group ℤ/p ⋊ ℤ/n_{l} which is branched at zero of order n_{l} and has ramification invariant σ = (p − 1 − 2l)/(p − 1). Let φ′ be the primitive SL_{2}(p)-tail cover obtained by induction. Using again Proposition 1.3, we see that there exists an SL_{2}(p)-cover f′ : Y′ → ℙ_{K}^{1} whose stable reduction gives rise to the special deformation datum (g, ω), the new tail covers f̄_{i} and the primitive tail cover φ′. Note that f′ has the same branch cycle description as f. It follows from Proposition 3.1 that f and f′ are outer isomorphic, i.e. become isomorphic as SL_{2}(p)-covers after twisting by an outer automorphism of SL_{2}(p). We conclude that N is an extension of ℤ/n_{l} by ℤ/p. □

Theorem 5.6 Let C = (C_{1}, C_{2}, C_{3}) be a triple of conjugacy classes of SL_{2}(p), where we suppose that the elements of the C_{i} are not contained in the center.

(a) If C_{i} contains an element of order p for some i, all covers with class vector C have bad reduction.

(b) Suppose C_{i} ∈ {C(l), C̃(l)} and put a_{i} = p − 1 − 2l if C_{i} = C(l) and a_{i} = p + 1 − 2l otherwise.

Write

Ni3^{bad}(C) = { f ∈ Ni_{2}^{in}(C) | f has bad reduction. }.

Then

| Ni3^{bad}(C)| = \{ \begin{matrix}2 & if a_{1} + a_{2} + a_{3} < p − 1, \\ 2 & if a_{1} + a_{2} + a_{3} = p − 1 and C_{i} = C̃(l) for some i, \\ 0 & otherwise. \end{matrix} .

Proof. Let f : Y → ℙ_{K}^{1} be an SL_{2}(p)-cover with class vector C = (C_{1}, C_{2}, C_{3}). If p divides one of the ramification indices of f, then f has bad reduction, so (a) is obvious.

Suppose that p does not divide the ramification indices of f. If f has bad reduction, then (7) implies that a_{1} + a_{2} + a_{3} ≤ p − 1. If a_{1} + a_{2} + a_{3} = p − 1, there are no new tails. Since Ȳ is connected, at least one of the primitive tails has decomposition group SL_{2}(p). Therefore Proposition 5.5 implies that C_{i} = C̃(l), for some i. This shows that if there exists a cover with bad reduction, then either a_{1} + a_{2} + a_{3} < p − 1 or a_{1} + a_{2} + a_{3} = p − 1 and C_{i} = C̃(l) for some i.

Now suppose that C_{1}, C_{2}, C_{3} are conjugacy classes of noncentral elements of order prime to p such that the condition of the theorem is satisfied. Let a_{1}, a_{2}, a_{3} be as in the statement of the theorem. Recall that a_{i} is even and different from p − 1. Let (g′, ω′) be the hypergeometric deformation datum of signature (a_{1}, a_{2}, a_{3}) constructed in Proposition 2.2 (ii). The cover g′ : Z′_{k} → ℙ_{k}^{1} is the (p − 1)/2-cyclic cover defined by

z′^{(p−1)/2} = x^{a_{1}/2}(x − 1)^{a_{2}/2} ∏_{i∈𝔹_{new}}(x − τ_{i}), (z, x) ↦ x.

We define a (p − 1)-cyclic cover g : Z_{k} → ℙ_{k}^{1}, factoring through g′ by

z^{p−1} = x^{b_{1}}(x − 1)^{b_{2}} ∏_{i∈𝔹_{new}}(x − τ_{i}), (z, x) ↦ x,

where b_{i} = a_{i}/2 + (p − 1)/2, i.e. b_{i} = p − 1 − l if C_{i} = C(l) and b_{i} = p − l otherwise. Then (g, ω) is a special deformation datum.

We define a set of tail covers, as follows. For i ∈ 𝔹_{new}, we define f̄_{i} ≃ h, where h is the new SL_{2}(p)-tail cover of ℙ_{k}^{1} with σ = (p + 1)/(p − 1) from Lemma 3.3. For i ∈ 𝔹_{prim} = {1, 2, 3}, we let f̄_{i} be the primitive SL_{2}(p)-tail cover with σ = a_{i}/(p − 1) whose existence is guaranteed by Lemma 5.1. As in the proof of Lemma 5.2, one checks that the datum (g, ω; f̄_{i}) satisfies the conditions of Proposition 1.3. Hence there exists an SL_{2}(p)-cover f : Y → ℙ_{K}^{1} with class vector C = (C_{1}, C_{2}, C_{3}) whose stable reduction gives rise to the datum (g, ω; f̄_{i}).

Since the outer automorphism group of SL_{2}(p) has order two, the number of liftings, up to isomorphism, is at least two. Proposition 3.1 implies that the number of elements of Ni_{3}^{in}(C) is at most two. We conclude that the number of liftings is exactly two. This proves that all SL_{2}(p)-covers over K with class vector C have bad reduction to characteristic p. □

Institut für Experimentelle Mathematik Max–Planck–Institut für Mathematik Universität GH Essen Vivatsgasse 7 Ellernstr. 29 53111 Bonn, Germany 45326 Essen, Germany wewers@mpim-bonn.mpg.de bouw@exp-math.uni-essen.de

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