Curves with infinite K-rational geometric fundamental group

Gerhard Frey, Ernst Kani and Helmut Völklein

1 Rational Geometric Fundamental Groups

Let K be a finitely generated field with separable closure K_{s} and absolute Galois group G_{K}. By a curve C/K we always understand a smooth geometrically irreducible projective curve. Let F (C) be its function field and let Π(C) be the Galois group of the maximal unramified extension of F (C). We have the exact sequence

1 → Π_{g}(C) → Π(C) → G_{K} → 1  (⋆)

where Π_{g}(C) is the geometric (profinite) fundamental group of C ×Spec(K_{s}) (i.e. Π_{g}(C) is equal to the Galois group of the maximal unramified extension of F (C) ⊗ K_{s}). This sequence induces a homomorphism ρ_{C} from G_{K} to Out(Π_{g}(C)) which is the group of automorphisms modulo inner automorphisms of Π_{g}(C). It is well known that ρ_{C} is an important tool for studying C. For instance, it determines C up to K−isomorphisms if the genus of C is at least 2 and K is a number field or even a 𝔭−adic field (see [Mo]).

So it is of interest to find quotients Π̄(C) of Π(C) such that the induced map of G_{K} is not the identity but the induced representation ρ̄_{C} becomes trivial. We give a geometric interpretation of such quotients. For this we assume that the sequence (⋆) is split and choose a section s which induces a homomorphism σ from G_{K} to Aut(Π_{g}(C)). Let U be a normal subgroup of Π(C) contained in Π_{g}(C). The representation ρ_{C} becomes trivial modulo U if and only if the map σ mod U has its image inside of Inn(Π_{g}(C)/U). Let Z be the center of Π_{g}(C)/U and Π̄ = (Π_{g}(C)/U)/Z. Our condition on U implies that there is exactly one group theoretical section s̄ from G_{K} to (Π̄(C)/U)/Z inducing the trivial action on Π̄ and so Π̄ occurs as Galois group of an unramified regular extension of F (C) in a natural way.

Thus, to find center free infinite factors of Π_{g} on which ρ_{C} becomes trivial is equivalent with finding infinite regular Galois coverings of C. Choosing as base point a geometric point of C we can say that “Π̄ is a factor of the geometric fundamental group of C over K”.

Remark 1.1 If s̄ can be extended to a (group theoretical) section into Π(C)/U then again it induces the trivial action on Π_{g}(C)/U and hence this group occurs as Galois group of a regular unramified extension of F (C). This is always true if K is a finite field or if Π/U is abelian.

But there is a difficulty: The sections s̄ corresponding to different quotients of Π_{g} may not be compatible ( i.e. the composite of the corresponding coverings are not regular) and so we do not have a “maximal unramified regular covering” of C over K.

So we specialize, assume that C has a K−rational point P and choose a splitting s_{P} of (⋆) corresponding to P by identifying G_{K} with the decomposition group of an extension of the place corresponding to P in F (C) to its separable closure . Now the finite quotients of Π_{g}(C) on which s_{P} (G_{K}) operates trivially correspond to unramified Galois coverings C′ of C on which the decomposition group of P operates trivially. Hence P has K-rational extensions to C′ and the choice of s_{P} corresponds to the choice of such an extension P′. (If we make another choice, then the corresponding section is replaced by a conjugate in Π(C).) We can regard C′ as etale covering of C with respect to the base points P resp. P′. Taking the limit we get the K-rational geometric fundamental group of C with base point P :

Π_{g}(C, P) := Π_{g}(C)/⟨(s_{P} (σ) − 1)Π_{g}(C)⟩_{σ∈G_{K}}.

Remark 1.2 The construction of unramified coverings with base points is also of interest for coding theory: Assume that g(C) ≥ 1. Since

g(C′) − 1 = [C′ : C](g_{C} − 1)

and #{P′ ∈ C′(K)} divisible by [C′ : C] we get (if C′(K) ≠ ∅) that

\frac{|C′(K)|}{g(C′) − 1} ≥ \frac{1}{(g_{C} − 1)}

and this means that C′ lies in a “good family” of coverings and so it is desirable to have infinite towers in this restricted sense over curves C (of small genus).

If K is a finite field such towers can be constructed by using class field theory of curves over any finite field (cf. Serre[Se1]) or over fields of square order by using Shimura curves (cf. [Ih]). By Ihara’s method one can even obtain curves of genus 2 which are optimal. The curve given by y^{2} = x^{6} − 1 is such an example. All these known examples are defined over finite fields and are of very special type.

In this paper we shall use either quartic coverings of the projective line or quadratic coverings of elliptic curves with enough ramification points (instead of quadratic coverings of the projective line as in the class field tower constructions) to find for every genus g ≥ 3 curves with infinite geometric fundamental group defined over ℚ(i) or over 𝔽_{q}(i) (where i is a forth root of unity) and we find even parametric families of such curves over every ground field containing i (cf. Theorem 5.22). ^{1}

More precisely we get the

Result: Let K be a field of characteristic ≠ 2, 3 which contains a fourth root of unity. Take a = \frac{2b^{2}}{1+b^{4}} with b ∈ K^{∗}, b^{4} ≠ 1. Let C be a cyclic covering of degree 4 of the projective line given by an equation

s^{4} = (1 + b^{4})t(t^{2} − 1)(t − a)g(t)

with a polynomial g(t) such that g(0)g(1)g(−1) ≠ 0 and g(a) = 1. Then C has an infinite K-rational geometric fundamental group with base point P_{a} corresponding to the place t = a.

We apply this and get the

Consequence: Let b and a be as above. Let E_{b} be the elliptic curve defined over K by the equation

E_{b} : y^{2} = (1+b^{4})x(x^{2}−1)(x−a).

Then for every natural number g ≥ 3 there exist quadratic coverings C of E_{b} defined over K of genus g_{C} = g with infinite K-rational geometric fundamental group with base point lying over P_{a}, the point of E_{b} corresponding to x = a.

1We remark that we do not have any example of a curve defined over ℚ with infinite ℚ-rational geometric fundamental group.

Example: Let K, b and a be as above. The curve

C_{a} : y^{4} = (1 + b^{4})x(x^{2} − 1)(x − a)

has an infinite K-rational geometric fundamental group with base point (a, 0).

We thus have examples of curves of genus g with an infinite K-rational geometric fundamental group for every g ≥ 3. No such example can exist for curves of genus g = 1, as will be explained in the next section. This leaves only the case g = 2. Here (cf. Example 4.13) we shall find very special curves with an infinite tower of regular unramified Galois coverings but we cannot decide there whether there are rational points in the tower.

There are other interesting aspects related to curves of genus 2 in the context of the questions discussed in this paper which will be investigated in more detail in the paper [FKV].

2 Abelian coverings

In this section we shall give a short review of the special case of abelian unramified coverings. The largest abelian factor of Π_{g}(C) which appears as the Galois group of a regular unramified extension of F (C) is Π_{g}(C)^{ab}/⟨(s_{P} (σ) − 1)Π_{g}(C)^{ab}⟩_{σ∈G_{K}} and so it is independent of the choice of P or of the choice of the splitting. (But the coverings will depend on the choices.) Moreover, one sees that every abelian group which can be realized as Galois group of a regular unramified covering of C can be realized in such a way that a given point P ∈ C(K) splits completely (cf. [V] or [Se]). So we do not have to distinguish between extensions with or without K-rational base points as long as we only want to describe the occurring Galois groups.

Let us begin by looking at elliptic curves. Here we have the following finiteness result:

Theorem 2.1 Let K be a finitely generated field. Then there is a number N = N(K) such that for all elliptic curves E defined over K we have |E(K)_{tor}| ≤ N.

Proof. We use induction with respect to the transcendence degree of K over its prime field K_{0}.

If K is a finite field the claim is obvious. If K is a number field the theorem of Merel gives a bound N depending on [K : ℚ] only (cf. [Me]). Let K be transcendental of degree d_{K} ≥ 1 over K_{0}. We fix a subfield K_{1} ⊂ K such that K_{1} is finitely generated over K_{0} and such that K is a function field of one variable over K_{1} of genus g_{K}. Let E/K be an elliptic curve with a K-rational point P of order n. Let K_{2} be the smallest extension field of K_{1} in K over which E is defined and over which P is rational. If K_{2} is equal to K_{1} we can use the induction hypothesis, otherwise K_{2} is a function field over K_{1} with a K_{1}-rational embedding of the function field of the modular curve X_{1}(n) into K_{2} and hence into K. Thus, g_{K} bounds the genus of X_{1}(n) which is of size O(n^{2}) (cf. Miyake[Mi], if (n, char(K)) = 1 and Igusa[Ig], for n = ℓ^{r}, ℓ = char(K)), and so n is bounded.

Corollary 2.2 There is a universal bound n = n(K) such that for all regular unramified Galois covers ϕ of elliptic curves E over K we have deg(ϕ) ≤ n.

Proof. If ϕ is an unramified Galois covering of E, then the covering curve E′ is an elliptic curve too, ϕ is an isogeny and the Galois group G_{K} acts on the kernel of ϕ trivially, i.e. Ker(ϕ) ≤ E′(K). Using the universal bound for torsion points obtained in the theorem the corollary follows.

Next we consider the case that the genus of C is larger than 1, and study as before the existence of regular unramified abelian Galois coverings of C.

Proposition 2.3 If ϕ : C_{1} → C is a regular abelian unramified covering, then deg ϕ is bounded by a number depending on K and on C.

Proof. By Serre [Se] we know that there is an abelian variety A/K such that ϕ induces an isogeny

ϕ_{∗} : A → J_{C}

with Ker(ϕ_{∗}) ⊂ A(K) and |Ker(ϕ_{∗})| = deg(ϕ). If K is finite we have |A(K)| = |J_{C}(K)|, and so we are done. If K is a number field or a field of transcendence degree larger than 1 over its prime field we can use reduction theory to get the result.

Remark: Since at present we do not have an analogue of Merel’s theorem for abelian varieties, we can only find a bound depending on the genus and on the conductor of C, and not only on K, even if K is a number field.

3 Coverings of ℙ^{1} with given ramification type

3.1 The Construction Principle

In the last section we had seen that for a given curve C over a finitely generated field K, the degree of the abelian unramified regular extensions of C/K is bounded. The situation changes if we look for non-abelian coverings. Indeed, in the next section we shall give examples of families of curves of arbitrary genus g ≥ 3 defined over a finitely generated field K which have infinite towers of unramified regular Galois coverings. However, it should be remarked that the dimension of these families is small compared with the dimension of the moduli space of curves of genus g and so it can be conjectured that for “general curves”, the degree of a regular unramified Galois covering is bounded by a number N(g, K).

The construction of these curves is based on the following principle. We begin with the projective line ℙ^{1} and choose a finite G_{K}-invariant set S_{0} of points of ℙ^{1}. Then we look for Galois coverings C′ of ℙ^{1} with a simple Galois group G (in our examples G is a projective linear group). We require that this covering is unramified outside of S_{0} and that the ramification in points P_{i} ∈ S_{0} has an order e_{P_{i}} which is small and prime to char(K). We take a Galois covering C_{0} of ℙ^{1} disjoint to C′ which ramifies in all P_{i} with an order which is a multiple of e_{P_{i}}. By Abhyankar’s Lemma it follows that C′×_{ℙ^{1}}C_{0}/C_{0} is unramified with Galois group G and, because of the simplicity of G, regular.

To find C′ we follow [FV] and use Hurwitz spaces which are moduli spaces for the ramification cycle configuration inside of G; this boils down to proving the existence of K-rational points on these spaces. This Diophantine problem leads to interesting relations between arithmetic geometry and group theory. Sometimes group theory or, to be more precise, the theory of rigidity implies the existence of rational points on the corresponding Hurwitz spaces. This is the case in our examples which are related to Thompson tuples, and so in this case the existence of such coverings is obtained by pure group theory (and Riemann’s existence theorem). However, this method fails to produce K-rational points on these coverings and so arithmetic-geometric methods have to be invoked in order to prove the existence of such points (which is necessary for our considerations of K-rational fundamental groups). In the case of the Thompson tuples, such a method is readily available, for behind their very construction (cf. [V3]) lies a deep arithmetic reason, viz. the existence of a family of abelian varieties with good reduction.

3.2 Rigidity

For the convenience of the reader we shall now give a very short overview over rigid systems of conjugacy classes. We shall assume that char(K) = 0, but actually it is enough to avoid fields with characteristics dividing the order of the occurring Galois groups and we shall use this fact in our examples. We want to describe the regular Galois coverings of the projective line. For this, we recall the situation over the complex numbers over which we can use as crucial tool the Riemann existence theorem.

We have the following invariants for a finite Galois extension L/ℂ(x): its Galois group G, the finite set \underline{P} = \{P_{1}, ..., P_{r}\} of branch points (which lie in ℙ^{1}(ℂ) = ℂ ∪ {∞}) and for each branch point P_{i} the associated conjugacy class C_{i} of G consisting of the distinguished inertia group generators over P_{i}. Define the ramification type of L/ℂ(x) to be the class of the triple (G, \underline{P} , C), where C = (C_{1}, ..., C_{r}). We say that two such triples are in the same class if the sets \underline{P} are the same and if there is an isomorphism between the groups that identifies the conjugacy classes associated with the same point of \underline{P}.

The extension L/ℂ(x) is defined over a subfield K of ℂ if there is a Galois extension L_{K}/K(x), regular over K, which becomes equivalent to L when tensored with ℂ. C is weakly rigid if any two generating systems σ_{1}, ..., σ_{r} of G with σ_{i} ∈ C_{i} and σ_{1} ⋯ σ_{r} = 1 are conjugate under an automorphism of G and if there is one such generating system. By Riemann’s existence theorem, C is weakly rigid if and only if there is exactly one Galois extension of ℂ(x) of type (G,\underline{P}, C) (up to equivalence), see [V], Th. 2.17. This uniqueness can be exploited to define the covering of ℙ^{1} over a smaller field. To do this, we sharpen the group theoretical conditions for the ramification type as follows.

C is quasi-rigid if it is weakly rigid and every automorphism of G fixing each C_{i} is inner.

Now suppose C is quasi-rigid and that L/ℂ(x) is a corresponding extension. Let L′ be the fixed field of the center Z(G) of G. Then the extension L′/ℂ(x) is defined over the field generated over ℚ by all branch points and by all roots of unity of order ord(σ), σ ∈ C_{1}∪...∪C_{r}. The exact minimal field of definition of L′ may be even smaller and can be determined using the branch cycle argument (see [V], 3.2 and 3.3.2). This construction principle for Galois extensions of rational function fields (“rigidity criterion”) has successfully been used to realize given groups as Galois groups.

4 Construction of infinite towers of unramified curve covers

4.1 Thompson tuples and Belyi triples

Let q be a power of the prime p, and n ≥ 3. We denote by 𝔽_{q}^{∗} the multiplicative group of the field 𝔽_{q} with q elements. For any σ ∈ GL_{n}(q), let c_{1}, ..., c_{n} be the eigenvalues of σ (counted with multiplicities) and χ_{σ}(T) = ∏_{i}(T − c_{i}) its characteristic polynomial (normalized to be monic). We call σ a perspectivity if it has an eigenspace of dimension n − 1.

Definition 4.1 Let r = n + 1 (respectively, r = 3). A Thompson tuple (respectively, a Belyi triple) in GL_{n}(q) is an r-tuple (σ_{1}, ..., σ_{r}) such that σ_{1}, ..., σ_{r} generate an irreducible subgroup of GL_{n}(q), their product satisfies σ_{1} ⋯ σ_{r} = 1 and σ_{i} is a perspectivity for all i = 1, ..., r (respectively, for i = 3).

In [V1] one finds

Theorem 4.2 Suppose that (σ_{1}, ..., σ_{r}) and (σ′_{1}, ..., σ′_{r}) are Thompson tuples (respectively, Belyi triples) in GL_{n}(q) with χ_{σ_{i}} = χ_{σ′_{i}} for all i. Then there is an element g ∈ GL_{n}(q) with σ′_{i} = g^{−1}σ_{i}g for all i. Thus σ_{1}, ..., σ_{r} are weakly rigid generators of G = ⟨σ_{1}, ..., σ_{r}⟩, and they are quasi-rigid generators if the normalizer of G in GL_{n}(q) equals 𝔽_{q}^{∗}G.

We can apply the theory of rigid ramification types presented in the last section to obtain (cf. [V], Theorems 2.17, 3.26 and 7.9):

Corollary 4.3 Let (σ_{1}, ..., σ_{r}) be a Thompson tuple or a Belyi triple in GL_{n}(q). Let C_{i} be the conjugacy class of σ_{i} in G = ⟨σ_{1}, ..., σ_{r}⟩, and let P_{1}, ..., P_{r} be distinct points of ℙ^{1}(ℂ). Put \underline{P} = \{P_{1}, ..., P_{r}\}, C_{P_{i}} = C_{i}, and C = (C_{P})_{P ∈\underline{P}}. Then we have:

(a) There is a unique Galois extension L′/ℂ(x) of ramification type [G, \underline{P} , C].

(b) Suppose that the normalizer of G equals 𝔽_{q}^{∗}G. If L = F ix(Z(G)) denotes the fixed field of the center Z(G) of G, then the extension L/ℂ(x) is defined over any subfield K ⊂ ℂ which contains all the (finite) P_{i} ∈ \underline{P} and all roots of unity of order ord(σ_{i}). In particular, there is a regular Galois extension L_{0} of K(x) with Galois group Ḡ = G/Z(G) with ramification type [Ḡ, \underline{P} , C̄], where C̄_{P} is the image of C_{P} in Ḡ.

Corollary 4.4 Suppose in addition that M is a regular extension of K(x) which is linearly disjoint to the above extension L_{0} over K, and which has the property that for each i = 1, …, r and each place m of M lying above the place corresponding to P_{i} of ℂ(x), the ramification index of m is a multiple of ord(σ_{i}). Then the composite L_{0} · M is an unramified Galois extension of the function field M and its Galois group is isomorphic to Ḡ.

We therefore see that to find such field extensions L, it is enough to find Thompson tuples or Belyi triples. These will now be investigated in more detail. The first result in this direction (taken from [V1]) shows that they can be characterized by their characteristic polynomials:

Theorem 4.5 Let f_{1}, ..., f_{r} ∈ 𝔽_{q}[T] be monic polynomials of degree n with ∏_{i} f_{i}(0) = (−1)^{rn}.

(a) There exists a Belyi triple (σ_{1}, σ_{2}, σ_{3}) in GL_{n}(q) with χ_{σ_{i}} = f_{i} if and only if f_{3}(T) = (T − c)^{n−1}(T − d) for elements c, d ∈ 𝔽_{q}^{∗} satisfying abc ≠ 1 for all a, b ∈ 𝔽̄_{q} with f_{1}(a) = f_{2}(b) = 0.

(b) There exists a Thompson tuple (σ_{1}, ..., σ_{r}) in GL_{n}(q) with χ_{σ_{i}} = f_{i} if and only if f_{i}(T) = (T − a_{i})^{n−1}(T − b_{i}) for elements a_{1}, ..., a_{r}, b_{1}, ..., b_{r} ∈ 𝔽_{q}^{∗} satisfying a_{1} ⋯ a_{r} ≠ 1 and b_{j} ∏_{i≠j} a_{i} ≠ 1 for all j = 1, ..., r.

From Theorem 4.2 we see that the tuples (σ_{1}, ..., σ_{r}) are classified by their characteristic polynomials χ_{σ_{i}}, and so the group G = ⟨σ_{1}, ..., σ_{r}⟩ depends only on the χ_{σ_{i}} (up to conjugation). In the case of Thompson tuples there is a complete classification of the related groups (cf. [V1] and [V4]). In the following Theorem 4.6 we shall state only that part of this classification which will be relevant for us below. In order to state the result, we require the following terminology. A Thompson tuple (σ_{1}, ..., σ_{n+1}) is called normalized if χ_{σ_{i}} = (T − 1)^{n−1}(T − b_{i}) for i = 1, ..., n.

Theorem 4.6 Let n be odd and ≥ 3. Suppose (σ_{1}, ..., σ_{n+1}) is a normalized Thompson tuple in GL_{n}(q) generating the group G with χ_{σ_{i}}(T) = (T − 1)^{n−1}(T − b_{i}) for certain b_{i} ∈ 𝔽_{q} for i = 1, ⋯, n. If n = 3 assume that some b_{i} has multiplicative order > 3. Suppose that 𝔽_{q} = 𝔽_{p}(a_{n+1}, b_{1}, ⋯ b_{n+1}), and assume that if q = q_{0}^{2} is a square, then not all the a_{i}, b_{i} have norm 1 over 𝔽_{q_{0}}. Then

SL_{n}(q) ≤ G ≤ GL_{n}(q).

4.2 Curves with infinite towers of unramified regular Galois extensions

We now apply the above results to construct unramified Galois G-covers of curves with group G = PSL_{n}(p) for suitable values of n and p. In order to be able to specify precisely which values are suitable here, we first introduce the following notation.

Notation. Let n ≥ 3 be an odd natural number, and let d_{1}, ..., d_{n+1} > 3 be integers. Then we denote by ℙ(n, d_{1}, ⋯, d_{n+1}) the set of all prime numbers p satisfying the following conditions:

(i) (n, p − 1) = 1;

(ii) there are elements b_{1}, …, b_{n+1} ∈ 𝔽_{p}^{∗} with b_{1} ⋯ b_{n+1} = 1 such that each b_{i} has multiplicative order d_{i} in 𝔽_{p}^{∗}.

A trivial but useful fact is

Lemma 4.7 If d_{1}, …, d_{n} are prime to n and d_{n+1} = lcm(d_{1}, …, d_{n}), then every prime p with p ≢ 1 mod l for all primes l | n and p ≡ 1 mod d_{n+1} is in ℙ(n, d_{1}, ⋯, d_{n+1}). Moreover, there is a constant c (depending only on n and d_{1}, …, d_{n+1}) such that for all primes l ≥ c there are infinitely many p ∈ ℙ(n, d_{1}, ⋯, d_{n+1}) such that l does not divide the order of GL_{n}(p).

From now on, suppose that the integers d_{1}, ..., d_{n+1} satisfy the conditions of this lemma, and that K is a field of characteristic 0 containing a primitive d^{th} root of unity, where d = lcm(d_{1}, …, d_{n+1}, 2) = lcm(d_{n+1}, 2). Moreover, let p ∈ ℙ(n, d_{1}, ⋯, d_{n+1}) and choose b_{i} ∈ 𝔽_{p}^{∗} such that condition (ii) holds. Then by Theorem 4.5 there is a normalized Thompson tuple (σ_{1}, ..., σ_{n+1}) in GL_{n}(p) corresponding to these b_{i} with a_{n+1} = −1. The order of σ_{i} equals d_{i} for i ≤ n and it equals lcm(d_{n+1}, 2) = d for i = n + 1. By Theorem 4.6 the group G = ⟨σ_{1}, ..., σ_{n+1}⟩ contains SL_{n}(p). Thus the group Ḡ = G/Z(G) is isomorphic to the simple group PSL_{n}(p) (because we assumed (n, p − 1) = 1 ). It remains to choose the ramification points. Since the automorphism group of ℙ^{1} acts sharply triple transitive we can assume without loss of generality that P_{n−1} = 0, P_{n} = 1 and P_{n+1} = ∞. The other ramification points are then denoted by t_{1}, ⋯, t_{n−2} ∈ K. Using Corollary 4.3 we see that for every p ∈ ℙ(n, d_{1}, ⋯, d_{n+1}), we get a regular Galois extension F′_{p} of K(X) with Galois group PSL_{n}(p) ramified only in 0, 1, ∞, t_{1}, …, t_{n−2} with ramification indices dividing d. Since for different primes p the Galois groups of the F′_{p}’s are non-isomorphic simple groups, the composite field

F′_{ℙ(n,d_{1},···,d_{n+1})} = ∏_{p∈ℙ(n,d_{1},···,d_{n+1})}F′_{p}

has the same regularity and ramification properties.

Next, let M_{d} be any regular extension field of K(X) such that every extension of the places 0, 1, ∞, t_{1}, ⋯, t_{n−2} is ramified with an order divisible by d. (For instance, take M_{d} as a cyclic cover of K(x) of degree d totally ramified at 0, 1, ∞, t_{1}, ⋯, t_{n−2}. In this case the genus of M_{d} is equal to (n−1)(d−1)/2.) Using Corollary 4.4, we get that F′_{ℙ(n,d_{1},···,d_{n+1})} · M_{d} is a regular unramified extension of M_{d} with Galois group equal to

∏_{p∈ℙ(n,d_{1},···,d_{n+1})}PSL_{n}(p).

Thus, if we take K = ℚ(ζ_{d}, t_{1}, …, t_{n−2}), where the t_{1}, …, t_{n−2} are algebraically independent over ℚ, then we obtain:

Theorem 4.8 Let n ≥ 3 be an odd number and let d_{1}, ⋯, d_{n} > 3 be integers prime to n, d_{n+1} := lcm(d_{1}, ⋯, d_{n}) and d = lcm(2, d_{n+1}). Then ℙ(n, d_{1}, ⋯, d_{n+1}) is an infinite set and there is an (n − 2)-dimensional rational family C of curves of genus (n − 1)(d − 1)/2 defined over ℚ(ζ_{d}) such that for all p ∈ℙ(n, d_{1}, ⋯, d_{n+1}) there is an unramified regular Galois cover C_{p} with Galois group PSL_{n}(p) and the composite of these covers is an infinite unramified regular Galois tower.

On the other hand, if we choose the t_{i}’s to be algebraic over ℚ, we obtain:

Theorem 4.9 Let K be a number field. Let n be odd and at least equal to 3 and let d be as in the previous theorem. Then there are infinitely many curves of genus (d − 1)(n − 1)/2 defined over K with infinite towers of unramified regular Galois covers defined over K(ζ_{d}).

As was mentioned above, the Hurwitz space theory and the theory of Thompson tuples works also over fields of characteristic l as long as l is prime to the order of the Galois group. Using the second part of Lemma 4.7 we see that Theorem 4.8 holds if we replace ℚ by 𝔽_{l} for almost all primes l. Hence we have

Corollary 4.10 Let n and d be as above, and let l be a sufficiently large prime such that there are infinitely many primes p with p^{i} − 1 prime to l for i = 1, …, n. Then there exist curves of genus (d − 1)(n − 1)/2 defined over K = 𝔽_{l}(ζ_{d}) which have an infinite tower of unramified regular Galois covers defined over K.

Now we discuss the simplest possibility and get the following example which will play an important role in Section 5.3.

Example 4.11 Let n = 3 and d_{1} = … = d_{4} = d = 4, and let K be a field of characteristic ≥ 5 which contains a fourth root of unity ζ_{4} = i. Furthermore, let t ∈ K be such that t ≠ 0, 1. Then for every p ≡ 5 mod 12 we get an extension of ℙ^{1} defined over K whose Galois group is equal to PSL_{3}(p) and which is ramified of order 4 in {0, 1, ∞, t}. Fix an integer g_{0} ≥ 1. If g_{0} = 1, let C_{0} be an elliptic curve which covers ℙ^{1} with ramification points {0, 1, ∞, t}, and if g_{0} > 1, let C_{0} be a hyperelliptic curve of genus g_{0} whose set of Weierstrass points contains {0, 1, ∞, t}. Let C be a quadratic extension of C_{0} ramified in the extensions of {0, 1, ∞, t}.

Then C has genus g := g_{0} + δ/2, where δ is the number of ramified primes in the cover C/C_{0}. For example, if we choose C as a cyclic cover of degree 4 over ℙ^{1} which is ramified in the Weierstrass points of C_{0} with ramification order 4 and in s additional points of ℙ^{1} with ramification order 2, then g = 3g_{0} + s.

Hence we get

Corollary 4.12 Let K be a field of characteristic ≠ 2, 3 which contains a primitive fourth root of unity i = ζ_{4}. Let t ∈ K, t ≠ 0, 1 and let C_{0} be either an elliptic curve (g_{0} = 1) or a hyperelliptic curve of genus g_{0} defined over K with at least four K-rational points of order 2 respectively, Weierstrass points equal to {0, 1, ∞, t}. Then for all non-negative integers s there are quadratic covers of C_{0} defined over K of genus 3g_{0} + s with an infinite tower of regular unramified Galois covers over K.

If we want to obtain examples as above for curves of genus 2, then we have to use Belyi triples, but in that case we find only isolated examples. Here is one of them:

Example 4.13 Let Φ_{m} be the cyclotomic polynomial of order m, and let (σ_{1}, σ_{2}, σ_{3}) be a Belyi triple in GL_{3}(p) with characteristic polynomials χ_{σ_{1}} = (T + 1)Φ_{6}, χ_{σ_{2}} = (T − 1)Φ_{3}, and χ_{σ_{3}} = (T − 1)^{2}(T + 1). Then the images σ̄_{1}, σ̄_{2}, σ̄_{3} generate the simple group PSL_{3}(p) for p ≡ −1 mod 3, and there is a corresponding extension L_{p}/ℚ(x) branched at 0, ∞, 1. Take x = t^{6} and let M/ℚ(t) be the genus 2 function field extension branched where t is a sixth root of 1, i.e. M corresponds to the curve with equation

y^{2} = t^{6} − 1.

Then L_{p} · M/M is unramified with group PSL_{3}(p) and defined over ℚ. It follows that the curve y^{2} = t^{6} − 1 has an infinite tower of regular unramified Galois extensions over ℚ and over 𝔽_{l} for all primes l ≥ 5.

We observe that although we obtain in this way the example of Ihara mentioned in the first section, we have actually realized different Galois groups by this method. At present, however, we cannot answer the question concerning the existence of rational points in the associated infinite tower.

Further examples (for PSL_{3}(p) again) are obtained by taking χ_{σ_{3}} as above, and χ_{σ_{1}} = (T − 1)Φ_{3}, χ_{σ_{2}} = (T + 1)Φ_{4}, or alternatively, χ_{σ_{1}} = (T + 1)Φ_{3}, χ_{σ_{2}} = (T − 1)Φ_{4}. We thank G. Malle for pointing out these examples.

4.3 Geometric Interpretation of Thompson tuples

In the last section, the examples of almost unramified covers of ℙ^{1} were constructed by using group theory. This construction did not provide any method for determining the existence of rational points in the covers, nor did it explain the arithmetical “reason” behind the existence of such examples. However, both these aspects are greatly elucidated by using the results of [V3] to relate the Thompson tuples to torsion points of Jacobians of curves which are cyclic covers of the projective line.

The basic idea here is the following. If ϕ : C → ℙ_{K}^{1} is a cyclic covering with group Z = Aut(ϕ) defined over a field K and λ : Z → 𝔽_{q}^{∗} is a faithful character, then the Galois group G_{K} acts 𝔽_{q}-linearly on the λ-component J_{C}[p]_{λ} of the group J_{C}[p] of p-torsion points of the Jacobian variety J_{C} of C. Now it turns out that if the ramification points t_{1}, …, t_{n+2} ∈ K of ϕ are in “sufficiently general position“, then the image of the Galois group G_{K} in GL_{n}(J_{C}[p]_{λ}) is (essentially) generated by a Thompson tuple, and conversely, every Thompson tuple arises in this fashion.

In order to make this connection between cyclic coverings and Thompson tuples more precise, it is useful to introduce the following alternate description of Thompson tuples, using the Braid group. For this, let s = n+2, and let Q_{1}, …, Q_{s−1} be the standard generators of the Artin braid group B_{s} on s strings. Mapping Q_{i} to the transposition (i, i + 1) yields a homomorphism κ : B_{s} → S_{s}. The kernel of κ is called the pure braid group, and is denoted by B^{(s)}. It is generated by the elements

Q_{ij} = Q_{j−1} ⋯ Q_{i+1} Q_{i}^{2} Q_{i+1}^{−1} ⋯ Q_{j−1}^{−1}, 1 ≤ i < j ≤ s.

The braid group B_{s} acts on tuples (g_{1}, …, g_{s}) of elements of any group by the rule that Q_{i} maps (g_{1}, …, g_{s}) to

(g_{1}, …, g_{i−1}, g_{i+1}, g_{i+1}^{−1}g_{i}g_{i+1} , …, g_{s})

Lemma 4.14 (a) If \underline{ζ} = (ζ_{1}, …, ζ_{n+2}) is an (n+2)-tuple of elements ζ_{i} ∈ 𝔽_{q}^{∗} satisfying the conditions

∏_{j=1}^{n+2} ζ_{j} = 1, ζ_{i} ≠ 1, for all i,  (1)

then there is a unique (up to conjugation) normalized Thompson tuple \underline{σ} = \underline{σ}\underline{ζ} = (σ_{1}, …, σ_{n+1}) in GL_{n}(q) such that

χ_{σ_{i}}(T) = (T − 1)^{n−1}(T − ζ_{i}^{−1}ζ_{n+2}^{−1}), 1 ≤ i ≤ n,
χ_{σ_{n+1}}(T) = (T − ζ_{n+2})^{n−1}(T − ζ_{n+1}^{−1}).

Moreover, every normalized Thompson tuple \underline{σ} in GL_{n}(q) is of the form \underline{σ} = \underline{σ}\underline{ζ}, for a unique (n + 2)-tuple \underline{ζ} satisfying (1).

(b) In the situation of (a) there is a homomorphism

Φ_{ζ} : B^{(s)} → GL_{n}(q)

such that for i = 1, …, s − 1 the elements τ_{i} := Φ_{ζ}(Q_{is}) are perspectivities with χ_{τ_{i}} = (T − 1)^{n−1}(T − ζ_{i}^{−1}ζ_{s}^{−1}) and τ_{1} ⋯ τ_{s−1} = ζ_{s}^{−1}· id. Thus (τ_{1}, …, τ_{s−2}, ζ_{s}τ_{s−1}) is a Thompson tuple associated with ζ.

Proof. (a) Put s = n + 2. Recall from Theorem 4.5 that a normalized Thompson tuple (σ_{1}, …, σ_{n+1}) is characterized by the characteristic polynomials of the elements σ_{i} and so by the s-tuples (b_{1}, ⋯, b_{n+1}, a) ∈ (𝔽_{q})^{n+2} which satisfy the additional properties that ∏_{i=1}^{n+1} b_{i}·a^{n−1} = 1 and that a ≠ 1, b_{n+1} ≠ 1 and a · b_{i} ≠ 1 for 1 ≤ i ≤ n. We associate to (σ_{1}, …, σ_{n+1}) the s−tuple (ζ_{1}, ⋯, ζ_{n+1}, ζ_{s}) with ζ_{i} = (ab_{i})^{−1} for 1 ≤ i ≤ n, ζ_{n+1} = b_{n+1}^{−1} and ζ_{s} = a. This tuple has the property that all ζ_{i} are different from 1 and that ∏_{i=1}^{s} ζ_{i} = 1. Conversely, if ζ_{1}, …, ζ_{n+2} ∈ 𝔽_{q}^{∗} satisfying condition (1) are given, then a := ζ_{n+2}, b_{n+1} := ζ_{n+1} and b_{i} := ζ_{i}^{−1}ζ_{n+2}^{−1}, for 1 ≤ i ≤ n satisfy the above conditions and hence give rise to a Thompson tuple \underline{σ}\underline{ζ} = (σ_{1}, …, σ_{n+1}) with the indicated characteristic polynomials.

(b) For the definition of Φ_{ζ} see [V4] and [MSV]. (It is essentially the Gassner representation of B^{(s)} associated with ζ). The first assertion follows from [V4], Lemma 3 and relation (5). The relation τ_{1} ⋯ τ_{s−1} = ζ_{s}^{−1}· id follows from the definition of Φ_{ζ} and the fact that the element

Q_{1s} ⋯ Q_{s−1,s} = Q_{s−1} ⋯ Q_{2} Q_{1}^{2} Q_{2} ⋯ Q_{s−1}

acts by sending (g_{1}, …, g_{s}) to (g_{1}, …, g_{s})^{g_{s}}. The τ_{i} generate an irreducible group by [V1], Lemma 3.3.

We thus see that ζ-tuples give rise to Thompson tuples and conversely. On the other hand we can also attach to such tuples an (essentially unique) cyclic covering ϕ : C → ℙ^{1} which is ramified at s = n + 2 prescribed points t_{1}, …, t_{s} ∈ K as follows.

Lemma 4.15 Suppose ζ_{1}, …, ζ_{s} ∈ 𝔽_{q}^{∗} satisfy ζ_{1} · . . . · ζ_{s} = 1 and d_{i} := ord(ζ_{i}) ≠ 1, 1 ≤ i ≤ s. Let K be a field containing a primitive d-th root of unity where d = lcm(d_{1}, …, d_{s}) and suppose that t_{1}, …, t_{s} ∈ K are s distinct points. Then there is a cyclic covering ϕ = ϕ_{\underline{t},\underline{ζ}} : C → ℙ_{K}^{1} and a faithful character λ : Z := Aut(ϕ) → 𝔽_{q}^{∗} such that ϕ is unramified outside of {t_{1}, …, t_{s}} and is ramified at t_{i} in such a way that the distinguished inertia group generator z_{i} at t_{i} satifies λ(z_{i}) = ζ_{i}, for 1 ≤ i ≤ s. Furthermore, all automorphisms of ϕ are defined over K.

Proof. This is elementary and well-known, but in order to be able to discuss the examples below it is useful to make this precise. Thus, fix a primitive d-th root ζ ∈ 𝔽_{q} (which exists by hypothesis). Then there exist unique numbers m_{1}, …, m_{s} with 0 < m_{i} < d such that ζ_{i} = ζ^{m_{i}}. Now consider the cyclic covering ϕ : C → ℙ^{1} defined by the equation

y^{d} = (x − t_{1})^{m_{1}}(x − t_{2})^{m_{2}} ⋯ (x − t_{s})^{m_{s}}.

Since (m_{1}, …, m_{s}) = 1, this is a covering of degree d and since m_{1}+…+m_{s} ≡ 0 mod d, it follows that ϕ is unramified at infinity and hence also outside of {t_{1}, …, t_{s}}. Now let σ be the unique generator of Z :=Aut(ϕ) such that σ(y) = e^{2πi/d}y. Then z_{i} = σ^{m_{i}} is the distinguished inertia group generator at t_{i}, and so if we let λ : Z → 𝔽_{q}^{∗} be the unique homomorphism such that λ(σ) = ζ, then λ(z_{i}) = ζ_{i}, as desired.

Now let J_{C} denote the Jacobian of the curve C of Lemma 4.15, and let p be a prime not dividing d. Then the group Z = Aut(ϕ) acts on the group J_{C}[p] of (K̄-rational) p-torsion points of J_{C}, and so J_{C}[p] is the direct sum of its λ-components J_{C}[p]_{λ^{k}} = J_{C}[p] ⊗_{𝔽_{p}[Z]} V_{λ^{k}}, where V_{λ^{k}} ≃ 𝔽_{q_{k}} denotes the representation space of the linear characters λ^{k} : Z → 𝔽_{q_{k}}^{∗} of Z. Since the automorphisms z ∈ Z are defined over K, the Galois action commutes with the Z-action, and so G_{K} acts 𝔽_{q_{k}}-linearly on each space J_{C}[p]_{λ^{k}} (which has a natural 𝔽_{q_{k}}-vector space structure). Since J_{C}[p]_{λ} has dimension n = s − 2 over 𝔽_{q} (see [V3], or equation(3) below), this yields a homomorphism G_{K} → GL_{n}(q).

Theorem 4.16 Let ζ = (ζ_{1}, …, ζ_{s}) be a ζ-tuple in 𝔽_{q} satisfying condition (1) with the property that 𝔽_{q} = 𝔽_{p}(ζ_{1}, …, ζ_{s}), and let \underline{σ}\underline{ζ} = (σ_{1}, …, σ_{s−1}) be the associated Thompson tuple in GL_{n}(q), where n = s − 2. In addition, let t_{1}, …, t_{s−1} be distinct points in k, and let t_{s} = X be transcendental over k. Put K = k(X), and choose accordingly the cyclic covering

ϕ = ϕ_{\underline{t},ζ} : C → ℙ_{K}^{1}

and character λ : Z → 𝔽_{q}^{∗} as in the previous Lemma. Let S = J_{C}[p]_{λ} be the associated λ-component of the p-torsion points of J_{C}, and let N = N_{λ,Z′} be the fixed field of the kernel of the homomorphism

ρ_{λ,Z′} : G_{K} → GL_{n}(q)/Z′

induced by the action of G_{K} on S ≃ 𝔽_{q}^{n}, where Z′ ≤ Z(GL_{n}(q)) is any subgroup containing λ(Z).

(a) The Galois group of the extension k̄N/k̄(X) is the subgroup

G′ = ⟨σ′_{1}, …, σ′_{s−1}⟩ ≤ GL_{n}(q)/Z′

generated by the images σ′_{i} of the Thompson tuple in GL_{n}(q)/Z′. Moreover, the ramification type of this extension is [G′, {t_{1}, …, t_{s−1}}, C′], where C′ = (C′_{1}, …, C′_{s−1}) and C′_{i} denotes the class of σ′_{i} in G′.

(b) If G′ is self-normalizing in GL_{n}(q)/Z′, then N is regular over k, and hence Gal(N/k(X)) ≃ Gal(k̄N/k̄(X)) ≃ G′.

Proof. The analogous result for the case that the branch points t_{1}, …, t_{s} are algebraically independent over k was proved in [V3], Theorem A. So we have to carry through a specialization argument which essentially boils down to replacing the pure braid group B^{(s)} by its normal subgroup generated by Q_{1,s}, …, Q_{s−1,s}. As in the proof of Theorem A of [V3], let O_{s} be the space of subsets of ℂ of cardinality s, and H(C) the Hurwitz space related to covers of type ζ; and let p be a point of H(C) over the point p̄ = {t_{1}, …, t_{s}} of O_{s} such that the corresponding cover of ℙ^{1} has ramification of type ζ_{i} over t_{i}. We can proceed as in that proof till to step 4. There we specialize: O has to be replaced by the curve L̃ defined as follows: Let L be the curve on O_{s} consisting of all {t_{1}, …, t_{n+1}, t}, t ∈ ℂ \ {t_{1}, …, t_{n+1}}, and L̃ the (absolutely) irreducible component of the inverse image of L in H_{s}(G)^{(A_{1})} that contains p_{1}; here A_{1} = V · Z′. The fundamental group π_{1}(O_{s}, p̄) can be identified with the braid group B_{s} in such a way that Q_{1,s}, …, Q_{s−1,s} correspond to loops around t_{1}, …, t_{s−1} on L that generate π_{1}(L, p̄). Let H be the stabilizer of L̃ in A_{0}/A_{1} ≅ GL_{n}(q)/Z′. Let Φ′_{ζ} : B^{(s)} → GL_{n}(q)/Z′ be the composition of Φ_{ζ} (see Lemma 4.14) with the canonical map GL_{n}(q) → GL_{n}(q)/Z′. As in [V5], Lemma 2.2 we see that H is conjugate in GL_{n}(q)/Z′ to the group G′ generated by Φ′_{ζ}(Q_{1,s}), …, Φ′_{ζ}(Q_{s−1,s}). We may assume H = G′. Then similarly as in Step 4, the Galois action of G_{k̄(X)} on the subspace S ≅ 𝔽_{q}^{n} of the p-division points of the Jacobian of C induces modulo Z′ the group H = G′. Furthermore, if we let N′ = k̄N (which is the fixed field of the kernel of the map G_{k̄(X)} → G′), then the extension N′/k̄(X) is ramified where t equals some t_{i}, and Φ′_{ζ}(Q_{i,s}) is a corresponding distinguished inertia group generator. This (together with Lemma 4.14) proves part (a) of the theorem.

(b) Note that G_{k̄(X)} is normal in G_{k(X)}. Thus, if G′ is self-normalizing in GL_{n}(q)/Z′, then the image of G_{k(X)} in GL_{n}(q)/Z′ also equals G′, and so the corresponding extension N/k(X) is regular over k.

Remark 4.17 Let K(S_{λ}) denote the extension of K obtained by adjoining the coordinates of the p-torsion points in S_{λ} ⊂ J_{C}[p] to K. Then clearly K(S_{λ}) = N_{λ,1} is the fixed field of the kernel of the representation

ρ_{λ} : G_{K} → Aut(S_{λ}) ≃ GL_{n}(q)

on S_{λ}, and hence is a Galois extension of K which contains the field(s) N = N_{λ,Z′} of Theorem 4.16. Moreover, K(S_{λ}) is cyclic over N with Galois group Gal(K(S_{λ})/N) ≤ Z′. We shall see below in Theorem 5.14 that the extension K(S_{λ})/N_{λ,Z′} is always an unramified extension.

The consequence of Theorem 4.16 is that the almost unramified families of covers of ℙ^{1} constructed in the previous section are obtained by adjoining Galois invariant subspaces of torsion points of families of abelian varieties to K(X). This imposes strong conditions on these abelian varieties. In the next section we shall discuss some consequences of these conditions in more detail.

5 A family of curves with infinite geometric fundamental group

5.1 The abelian variety J_{C}^{new}

In order to interpret the results of the previous section geometrically, it is useful to introduce the following “new part” J_{C}^{new} of the Jacobian J_{C} of a cyclic covering π : C → C′ of curves, which is (partially) analogous to the new part of the Jacobian of the modular curve X_{0}(N).

Definition 5.1 Let π : C → C′ be a cyclic covering of curves defined over an arbitrary field K, and let Z = Aut(π) ≃ ℤ/Nℤ denote its covering group. For each subgroup H ≤ Z let π_{H} : C → C_{H} = C/H denote the quotient map. Then we call the abelian subvariety

J_{C}^{old} := ∑_{{H≤Z \atop H≠1}} π_{H}^{∗}J_{C/H}

the old part of J_{C}, and its orthogonal complement (with respect to the canonical polarization on J_{C}) the new part J_{C}^{new}.

For our purposes it is important to observe that J_{C}^{new} and J_{C}^{old} can be expressed in terms the following idempotent ε_{new} of the group ring ℚ[Z]:

ε_{new} = ∑_{d|N} μ(d)ε_{d},

where, for a subgroup H ≤ Z of order d,

ε_{d} = ε_{H} = \frac{1}{d} ∑_{h∈H} h ∈ ℚ[Z].

Lemma 5.2 If χ : Z → ℂ^{∗} is a fixed character of order N = |Z|, then

ε_{d} = ∑_{{k=0 \atop d|k}}^{N−1} ε_{χ^{k}} and ε_{new} = ∑_{{k=1 \atop (k,N)=1}}^{N−1} ε_{χ^{k}},  (2)

where ε_{χ^{k}} = \frac{1}{N} ∑_{g∈Z} χ(g)^{k}g^{−1} ∈ ℂ[Z] denotes the primitive idempotent associated to the character χ^{k}. In particular, ε_{new} is a symmetric idempotent of ℚ[Z] and

ε_{d} · ε_{new} = 0, for all d|N, d ≠ 1.

Proof. Since the χ^{k} form a dual basis of the group ring ℂ[Z], it is enough to verify (2) after evaluating both sides by χ^{k}, for k = 0, …, N −1. Now for any class function ψ on Z we have ψ(ε_{H}) = (1_{H}, ψ_{|H}), ψ(ε_{χ^{k}}) = (χ^{k}, ψ). Since for a subgroup H of order d we have χ_{|H}^{k} = 1_{H} ⟺ Ker(χ^{k}) ≥ H ⟺ d|k, it follows that χ^{k}(ε_{d}) = 1 if d|k and χ^{k}(ε_{d}) = 0 otherwise. From this, the first formula of (2) is obvious and the second follows readily, using the fact that s := ∑_{d|(k,N)} μ(d) = 1, if (k, N) = 1 and s = 0 otherwise. Thus, by (2) we see that ε_{new} is a sum of pairwise orthogonal idempotents, and hence is also an idempotent. Furthermore, ε_{new} is symmetric since all ε_{d} are symmetric. Finally, if d ≠ 1, then (2) shows that ε_{d} and ε_{new} have no common components ε_{χ^{k}}, and hence are orthogonal.

Corollary 5.3 Put ε̃_{new} = Nε and ε̃_{old} = N − ε̃. Then ε̃_{new}, ε̃_{old} ∈ End(J_{C}) and we have

J_{C}^{old} = ε̃_{old}(J_{C}), J_{C}^{new} = ε̃_{new}(J_{C}).

Furthermore, J_{C} = J_{C}^{old} + J_{C}^{new}, and J_{C}^{old} ∩ J_{C}^{new} ≤ J_{C}[N]; in particular, J_{C} ∼ J_{C}^{old} × J_{C}^{new}.

Proof. By definition, Nε_{new} ∈ ℤ[Z] ⊂ End(J_{C}), and so ε̃_{new}, ε̃_{old} ∈ End(J_{C}); note that ε̃_{old} = N(1 − ε_{new}) = ∑_{1≠d|N} μ(d)Nε_{d}. Put ε̃_{d} = dε_{d} ∈ ℤ[Z]. Then ε̃_{d}(J_{C}) = π_{H_{d}}^{∗}(J_{C/H_{d}}), so it follows from the definition that ε̃_{old}(J_{C}) ⊂ ∑_{1≠d|N} π_{H_{d}}^{∗}(J_{C/H_{d}}) = J_{C}^{old}. On the other hand, by Lemma 5.2 we have ε̃_{old}ε̃_{d} = Nε̃_{d}, if d ≠ 1, so π_{H_{d}}^{∗}(J_{C/H_{d}}) = ε̃_{d}(J_{C}) = ε̃_{old}(ε̃_{d}(J_{C})) ⊂ ε̃_{old}(J_{C}) and hence J_{C}^{old} = ε̃_{old}(J_{C}). By construction, ε̃_{new} + ε̃_{old} = N, ε̃_{new} · ε̃_{old} = 0, so J_{C}^{new} := ε̃_{new}(J_{C}) is the orthogonal complement of J_{C}^{old}, and hence the isogeny relation follows.

Corollary 5.4 Suppose p ≠ char(K) is a prime with p ≡ 1 mod N. Then there exist primitive characters λ : Z → 𝔽_{p}^{∗} and λ̃ : Z → ℤ_{p}^{∗} of order N and we have natural identifications (of Z-modules and G_{K}-modules)

J_{C}^{new}[p] = ⨁_{{1≤k<N \atop (k,N)=1}} J_{C}[p]_{λ^{k}}, and T_{p}(J_{C}^{new}) = ⨁_{{1≤k<N \atop (k,N)=1}} T_{p}(J_{C})λ̃^{k},

where T_{p}(A) = lim_{⟵}A[p^{n}] denotes the Tate-module of an abelian variety A.

In particular, dim J_{C}^{new} = \frac{1}{2}ϕ(N)(2g_{C′} − 2 + r), where r = #\{P ∈ C′(K̄) : e_{P} (π) > 1} denotes the number of ramified primes.

Proof. By the hypothesis on p, both 𝔽_{p} and ℤ_{p} contain a primitive N-th root of unity and so λ and λ̃ exist. Thus, if we replace χ by λ and λ̃ in (2), the analogous formulae of (2) hold in 𝔽_{p}[Z] and in ℤ_{p}[Z]. From this the above decomposition follows because J_{C}^{new}[p] = ε_{new}(J_{C}[p]) and J_{C}[p]_{λ^{k}} = ε_{λ^{k}}(J_{C}[p]), and similarly, T_{p}(J_{C}^{new}) = ε_{new}T_{p}(J_{C}) and T_{p}(J_{C})λ̃^{k} = ελ̃^{k}T_{p}(J_{C}). (Note that this is also a decomposition of G_{K}-modules since the Z-action commutes with the G_{K}-action.) To work out the dimension of J_{C}^{new}, we use the well-known fact (cf. Serre[Se2], p. 106) that the character h_{1} of the representation of Z on T_{p}(J_{C}) ⊗ ℚ_{p} is given by

h_{1} = 2 · 1_{Z} + (2g_{C′} − 2)reg_{Z} + a_{Z}

where a_{Z} = ∑_{P′∈C′} a_{P′}, and a_{P′} = (reg_{Z} − 1_{D_{P}}^{∗}) denotes the (tame) Artin character at P′. (Here D_{P} is the decomposition group of any P ∈ π^{−1}(P′), and ^{∗} denotes induction.) From this it follows easily (by Frobenius reciprocity) that if k ≢ 0 mod N, then the rank of T_{p}(J_{C})λ̃^{k} (= dim_{𝔽_{p}}(J_{C}[p]_{λ^{k}})) is

(h_{1}, λ̃^{k}) = (2g_{C′} − 2) + r_{k}, where r_{k} := (a_{Z}, λ̃^{k}) = #{P′ ∈ C′ : e_{P′} |̸ k}. (3)

Since rank(T_{p}(J_{C}^{new})) = 2 dim(J_{C}^{new}), the asserted dimension formula follows.

Using the abelian variety J_{C}^{new}, we can now give the following geometric interpretation of the main result (Theorem 4.16) of the previous section.

Theorem 5.5 Let K = k(t), where t is transcendental over k and k contains the N-th roots of unity. Suppose that t_{1}, …, t_{n+1} are n + 1 distinct elements in k, and put t_{n+2} = t. In addition, suppose that m_{1}, …, m_{n+2} are integers with 1 ≤ m_{i} < N such that gcd(m_{1}, …, m_{n+2}, N) = 1, m_{1} + … + m_{n+2} ≡ 0 mod N and m_{i} ≠ N − m_{n+2}, for 1 ≤ i ≤ n + 1. Let π : C → ℙ^{1} denote the cyclic covering of degree N defined by the equation

y^{N} = c(x − t_{1})^{m_{1}} ⋯ (x − t_{n+2})^{m_{n+2}},  (4)

where c ∈ k^{∗}, and let A = J_{C}^{new} denote the new part of the Jacobian J_{C} of C. Then dim A = \frac{1}{2}ϕ(N)n, and for any prime p ≡ 1 mod N we have:

(a) The group A[p] of p-torsion points of A is the direct sum of G_{K}-invariant subspaces S_{i} = J_{C}[p]_{λ^{i}}, where (i, N) = 1, and λ : Z = Aut(π) → 𝔽_{p}^{∗} is a primitive character. Moreover, G_{K} acts irreducibly on each S_{i}, and hence semi-simply on A[p].

(b) Let L_{i} (respectively, M̃_{i} and M_{i}) denote the fixed field of the kernel of the action of G_{K} on S_{i} (respectively, on S_{i}/λ^{i}(Z) and on ℙ(S_{i})), so L_{i} ⊃ M̃_{i} ⊃ M_{i}. Then L_{i}/M̃_{i} is a cyclic extension of order dividing N, and M̃_{i}/K (respectively, M_{i}/K) is a Galois extension whose geometric part G̃ := Gal(k̄M̃_{i}/k̄K) ≤ GL_{n}(p)/λ^{i}(Z) (respectively, Ḡ := Gal(k̄M_{i}/k̄K) ≤ PGL_{n}(p)) is generated by the image of the Thompson tuple attached to the covering. In addition, M̃_{i} (and hence also M_{i}) is ramified only at t_{1}, …, t_{n+1} with ramification order dividing N.

(c) The field K(A[p]) (which is generated over K by the coordinates of all p-torsion points of A) is an abelian extension of the compositum M̃ of all the M̃_{i}’s of degree dividing N^{ϕ(N)}. Moreover, M̃/K is ramified only at t_{1}, …, t_{n+1} with ramification order dividing N.

(d) If (n, p − 1) = 1, (and N |̸ 6 if n = 3), then M_{i} is regular over k and

Gal(M_{i}/K) = PGL_{n}(p) = PSL_{n}(p).

Proof. (a) The first assertion follows directly from the decomposition of Corollary 5.4. Next, choose a “canonical generator” σ of Z as in the proof of Lemma 4.15. Then for any i with (i, N) = 1, the hypotheses on the m_{j}’s guarantee that the (n + 2)-tuple ζ_{1} = λ^{i}(σ^{m_{1}}), …, ζ_{n+2} = λ^{i}(σ^{m_{n+2}}) satisfies condition (1) of Lemma 4.14 and so by Theorem 4.16(a) (with Z′ = Z(GL_{n}(p))) it follows that the image of the Galois group G_{k̄K} on ℙ(S_{i}) is generated by a Thompson tuple and hence G_{K} ≥ G_{k̄K} acts irreducibly on S_{i}.

(b) Since Gal(L_{i}/M̃_{i}) = Gal(L_{i}/K) ∩ λ^{i}(Z) ≤ λ^{i}(Z) (viewed as subgroups of Aut_{𝔽_{p}}(S_{i}) ≃ GL_{n}(p)), the first assertion is clear. The other assertions follow directly from Theorem 4.16(a) by taking Z′ = λ^{i}(Z) (respectively, Z′ = Z(GL_{n}(p))).

(c) Since A[p] is the direct sum of the S_{i}’s (cf. part (a)), we have K(A[p]) = ∏ L_{i}, and so both assertions follow from part (b).

(d) The hypothesis implies that n is odd and that PSL_{n}(p) = PGL_{n}(p). Thus, from Theorem 4.6 it follows that the image Ḡ of the Thompson tuple in PGL_{n}(p) generates the whole group. In particular, Ḡ is self-normalizing, and so M_{i} is regular over k by Theorem 4.16(b).

Remark 5.6 We shall see below in Theorem 5.14 that the extension L_{i}/M_{i} is always unramified; in particular, K(A[p]) is unramified over M̃.

By applying a variant of the Serre-Tate criterion of potentially good reduction (see below), we obtain Theorem 5.7 If C/K is as in Theorem 5.5, then the new part A = J_{C}^{new} of its Jacobian J_{C} has potentially good reduction over K. In other words, there is a finite extension K_{0} of K such that A_{K_{0}} = A ⊗ K_{0} extends to an abelian scheme A over the projective curve T_{0} with function field K_{0}, i.e. for each geometric point P ∈ T_{0}(K̄) the fibre of A over P is an abelian variety.

Proof. By Theorem 5.5 (c), we see that for all primes p ≡ 1 mod N, the ramification degrees of the extension K(A[p])/K divide N^{ϕ(N)+1}, and so the assertion follows from the following criterion:

Proposition 5.8 (Serre-Tate Criterion) Let K be a field with a discrete valuation v, and let A be an abelian variety over K. Then the following conditions are equivalent:

(i) A has potentially good reduction at v;

(ii) the ramification degree of v in K(A[m]) is bounded for all m prime to char(κ(v));

(iii) there is a constant c such that the ramification degree e_{v}(K(A[p])/K) ≤ c, for infinitly many primes p.

Proof. This is well-known, but for convenience of the reader we present the proof.

(i) ⇒ (ii): This follows from the criterion of Néron-Ogg-Shafarevich for good reduction; cf. Serre-Tate[ST], Theorem 2. (ii) ⇒ (iii): Trivial. (iii) ⇒ (i): By replacing K by a suitable finite extension K′, we may assume without loss of generality that A has semi-stable reduction at v. (For example, we could take K′ = K(A[p]), for any prime p ≥ 3, p ≠ char(κ(v)); cf. Grothendieck[Gro], Prop. 4.7). In addition, we may assume that K is henselian with separably closed residue field. Then the implication (iii) ⇒ (i) follows once we have verified:

Claim: If A does not have good reduction at v, then p|e_{v}(K(A[p])/K), for every prime p |̸ #(Φ(A_{K})) (p ≠ char(κ(v))), where Φ(A_{K}) denotes the (finite) group of components of the Néron model of A/K with respect to v.

To see this, we first note that for p as above, #(A[p](K)) = #(A^{0}[p](κ(v)) = p^{t+2a}, where t denotes the toric rank and a the abelian rank of the connected component A^{0} of the identity of the reduction of the Néron model. (Note that t + a = d := dim A, and that t > 0 since A has bad reduction.) On the other hand, for K′ = K(A[p]) we clearly have #(A[p](K′)) = p^{2d}, and so p^{t}|#Φ(A_{K′}). Thus, p|#(Φ(A_{K′})/Φ(A_{K})). On the other hand, the quotient group Φ(A_{K′})[p]/Φ(A_{K})[p] has exponent e_{v}(K′/K); this follows from [Gro], Th. 11.5, together with formula (10.3.5). Thus p|e_{v}(K′/K), as asserted.

Remark. In the above implication (iii) ⇒ (i) we needed several deep results from Grothendieck’s article [Gro]. However, all these can be avoided if in condition (iii) we can assume in addition that K(A[p])/K is tamely ramified for the primes p in question. (This is always the case if |GL_{2d}(p)| is prime to the residue characteristic char(K), but for the above application we need to assume only that (char(K), N) = 1 since we already know that the ramification degree divides a power of N.) In that case the implication (iii) ⇒ (i) can be proved as follows: Again, it is enough to prove this in the case that K is henselian with separably closed residue field. If L denotes the compositum of the K(A[p])’s for the primes in question, then the hypothesis (and tame ramification) implies that L is a finite (cyclic) extension of K. By construction, L(A[p]) = L for infinitely many p’s, so by the Néron-Ogg-Shafarevich criterion for good reduction (cf. [ST], Theorem 1), A has good reduction over L and hence potentially good reduction over K.

Let us now come back to Theorem 5.7. By imposing further restrictions on the m_{i}’s, we can conclude that all of J_{C} has potentially good reduction:

Corollary 5.9 Suppose that C/K is as above and satifies in addition the condition that m_{i} ≢ −m_{n+2} mod d, for every d|N with e_{n+2} |̸ d ≠ 1, where e_{n+2} = \frac{N}{(m_{n+2},N)}. Then J_{C} has potentially good reduction over K.

Proof. Induct on the number of the divisors of N. If N is prime, then J_{C} = J_{C}^{new} and so the assertion follows from the theorem. If N is composite, then by the theorem it is enough to show that J_{C}^{old} has potentially good reduction or, equivalently, that every J_{d} := J_{C/H_{d}} has potentially good reduction (for d|N, d ≠ 1). This is clear if e := e_{n+2}|d, for then C_{d} := C_{H_{d}} is unramified at t_{n+2} and hence the covering is already defined over K. Thus, assume e |̸ d. Then the ramification of C_{d} → ℙ^{1} satisfies the same hypotheses as C → ℙ^{1}, and so by the induction hypothesis we have that J_{C_{d}} has good reduction.

5.2 Rational points on K(ℙ(S_{i}))

One of the advantages of the above geometric description of the extensions generated by Thompson tuples is that it is possible to determine whether or not the extension field has a rational point. This is based on the following (well-known) fact.

Proposition 5.10 Let A be an abelian variety over K which has potentially good reduction with respect to a discrete valuation v on K. Suppose that v is unramified in K′ = K(A[m]), where m ≥ 3 is an integer which is relatively prime to the characteristic of the residue field k = κ(v) of v. Then:

(i) the reduction A_{v} of A at v is an abelian variety over k;

(ii) if v′ denotes any extension of v to K′, then its residue field is κ(v′) = k(A_{v}[m]).

Proof. The first assertion follows from [ST], Theorem 2, Corollary 2(b). To prove the second, let k′ = κ(v′). Then the reduction map r_{v′} : A(K′) → A_{v′}(k′) induces an isomorphism A[m](K′) ≃ A_{v′}[m](k′), so all m-torsion points of A_{v} are rational over k′. Thus k(A_{v}[m]) ⊂ k′. To prove that equality holds, let σ ∈ Gal(k′/k(A_{v}[m])). Then σ lifts uniquely to an automorphism σ̃ ∈ D_{v′}(K′/K), where D_{v′}(K′/K) ≤ Gal(K′/K) denotes the decomposition group of v′. Since the action of σ̃ on A[m] is (via r_{v′}) the same as that of σ on A_{v′}[m] = A_{v}[m], this means that σ̃ acts trivially on A[m]. But then σ̃ and hence σ are both trivial, so k′ = k(A_{v}[m]).

Corollary 5.11 In the above situation, suppose that S ⊂ A[m] is a G_{K}- invariant subspace. Let S̄ denote the image of S in A_{v′}[m]. Then κ(v′_{|K(S)}) = k(S̄) and κ(v′_{|K(ℙ(S))}) = k(ℙ(S̄)). In particular, if m = p is a prime, then v splits completely in K(ℙ(S)) if and only if every k̄-isogeny of A_{v} with kernel in S̄ is k-rational.

Proof. By the same argument as in the proof of the proposition, we see that Gal(k′/k(S̄)) ≃ D_{v′}(K′/K) ∩ Gal(K′/K(S)) = D_{v′}(K′/K(S)), and so the first assertion follows. To prove the second, we note that the group Gal(K(S)/K(ℙ(S))) consists of those σ ∈ Gal(K(S)/K) which act diagonally on S. Since Gal(k(S̄)/k(ℙ(S̄))) is defined similarly, we see that Gal(k(S̄)/k(ℙ(S̄))) ≃ D_{v′_{|K(S)}}(K(S)/K(ℙ(S))), and so the second assertion follows.

To prove the last assertion, we observe that by the above result v splits completely in K(ℙ(S)) ⟺ k(ℙ(S̄)) = k ⟺ every σ ∈ Gal(k(S̄)/k) acts diagonally on S̄ ⟺ σ(𝔽_{p}s̄) = 𝔽_{p}s̄, for all s̄ ∈ S̄ and σ ∈ Gal(k(S̄)/k) ⟺ every isogeny φ : A_{v} → A′ with kernel Ker(φ) = 𝔽_{p}s̄ (where s̄ ∈ S̄−{0}) is k-rational ⟺ every k̄-isogeny of A_{v} with kernel in S̄ is k-rational.

The main difficulty in applying the above criterion to our situation is the fact that we need to guarantee that v is unramified in K(A[p]) for some p. Although we already know by Theorem 5.5 the ramification behaviour of the extension K(ℙ(S_{i}))/K, there remains the problem of understanding that of K(S_{i})/K(ℙ(S_{i})). This will be analyzed next by using the following criterion.

Proposition 5.12 Let A/K be an abelian variety and let v be a discrete valuation of K. Moreover, let I = I_{v′} denote the inertia group of an extension v′ of v to K′ = K(A[p]), where p ≠ char(κ(v)) is a prime. If S ⊂ A[p] is a non-zero G_{K}-invariant subspace and H = Gal(K′/K(ℙ(S))), then v′_{|K(S)} is unramified over K(ℙ(S)) if and only if S^{I∩H} ≠ {0}. In particular, if S^{I} ≠ {0}, then v′_{|K(S)} is unramified over K(ℙ(S)).

Proof. Let H_{0} = Gal(K′/K(S)) ≤ H. Then v′_{|K(S)} is unramified over K(ℙ(S)) ⟺ I ∩ H ≤ H_{0} ⟺ I ∩ H acts trivially on S ⟺ S^{I∩H} = S. Thus, if v′_{|K(S)} is unramified, then clearly S^{I∩H} = S ≠ {0}. Conversely, if S_{0} := S^{I∩H} ≠ {0} and g ∈ I ∩ H, then g acts diagonally on S and trivially on S_{0}, so g acts trivially on all of S. Thus I ∩ H ≤ H_{0}, and so v′_{|K(S)} is unramified.

Corollary 5.13 Suppose in addition that S = A[p]_{λ} where λ : Z → 𝔽_{p}^{∗} is a character on a finite subgroup Z ≤ Aut(A). If either A[p]_{λ}^{I} ≠ {0} or if T_{p}(A)Ĩ ; λ̃ ≠ {0}, where λ̃ : Z → ℤ_{p}^{∗} is a lift of λ and Ĩ ≤ G_{K} denotes the absolute inertia group of an extention of v (and of v′) to K^{sep}, then v′_{|K(S)} is unramified over K(ℙ(S)). In particular, if the reduction A_{v} of the Néron model of A/K at v is not unipotent (i.e. the connected component of A_{v} is not an extension of additive groups), then v′ is unramified over K(ℙ(A[p])).

Proof. If the second hypothesis holds, then also V := T_{p}(A)Ĩ ; λ̃ ⊗ 𝔽_{p} ≠ {0}. But then {0} ≠ V ⊂ (T_{p}(A)λ̃ ⊗ 𝔽_{p})Ĩ = A[p]_{λ}^{I}, which means that the second hypothesis implies the first. Now if the first hypothesis holds, then S^{I∩H} ⊃ S^{I} ≠ {0}, and so the conclusion follows directly from the proposition. In particular, taking for λ̃ the trivial character (on Z = {1}), then by [Gro], Prop. 2.2.5 (and (2.1.11)) we have rank(T_{p}(A)Ĩ) = n − λ(A), where n = dim(A) and λ(A) denotes the unipotent rank of A_{v}. Thus, T_{p}(A)Ĩ = {0} ⟺ A_{v} is unipotent, and so the assertion follows from the previous criterion.

The above criterion can be used in our situation in the following way.

Theorem 5.14 In the situation and notation of Theorem 5.5, let v_{i} denote the place of K/k corresponding to the specialization t → t_{i}, for 1 ≤ i ≤ n+1. Then:

(a) If v is any place of K/k with v ∉ R := {v_{1}, …, v_{n+1}}, then C and hence J_{C} and A = J_{C}^{new} have good reduction at v.

(b) For each i with 1 ≤ i ≤ n + 1, let C̄_{i} denote the normalization of the curve

y^{N} = c(x − t_{1})^{m_{1}} ⋯ (x − t_{n+1})^{m_{n+1}} · (x − t_{i})^{m_{n+2}},

and let π̄_{i} : C̄_{i} → ℙ_{k}^{1} denote the associated cyclic covering of degree N. Then C̄_{i} is the normalization of the reduction of C at v_{i}, and the new part J new ; C̄_{i} of its Jacobian has dimension \frac{1}{2}ϕ(N)(n − 1).

(c) Let J_{v_{i}} and A_{v_{i}} denote the reductions of the Néron models of J_{C} and A = J_{C}^{new} at v_{i}, and let J_{v_{i}}^{0} and A_{v_{i}}^{0} denote their respective connected components of the identity. Then there are natural surjections of algebraic groups f_{i} :

J_{v_{i}}^{0} → JC̄_{i} and f_{i}^{new} : A_{v_{i}}^{0} → J new ; C̄_{i}.

(d) Let Ĩ = Ĩ_{v_{i}} ≤ G_{K} denote the inertia group of an extension of v_{i} to K^{sep}. Then for any prime p ≡ 1 mod N (with p ≠ char(K)) and any primitive character λ̃ : Z → ℤ_{p}^{∗}, we have that rank(T_{p}(J_{C})Ĩ ; λ̃) ≥ n − 1.

(e) For p as above, let λ : Z → 𝔽_{p}^{∗} be a primitive character. Then for any j with (j, N) = 1, the extension L_{j} = K(S_{j}) is everywhere unramified over M_{j} = K(ℙ(S_{j})), and hence K′ = K(A[p]) is ramified over K only at the places of R with ramification index dividing N and is unramified over M = ∏ M_{j}.

Proof. Recall that C is by definition the normalization of the projective plane curve C′ ⊂ ℙ_{K}^{2} defined by equation (4). Let C′ denote the closure of C in ℙ_{ℙ^{1}}^{2} = ℙ^{2} × ℙ^{1}, and let ν : C → C′ denote its normalization (in κ(C′) = κ(C)).

Then, if v is a place of K/k with v ≠ v_{∞}, the place at infinity, then the fibre C′_{v} of C′ is the projective plane curve over κ(v) defined by the equation

y^{N} = c(x − t_{1})^{m_{1}} ⋯ (x − t_{n+1})^{m_{n+1}} · (x − t̄)^{m_{n+2}},

where t̄ ∈ κ(v) ⊃ K denotes the image of t in the residue field of v. Since this curve is reduced and geometrically irreducible, it follows that the same is true for the fibres C_{v} of C, and so the normalization ν̃_{v} : C̃_{v} → C′_{v} of C′_{v} factors over the finite map ν_{v} : C_{v} → C′_{v}. Note that by considering a different affine model, the above argument extends to show that the last assertions are also true for v = v_{∞}.

(a) If v ≠ v_{∞}, then the hypothesis on v means that t̄≠ t_{1}, t_{2}, … t_{n+1}. Thus, by Riemann-Hurwitz, the genus of C̃_{v} is the same as that of C, and so C has good reduction at v. Then J_{C} also has good reduction (cf. [BLR], 9.4/4) and hence so does any quotient A of J_{C} (cf. [BLR], 7.5/3). Since the argument for v = v_{∞} is similar, this proves assertion (a).

(b) Now suppose that v = v_{i}, so t̄ = t_{i} (and κ(v) = k). Then C′_{v} is the projective plane curve defined by the given equation and so by what was said above, C̄_{i} = C̃_{v}, which proves the first assertion. The second assertion follows directly from Corollary 5.4, for π̄_{i} : C̄_{i} → ℙ_{k}^{1} is a cyclic covering of degree N which is ramified at precisely r = n + 1 places.

(c) Let δ : C̃ → C denote a desingularization of C (which exists since the base ℙ_{K}^{1} is clearly excellent). Then, since C is normal, C̄_{i} is the normalization of a component of the fibre C̃_{v} of C̃ at v = v_{i}. Thus, we have a natural surjection Pic_{C̃_{v}/k}^{0} → Pic_{C̄_{i}/k}^{0} (cf. [BLR], 9.2/13). On the other hand, by [BLR], Th. 9.5/4, Pic_{C̃_{v}/k}^{0} is canonically isomorphic to the connected component J_{i}^{0} of the identity of the reduction J_{i} = J_{v_{i}} of the Néron model of J_{C} at v = v_{i}, and so we have a canonical surjection f_{i} : J_{i}^{0} → JC̄_{i}. Next we observe that the covering automorphisms σ ∈ Aut(π) map isomorphically onto those of π̄_{i} : C̄_{i} → ℙ_{k}^{1}, and so these extend to automorphisms of J_{i}^{0} and of JC̄_{i} in such a way that f_{i} becomes equivariant. From this it follows that if ε̄_{i} : JC̄_{i} → J new ; C̄_{i} denotes the projection map onto the new part of JC̄_{i} (cf. Corollary 5.3), then the composition ε̄_{i} ∘ f_{i} factors over the map ε̃_{new,i} : J_{i}^{0} → A_{v_{i}}^{0} which is induced by the universal property of Néron models; i.e. there is a homomorphism f_{i}^{new} : A_{v_{i}}^{0} → J new ; C̄_{i} such that ε̄_{i}∘f_{i} = f_{i}^{new}∘ε̃_{new,i}. Thus, since f_{i} and ε̄_{i} are surjective, so is f_{i}^{new}, which proves the assertion.

(d) By Grothendieck [Gro], Prop. 2.2.5 (and (2.2.3.3)), we have a natural identification T_{p}(A_{i}^{0}) ≃ T_{p}(A)Ĩ (after passing to the henselization of v_{i}). Since the action of Z commutes with the Galois action, this induces an isomorphism T_{p}(A_{i}^{0})λ̃ ≃ T_{p}(A)Ĩ ; λ̃. Now by (c), the map f_{i}^{new} is surjective, hence so is the induced map on the Tate modules, and therefore T_{p}(A_{i}^{0})λ̃ → T_{p}(JC̄_{i})λ̃ is surjective as well. Thus, dim T_{p}(A)λ̃ ≥ dim T_{p}(JC̄_{i})λ̃ = n − 1, the latter by formula (3) of Corollary 5.4.

(e) Let v be a place of K/k. If v ≠ v_{i}, then A has good reduction at v, so K(A[p]) is unramified over K at v, hence a fortiori so is K(S_{j}) over K(ℙ(S_{j})). On the other hand, if v = v_{i}, then in in view of (d) we can apply the criterion of Corollary 5.13 to see that v′_{|K(S_{j})} is unramified over K(ℙ(S_{j})) for every extension v′ of v. Thus, L_{j} = K(S_{j}) is everywhere unramified over M_{j} = K(ℙ(S_{j})) and hence K(A[p]) = ∏ L_{j} is unramified over M = ∏ M_{j} as well. On the other hand, it follows from Theorem 5.5 that each M_{j} and hence M is unramified outside R and ramified at each v_{i} ∈ R of order dividing N, so the same is true for K′.

By using the above proposition, we can now make the assertion of Theorem 5.7 much more precise.

Theorem 5.15 Let K_{0} be a finite Galois extension of K = k(t) such that the ramification index e_{i} = e_{v_{i}}(K_{0}/K) at v_{i} is divisible by N, for 1 ≤ i ≤ n + 1. Then A = J_{C}^{new} has good reduction everywhere over K_{0}, and hence K_{0}(A[m]) is unramified over K_{0} for every integer m relatively prime to char(K).

Proof. If p ≡ 1 mod N is any prime (p ≠ char(K)), then by Abhyankar’s Lemma it follows from Theorem 5.14(e) that K_{0}(A[p]) = K(A[p]) · K_{0} is unramified over K_{0}. By the criterion of Néron-Ogg-Shafarevich, this means that A_{K_{0}} = A ⊗ K_{0} has good reduction everywhere, and hence K_{0}(A[m]) is unramified over K_{0} for all m which are prime to char(K).

Thus, by above theorem we can now apply Corollary 5.11 to obtain following criterion for the existence of rational points:

Corollary 5.16 Suppose in addition that v = v_{i_{0}} is totally ramified in K_{0}, and let Aṽ denote the reduction of A ⊗ K_{0} at ṽ, where ṽ denotes the (unique) extension of v to K_{0}. If p ≠ char(K) is any prime and S ⊂ A[p] is a G_{K}-invariant subspace such that

(⋆⋆) every isogeny of the reduction Aṽ with kernel contained in S̄ (= the image of S in A_{v}) is rational over K,

then every extension of ṽ to M_{S} := K_{0}(ℙ(S)) is a K-rational point of M_{S}.

5.3 Example

We now return to Example 4.11, but change the notation slightly to conform with that of this section. Thus, let k be a field containing a primitive fourth root of unity i (so in particular char(k) ≠ 2). As before, take n = 3 and (in the notation of Theorem 5.5) N = 4, m_{1} = … = m_{3} = 1, m_{4} = 3 and m_{5} = 2. Moreover, let the ramification points be t_{1} = 0, t_{2} = 1, t_{3} = −1, t_{4} = a, t_{5} = t where a ∈ k \ {0, ±1} and t is transcendental over k, and put K = k(t). Then a corresponding cyclic cover π : C → ℙ_{K}^{1} is given by the equation

Y^{4} = cX(X − 1)(X + 1)(X − a)^{3}(X − t)^{2} with c ∈ k^{∗}.  (5)

Clearly, C has genus 4 and deg(π) = 4. Moreover, π has a subcover of degree 2 whose quotient curve is an elliptic curve E given by the equation

Z^{2} = cX(X − 1)(X + 1)(X − a) with Z = Y^{2}/((X − a)(X − t)). (6)

Note that E is a constant curve, i.e. E is already defined over k. Moreover, E maps injectively into the Jacobian J_{C} of C and can be identified with the old part J_{C}^{old} of J_{C} (cf. section 5.1). Its complement in J_{C} is therefore the new part A = J_{C}^{new}; thus J_{C} ∼ E × A and so dim A = 3.

Next we choose a cyclic extension K_{0} of K of degree 4 which is totally ramified at each of the points of R = {0, 1, −1, a}. For example, we can take K_{0} = k(t, s) where

s^{4} = t(t^{2} − 1)(t − a)g(t),  (7)

and g(t) ≠ 0 is any nonzero polynomial of which does not vanish at R. Thus, if T_{0} denotes the smooth curve over k defined by this equation (whose function field is K_{0}), then by Theorem 5.15 we obtain

Proposition 5.17 The abelian variety A_{K_{0}} = A ⊗ K_{0} extends to an abelian scheme A over the curve T_{0}.

Now we want to verify that the tower of unramified covers K_{0}(ℙ(S_{p})) of K_{0} attached to the subspaces S_{p} ⊂ A[p] for p ≡ 1 mod 4 has rational points which all lie over the same base-point ṽ of K_{0}. For this we shall use the criterion (⋆⋆) of Corollary 5.16 applied to the place v = v_{4} corresponding to the specialization of t to t_{4} = a. However, in order to do this, we need to determine the structure of the reduction A_{0} := Aṽ of A_{K_{0}} at the unique extension ṽ of v to K_{0}. (Recall that by the above proposition we know that A_{0} is an abelian variety over k.) As a first step towards this end, we shall prove:

Proposition 5.18 Let Jṽ denote the reduction of the Jacobian J = J_{C} ⊗ K_{0} at ṽ. Then

Jṽ ≃ J_{0} × E_{d},

where J_{0} is the Jacobian of the curve C_{0} which is the normalization of the curve defined by the equation

Ȳ^{4} = cX̄(X̄ − 1)(X̄ + 1)(X̄ − a)

and E_{d} denotes the elliptic curve given by the equation

Y^{2} = X(X^{2} − d),  (8)

where d = c/g(a) and g is as in (7).

Proof. First note that since J_{C} ∼ E×A, and E is a constant curve, it follows from Proposition 5.17 that J has good reduction over T_{0} and in particular at ṽ. Thus, if C denotes the minimal model of C at ṽ, then C has semi-stable reduction (use [BLR], 9.5/4, 9.2/5 and 9.2/12). Thus, if C_{1}, …, C_{r} denote the irreducible components of the reduction Cṽ of C, then each C_{i} is smooth (by [BLR], 9.2/12) and we have (by [BLR], 9.5/4 and 9.2/8)

Jṽ = J_{C_{1}} × … × J_{C_{r}}.

Thus, the assertion follows once we have shown that C_{0} and E_{d} occur as components of Cṽ, for then all other components must have genus 0 (since g(C) = 4, g(C_{0}) = 3 and g(E_{d}) = 1). For this it is enough to show: Claim: There are normal models C′ and C′′ of C over 𝔒ṽ whose reductions C′ ; ṽ and C′′ ; ṽ each have an irreducible component C′_{ṽ,1} and C′′_{ṽ,1} with function field κ(C′_{ṽ,1}) ≃ κ(C_{0}) and κ(C′′_{ṽ,1}) ≃ κ(E_{d}). Indeed, if such a model C′ exists, then its desingularization and hence the minimal model C have the same property (because g(C_{0}) > 0). But since all components of C are smooth, there is a component C_{1} of C with C_{1} ≃ C_{0}, and similarly, if such a model C′′ exists, then C has a component C_{2} ≃ E_{d}. Proof of claim: a) The construction of C′: Let C̃ denote the normal model constructed in the proof of Theorem 5.14. Its fibre C̃_{v} at v is integral and has C_{0} as its normalization (because Y^{4} = cX(X^{2} − 1)(X − a)^{5} also has normalization C_{0}). Thus, if we let C′ denote (the normalization of) C̃ ⊗ 𝔒ṽ, then its fibre C′ ; ṽ has normalization C_{0}, as desired. b) The construction of C′′: Let F_{0} = κ(C ⊗ K_{0}) denote the function field of C ⊗ K_{0}; thus F_{0} = K_{0}(X, Y) = k(t, s, X, Y), where X and Y are related by equation (5) and t and s by equation (7). Put x = (X − a)/s^{4} and y = Y/s^{5}. Then F_{0} = K_{0}(x, y) and equation (5) becomes

y^{4} = x^{3}(x − B)^{2}g_{1}(x),

where B = (t − a)/s^{4} = (t(t^{2} − 1)g(t))^{−1} ∈ K = k(t) and g_{1}(x) = cX(X^{2} − 1) = c(s^{4}x + a)((s^{4}x + a)^{2} − 1). Let 𝔄 denote the integral closure of 𝔅 = 𝔒ṽ[x] in F_{0}, and let C′′ = Spec(𝔄). Fix an irreducible component of the reduction of C′′, and let V denote the associated (normalized) valution of F_{0}. We then have:

V (∑a_{i}x^{i}) = e min(ṽ(a_{i})), if a_{i} ∈ K_{0},

for some integer e ≥ 1. Since by construction ṽ(s) = 1, ṽ(t − a) = 4 and ṽ(g(t)) = 0, we see that V (x − B) = V (g_{1}(x)) = 0, and so also V (y) = 0. Thus, if x̄ and ȳ denote the images of x and y in the residue field κ(V) = 𝔒ṽ/𝔐ṽ of V, then the above relation specializes to

ȳ^{4} = a_{0}x̄^{3}(x̄ − b_{0})^{2},

where a_{0} = \overline{g_{1}(x)} = ca(a^{2} − 1) and b_{0} = B̄ = (a(a^{2} − 1)g(a))^{−1}. Since this equation is irreducible over k(x̄) = κ(V_{|K_{0}(x)}), we see that [k(x̄, ȳ) : k(x̄)] = 4. Thus, since [κ(V) : κ(V_{|K_{0}(x)})] ≤ [F_{0} : K_{0}(x)] = 4, it follows that κ(V) = k(x̄, ȳ). (It also follows that e = 1 and that V is the unique valuation above V_{|K_{0}(x)}, i.e. that the fibre of C′′ at ṽ is integral, but we don’t need this.) It remains to show that k(x̄, ȳ) = κ(E_{d}). For this, put u = ȳ^{2}/(x̄(x̄ − b_{0})) and v = a_{0}ȳ/u. Then by the above relation we have u^{2} = a_{0}x̄, so k(x̄, ȳ) = k(u, v). Moreover, v^{2} = a_{0}ȳ^{2}/x̄ = u(u^{2} − a_{0}b_{0}), so κ(V) = κ(E_{d}), with d = a_{0}b_{0} = c/g(a), as desired.

Remark 5.19 It follows from the above proof that reduction Cṽ of the minimial model C has the form Cṽ = C_{1} ∪ … ∪ C_{r}, where C_{1} ≃ C_{0}, C_{r} ≃ E_{d}, and C_{i} ≃ ℙ^{1}, for 2 ≤ i ≤ r − 1, and each C_{i} meets C_{i−1} transversally in a unique point for 2 ≤ i ≤ r.

Next we want to determine the abelian variety J_{0} up to k-isogeny. To simplify matters, we shall assume in the following that 1 − a^{2} is a square in k or, equivalently, that a = \frac{2b}{1+b^{2}} for some b ∈ k, where b ≠ 0, ±1. (In fact, b = (1 + \sqrt{(1 − a^{2})})/a.) In addition, we shall choose c = (1 + b^{2}).

Lemma 5.20 Let C_{0} be the curve defined over k by the equation

Y^{4} = cX(X − 1)(X + 1)(X − a), where a = \frac{2b}{1 + b^{2}} and c = 1 + b^{2},

for some b ∈ k\{0, ±1}. Then its Jacobian J_{0} is up to an isogeny of 2−power degree isomorphic to E × E_{1} × E_{b^{2}}, where E is given by equation (6) and E_{1} and E_{b^{2}} by equation (8) by putting d = 1 and d = b^{2}, respectively. In particular, if b = b_{1}^{2} is a square in k, then J_{0} is k-isogenous to E ×E_{1}×E_{1}.

Proof. Let F = κ(C_{0}) = k(X, Y) denote the function field of C_{0}. Then F′ := k(X, Y^{2}) = κ(E) is the function field of E. We shall show that there are two subfields F_{1} ≃ κ(E_{1}) and F_{2} ≃ κ(E_{b^{2}}) of F which, for suitable U, V ∈ F, fit into the following field diagram of quadratic extensions:

F
⧸ | ⧹
F_{1} F_{2} F′
⧹ | ⧸  ⧹
k(V)  k(X)
⧹  ⧸
k(U)

From this the assertion follows because the fact that F/k(V) is a ℤ/2ℤ × ℤ/2ℤ-extension implies that J_{0} = J_{F} ∼ J_{F′} ×J_{F_{1}} ×J_{F_{2}} = E ×E_{1} ×E_{b^{2}} where the indicated isogeny has 2-power degree; cf. [KR]. To find these subfields, put U = cX(X − a)/(X^{2} − 1), so [k(X) : k(U)] = 2. Then

(U − 1)(X^{2} − 1) = cX(X − a) − (X^{2} − 1) = (1 − bX)^{2}

because c − 1 = b^{2} and ca = 2b by hypothesis. Thus, if we put Z_{1} = Y (1 − bX)/(X^{2} − 1), then

Z_{1}^{4} = U(U − 1)^{2},

and so, if we put F_{1} := k(U, Z_{1}), then [F_{1} : k(U)] = 4 and F_{1}(X) = F. In particular, since [F : k(X)] = 4, we see that F_{1} ∩ k(X) = k(U) and that [F : F_{1}] = 2.

Next, let V = Z_{1}^{2}/(U − 1). Then by the above equation V^{2} = U, so Z_{1}^{2} = V (U − 1) = V (V^{2} − 1). From this we see that k(V) = k(U, Z_{1}^{2}), so [F_{1} : k(V)] = 2, and hence F_{1} = k(V, Z_{1}) ≃ κ(E_{1}). In addition, it follows that k(V, X) = F′ because F′ is the unique proper intermediate field of F/k(V). In addition, let us observe that

(U − b^{2})(X^{2} − 1) = cX(X − a) − b^{2}(X^{2} − 1) = (X − b)^{2},

because (c − b^{2}) = 1 and ca = 2b by hypothesis. Therefore, if we put X_{1} = (X − b)/(1 − bX) and Z_{2} = Z_{1}X_{1}, then X_{1}^{2} = (U − b^{2})/(U − 1) and Z_{2}^{2} = V (U − 1) · (U − b^{2})/(U − 1) = V (V^{2} − b^{2}). From this we obtain for F_{2} := k(V, Z_{2}) that [F_{2} : k(V)] = 2 and that F_{2} ≃ κ(E_{b^{2}}). We therefore see that F_{1}, F_{2}, F′, k(V) and k(U) fit into the above field diagram as indicated, and that F_{1} ≃ κ(E_{1}) and F_{2} ≃ κ(E_{b^{2}}).

We can now determine the structure of the reduction A_{0} = Aṽ of the abelian variety A = J_{C}^{new} at ṽ.

Proposition 5.21 Assume again that a = \frac{2b}{1+b^{2}} and c = 1 + b^{2} for some b ∈ k^{∗}, and let d = c/g(a). Then:

(a) There is a k-isogeny φ : A_{0} → A′_{0} := E_{1}×E_{b^{2}} ×E_{d} of 2-power degree, and hence A_{0} is k′-isogenous to E_{1}×E_{1}×E_{1}, where k′ = k(\sqrt{b}, ^{4} \sqrt{d}). Moreover, if σ_{0} ∈ Aut(A_{0}) denotes the automorphism induced by a generator σ ∈ Aut(π) of the covering group, then we have φ ∘ σ_{0} = σ′_{0} ∘ φ for some σ′_{0} ∈ Aut(A′_{0}).

(b) Let S̄_{p} ≤ A_{0}[p] be an eigenspace of σ_{0}, where p ≡ 1 mod 4 is a prime. Then every cyclic k̄-isogeny of A′_{0} with kernel contained in φ(S̄_{p}) is an endomorphism of A′_{0}. In particular, if k′ = k, then every k̄-isogeny of A_{0} with kernel contained in S̄_{p} is defined over k.

Proof. (a) Since A = J_{C}^{new} and E = J_{C}^{old}, we have an isogeny φ_{1} : E × A → J = J_{C} of 2-power degree (cf. Corollary 5.3) whose reduction \overline{φ}_{1} : E × A_{0} → Jṽ is a k-isogeny of the same degree. Moreover, by Proposition 5.18 we have an isomorphism φ_{2} : Jṽ →^{∼} J_{0} × E_{d}, and by Lemma 5.20 we have a k-isogeny φ_{3} : J_{0} → E × E_{1} × E_{b^{2}} of 2-power degree. We thus see that φ̃ = (φ_{3} × id_{E_{d}}) ∘ φ_{2} ∘ \overline{φ}_{1} : E × A_{0} → E × E_{1} × E_{b^{2}} × E_{d} = E × A′_{0} is an isogeny of 2−power degree, and so the first assertion follows (by taking φ = φ̃_{|{0}×A_{0}}) once we have shown that φ̃({0} × A_{0}) = {0} × A′_{0}.

To prove this, it is enough to find automorphisms σ_{0} ∈ Aut(A_{0}) and σ′_{0} ∈ Aut(A′_{0}) with σ_{0}^{2} = [−1]_{A_{0}} and (σ′_{0})^{2} = [−1]_{A′_{0}} such that

φ̃ ∘ (σ̃_{0}) = (σ̃′_{0}) ∘ φ̃,

where σ̃_{0} = [−1]_{E} × σ_{0} and σ̃′_{0} = [−1]_{E} × σ′_{0}. Indeed, since 1+σ_{0} is an isogeny of A_{0} (because (1+σ_{0})(1−σ_{0}) = 1−[−1]_{A_{0}} = [2]_{A_{0}}), we see that (1 + ([−1]_{E} × σ_{0}))(E × A_{0}) = {0} × A_{0}, and similarly (1+σ̃′_{0})(E ×A′_{0}) = {0}×A′_{0}, and hence φ̃({0}×A_{0}) = φ̃((1+σ̃_{0})(E ×A_{0})) = (1 + σ̃′_{0})φ̃(E × A_{0}) = (1 + σ̃′_{0})(E × A′_{0}) = {0} × A′_{0}, as desired. To construct σ_{0} and σ′_{0} with these properties, let σ_{J} denote the automorphism (of order 4) on the Jacobian J induced by σ ∈ Aut(π). Then σ_{J} maps the subvarieties E and A of J_{C} into themselves, and (σ_{J})_{|E} = [−1]_{E}. Thus, if σ_{A} = (σ_{J})_{|A}, then we have φ_{1} ∘ ([−1] × σ_{A}) = σ_{J} ∘ φ_{1}. Thus, if σ̄ (resp. σ_{0}) denotes the automorphism induced by σ_{J} (resp. σ_{A}) on the reduction Jṽ (resp. on A_{0}), then we also have \overline{φ}_{1} ∘ ([−1]_{E} × σ_{0}) = σ̄ ∘ \overline{φ_{1}}. Next we note that the proof of Proposition 5.18 shows that σ induces automorphisms σ_{d} and σ_{C_{0}} on E_{d} and on C_{0} respectively, from which we see that φ_{2} ∘ σ̄ = (σ_{J_{0}} × σ_{d}) ∘ φ_{2}. Similarly, the proof of Lemma 5.20 shows that φ_{3} ∘ σ_{J_{0}} = ([−1]_{E} × σ_{1} × σ_{b^{2}}) ∘ φ_{3}, for certain automorphisms σ_{i} ∈ Aut(E_{i}) where i = 1, b^{2}. We thus see that if we put σ′_{0} = σ_{1} × σ_{b^{2}} × σ_{d} ∈ Aut(A′_{0}), then the above commutation relation holds. It remains to show that σ_{0}^{2} = [−1]_{A_{0}} and that (σ′_{0})^{2} = [−1]_{A′_{0}}. The latter is clear, for by construction σ′_{0} = σ_{1} × σ_{b^{2}} × σ_{d}, where (for i = 1, b^{2}, d) the automorphism σ_{i} ∈ Aut(E_{i}) has order 4 and so σ_{i}^{2} = [−1]_{E_{i}} because char(k) ≠ 2, 3. To prove the former, recall that A = ε̃_{new}J, where ε̃_{new} = 1− σ_{J}^{2}, so σ_{J}^{2} acts on A like [−1]_{A}. Thus, σ_{A}^{2} = [−1]_{A} and hence by functoriality we have σ_{0}^{2} = [−1]_{A_{0}}, as desired.

(b) Write E_{2} = E_{b^{2}} and E_{3} = E_{d}, so A′_{0} = E_{1} × E_{2} × E_{3}. Let S̄′_{p} = φ(S̄_{p}) ≤ A′_{0}[p], which is an eigenspace under the action of σ′_{0} = σ_{1} × σ_{2} × σ_{3} . Since ^{4} \sqrt{−1} ∈ k and j(E_{i}) = 1728, we have for i = 1, 2, 3 that End(E_{i}) ⊃ ℤ[ ^{4} \sqrt{−1}], and equality holds unless E_{i} is supersingular (i.e. unless char(k) ≡ 3 mod 4). Thus, since p ≡ 1 mod 4, there is an α_{i} ∈ End(E_{i}) of degree p such that Ker(α_{i}) = S̄′_{p} ∩ e_{i}(E_{i}), where e_{i} : E_{i} → A′_{0} is the embedding of E_{i} as the i-th factor of A′_{0}. Thus, S̄′_{p} = Ker(α_{1}) × Ker(α_{2}) × Ker(α_{3}). Now since E_{2} and E_{3} are twists of E_{1}, there are k′-isomorphisms f_{1i} : E_{1} → E_{i}, for i = 2, 3, which we can choose such that f_{1i} ∘ α_{1} = α_{i} ∘ f_{1i}. Thus, if P̃_{1} ∈ E_{1}(k̄) is a generator of Ker(α_{1}), and P̃_{i} = f_{1i}(P_{1}) then {P_{1}, P_{2}, P_{3}} is a basis of S̄′_{p}, where P_{i} := e_{i}(P̃_{i}).

Now let h be a cyclic k̄-isogeny of A′_{0} with Ker(h) ≤ S̄′_{p}. Then Ker(h) is generated by P = aP_{1} + bP_{2} + cP_{3}, for some a, b, c ∈ 𝔽_{p}. W.l.o.g. we may assume that a ≠ 0 (otherwise we interchange the roles of E_{1}, E_{2} and E_{3}), and hence that a = 1. Then the k′-endomorphism h′ : A′_{0} → A′_{0}, defined by the

matrix ( \begin{matrix}(1 − b)α_{1} & α_{1}f_{12}^{−1} & 0 \\ −bf_{12} & id_{E_{2}} & 0 \\ −cf_{13} & 0 & id_{E_{3}} \end{matrix} ) has kernel Ker(h′) = ⟨P_{1} +bP_{2} +cP_{3}⟩ =

Ker(h), which proves the first assertion. From this the second assertion is immediate. Indeed, if k′ = k, then by what was just shown, every k̄-subgroup of S̄′_{p} is k-rational, and hence the same is true for S̄_{p}.

We can now summarize the results obtained above in the following theorem.

Theorem 5.22 Suppose k contains a primitive fourth root of unity, and let a = \frac{2b^{2}}{1+b^{4}} , for some b ∈ k^{∗} with b^{4} ≠ 1. Let K = k(t, s) be the function field of any curve of the form

s^{4} = t(t^{2} − 1)(t − a)g(t),

where g(t) ∈ k[t] is any polynomial which does not vanish at {0, 1, −1, a}, and which has been normalized in such a way that (1 + b^{4})/g(a) is a fourth power in k. Moreover, let 𝔭 be the unique place of K/k inducing the specialization t ↦ a. Then the k-rational geometric fundamental group of K with base point 𝔭 is infinite; more precisely, for every prime p ≡ 5 mod 12 (with p ≠ char(k)), there is an unramified extension K_{p}/K with Gal(K_{p}/K) ≃ PSL_{3}(p) in which 𝔭 splits completely.

Proof. It is clearly enough to prove the last statement. For this, let C be the curve defined over k(t) by the equation Y^{2} = (1 + b^{4})X(X^{2} − 1)(X − a)^{3}(X − t)^{2}, and let J denote its Jacobian and A = J^{new} its new part (cf. section 5.1). Since the hypotheses on a, c = 1 + b^{2} and g ensure that k′ := k(\sqrt{b^{2}}, ^{4} \sqrt{c/g(a)}) = k, it follows from Proposition 5.21a) that the reduction A_{0} of A ⊗ K at 𝔭 is k-isogenous to E_{1} × E_{1} × E_{1}. Now let p ≡ 5 mod 12 be any prime (and ≠ char(k)), and let S_{p} ≤ A[p] denote an eigenspace of the automorphism σ_{A} (cf. proof of Proposition 5.21). By Theorem 5.5(b),(d) (and Abhyankar’s Lemma) we know that K_{p} := K(ℙ(S_{p})) is unramified over K and that Gal(K_{p}/K) = PSL_{3}(p). Let S̄_{p} denote the image of S_{p} in A_{0}[p] under the reduction map at 𝔭; clearly, S̄_{p} is identical to the set S̄_{p} as defined in Proposition 5.21(b). Thus, by that proposition, condition (⋆⋆) of Corollary 5.16 holds, and so it follows that 𝔭 splits completely in K_{p}.

Remark 5.23 The simplest examples of fields K satifying the hypothesis of Theorem 5.22 are clearly the fields K = k(s, t), where

s^{4} = (1 + b^{4})t(t^{2} − 1) (t − \frac{2b^{2}}{1+b^{4}} )

and b ∈ k^{∗} is any element with b^{4} ≠ 1; recall that fields of this type were studied in Lemma 5.20. Clearly, all these fields have genus 3. More generally, by choosing g(t) suitably, we can obtain examples of fields of any genus g ≥ 3 which satisfy the hypotheses of Theorem 5.22.

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