Elliptic Subcovers of Hyperelliptic Curves

Ernst Kani

1 Introduction

Let C be a curve over an arbitrary field K. An elliptic subcover is a finite K-morphism f : C → E to an elliptic curve E/K such that its base-change f\overline{K} : C\overline{K} → E\overline{K} to the algebraic closure does not factor over a non-trivial isogeny of E\overline{K}. If f′ : C → E′ is another elliptic subcover, then f′ is said to be equivalent to f if there is an isomorphism φ : E →^{∼} E′ such that f′ = φ ∘ f. Let E(C) denotes the set of all equivalence classes of elliptic subcovers of C.

If C(K) ≠ ∅, then it is well-known that there is a natural bijection between the set E(C) and the set S(J_{C}) of elliptic subgroups of the Jacobian J_{C} of C.

The purpose of this paper is to show that this result is also true when C(K) = ∅, provided that C is a hyperelliptic curve whose genus is even. Thus, we prove in §4:

Theorem 1.1. Let C/K be a hyperelliptic curve of genus g_{C} ≡ 0 (2). Then the rule f ↦ f^{∗}J_{E} induces a bijection

Φ_{C} : E(C) →^{∼} S(J_{C}).

Note that this result does not extend to hyperelliptic curves of odd genus, as counterexamples show; cf. Example 2.5 and Remark 2.4.^{1}

Theorem 1.1 implies:

Corollary 1.2. If C is as above, then every finite morphism f : C → E to an elliptic curve E/K factors over an elliptic subcover f_{1} : C → E_{1}, and f_{1} is uniquely determined by f up to equivalence. In particular, f is an elliptic cover if and only if f does not factor over a non-trivial K-isogeny.

This result can also be reformulated in terms of elliptic subfields of the function field F = κ(C) of C:

Corollary 1.3. If C is as above, and E/K is an elliptic curve, then a finite morphism f : C → E is an elliptic cover if and only if f^{∗}κ(E) is a maximal elliptic subfield of F = κ(C). Thus, the map f ↦ f^{∗}κ(E) induces a bijection between E(C) and the set of M_{1}(F) of a maximal elliptic subfields of F.

We note in passing that throughout we could have replaced “elliptic curve” by “curve of genus 1”. Indeed, the following result is an easy consequence of Theorem 1.1.

Corollary 1.4. If E/K is a curve of genus 1 which admits a finite cover f : C → E by a hyperelliptic curve C/K of even genus, then E(K) ≠ ∅, and so E/K is an elliptic curve.

1This answers a question posed by the referee.

In the case that char(K) ≠ 2 and g_{C} = 2, Theorem 1.1 was already proven in [4]. The proof there uses the key fact that in that case one can “normalize” an elliptic subcover f : C → E, and that normalized subcovers possess a Galois descent. Here in §4 we generalize this concept to hyperelliptic curves of higher genus (and to the case of an arbitrary base field K) and show that a similar result holds when g_{C} is even.

Theorem 1.5. Let C/K be a hyperelliptic curve whose genus is even. If f : C → E is an elliptic subcover, then there exists an x ∈ E(K) such that T_{x} ∘ f is normalized.

By combining this result with the fact that normalized covers posses a Galoisdescent (cf. Corollary 4.4), it is easy to complete the proof of Theorem 1.1.

2 Elliptic subcovers

Let C/K be a (smooth, projective, geometrically connected) curve of genus g ≥ 2 over an arbitrary field K. We review here some basic facts about elliptic subcovers. For technical reasons, however, it is useful consider a slightly more general class of subcovers.

Definition 2.1. A quasi-elliptic subcover of C/K is a finite K-morphism f : C → E to a curve E/K of genus 1 such that its base-change f\overline{K} : C\overline{K} → E\overline{K} to the algebraic closure \overline{K} of K does not factor over an non-trivial isogeny of E\overline{K}. If, in addition, E has a K-rational point, then we call f an elliptic subcover.

As we shall see below, in many cases quasi-elliptic subcovers are automatically elliptic subcovers (cf. Proposition 2.3 and Corollary 1.4). However, there do exist quasi-elliptic subcovers which are not elliptic; cf. Example 2.5 below.

For much of what follows, it is important to note that quasi-elliptic covers can be characterized by the condition that the induced map f^{∗} : J_{E} → J_{C} on the Jacobians is a closed immersion. Here and below, we use the notations and basic facts about Jacobians as in [4]. (See also [9].)

Proposition 2.2. Let f : C → E be a morphism to a curve E/K of genus 1. Then f is a quasi-elliptic subcover if and only f^{∗} : J_{E} → J_{C} is a closed immersion. If this is the case, then f is separable.

Proof. The first assertion follows immediately from [2], Proposition 4.2, because (f\overline{K})^{∗} = (f^{∗})\overline{K} is a closed immersion if and only if f^{∗} is a closed immersion. To prove the second assertion, suppose the contrary, i.e., that f is not separable. Then also f\overline{K} is not separable, and hence the function field F := κ(C\overline{K}) of C\overline{K} is an inseparable extension of F_{0} = f^{∗}κ(E\overline{K}), and so F_{0} ⊂ F^{p}, where p = char(K) ≠ 0; cf. [11], Prop. III.9.2. Thus F_{1} := F_{0}^{1/p} ⊂ F is an elliptic subfield of F, and so f\overline{K} factors as f\overline{K} = f_{1} ∘f_{2}, where f_{1} : E_{1} → E\overline{K} is purely inseparable of degree p. Thus f ∗ ; \overline{K} = f_{2}^{∗} ∘f_{1}^{∗} factors over the (inseparable) isogeny f_{1}^{∗} of degree p, and hence f^{∗} cannot be a closed immersion, contradiction. Thus f is separable.

As in the introduction (or as in [2]), two quasi-elliptic subcovers f_{i} : C → E_{i} are said to be equivalent if there exists an isomorphism φ : E_{1} →^{∼} E_{2} (of genus 1 curves) such that f_{2} = φ ∘ f_{1}. We denote the set of equivalence classes of quasi-elliptic subcovers of C/K by Ẽ(C) = Ẽ(C/K). Clearly, the set E(C) of equivalence classes of elliptic subcovers of C/K is naturally a subset of Ẽ(C). We then have the following fact which generalizes Corollary 4.3 of [2].

Proposition 2.3. The rule f ↦ f^{∗}J_{E} defines an injection

Φ_{C} : Ẽ(C) ↪ S(J_{C})

of Ẽ(C) into the set S(J_{C}) of elliptic subgroups E ≤ J_{C} of J_{C}. Moreover, if C/K has a divisor of degree 1, then every quasi-elliptic cover is an elliptic cover, and Φ_{C} is a bijection.

Remark 2.4. If K is a finite field, then C/K always has has a divisor of degree 1 by a theorem of F. K. Schmidt; cf. [11], Corollary V.1.11.

Moreover, if C/K has an elliptic subcover f : C → E of odd degree, then C/K has a divisor of degree 1 (because f^{∗}(0_{E}) has odd degree and the canonical divisor of C has even degree), and hence it follows from Proposition 2.3 that every quasi-elliptic subcover is elliptic and that Φ_{C} is a bijection.

However, quasi-elliptic covers which are not elliptic do exist for curves over ℚ or ℝ, as the following general Example 2.5 shows. Thus, for these examples the restriction of the map Φ_{C} to E(C) is not surjective. In particular, Theorem 1.1 does not extend to hyperelliptic curves C/ℚ of odd genus.

Example 2.5. Let K ⊂ ℝ be a subfield of ℝ, and consider the curve E/K whose function field is F_{0} := K(x, y), where y^{2} = −(x^{4} +1). Since the double cover φ̄ : E → ℙ^{1} is unramified at the point P_{∞} at infinity and has relative residue degree 2 there (because −1 ∉ (K^{×})^{2}), we see that E/K is a curve of genus 1 with E(K) = ∅.

Next, consider integers n ≥ 2 and d ≥ 1 with gcd(n, d) = 1, and let F̄ = F̄_{n,d} = K(x, z) be the function field with equation z^{n} = ∏_{i=1}^{d}(x − a_{i}), where the a_{i} ∈ K are distinct. The associated curve C̄ = C̄_{n,d} has genus gC̄ = (n − 1)(d − 1) and has a subcover f̄ = f̄_{n,d} : C̄_{n,d} → ℙ^{1} of degree n; cf. [11], Prop. VI.3.1.

Finally, let C = C_{n,d} be the curve with function field F = F̄_{n,d}F_{0}. Since F̄ ∩ F̄_{0} = K(x) (because the points P_{a_{i}} are totally ramified in F̄ and are unramified in F_{0}), we see that C is the normalization of C̄ ×_{ℙ^{1}} E and hence we have morphisms f = f_{n,d} : C → E and φ : C → C̄ of degrees n and 2, respectively, such that f̄∘ φ = φ̄ ∘ f. Note that f : C → E is a quasi-elliptic subcover because the points on E over P_{a_{i}} are totally ramified in F, so f\overline{K} cannot factor over a nontrivial isogeny of E\overline{K}.

Since the cover φ̄\overline{K} ∘ f\overline{K} : C\overline{K} → ℙ1 ; \overline{K} is abelian of degree 2n and has d + 1 points of ℙ^{1} which are ramified of degree n and 4 points which are ramified of degree 2, we see that the genus of C is g_{C} = 1 + (n − 1)(d + 1). In particular, when d = 1, the curve C = C_{n,1} is hyperelliptic (because gC̄ = 0) of odd genus g_{C} = 1 + 2(n − 1), for n = 2, 3, …, so we have a hyperelliptic example for every odd genus g ≥ 3.

Proof of Proposition 2.3. Note first that this map is well-defined because if f_{1} ∼ f_{2} are two equivalent quasi-elliptic subcovers, then f_{2} = φ ∘ f_{1}, for some isomorphism φ : E_{1} →^{∼} E_{2}, and hence f_{2}^{∗}J_{E_{2}} = f_{1}^{∗}φ^{∗}J_{E_{2}} = f_{1}^{∗}J_{E_{1}}, so both define the same elliptic subgroup of J_{C}.

We first consider the case that C/K has a divisor D ∈ Div(C) of degree 1. Now if f : C → E is a quasi-elliptic subcover, then f is in fact an elliptic subcover because f_{∗}D ∈ Div(E) is also a divisor of degree 1, and so f_{∗}D ∼ P_{0} ∈ E(K) by Riemann-Roch. Thus, E is an elliptic curve and f is an elliptic subcover.

We next show that Φ_{C} is bijective by constructing an inverse map Ψ_{C} = Ψ_{C,D}. For this, let j_{D} : C → J_{C} be the closed immersion given by j_{D}(P) = cl(P − D), for P ∈ C(\overline{K}), and let E ≤ J_{C} be an elliptic subgroup with inclusion h_{E} : E ↪ J_{C}. Put

Ψ(h_{E}) := [−1]_{E} ∘ j_{E}^{−1} ∘ ĥ_{E} ∘ λ_{C} ∘ j_{D} : C → E, (1)

where j_{E} = j_{0_{E}} : E →^{∼} J_{E} is the canonical isomorphism and λ_{C} : J_{C} → Ĵ_{C} denotes the canonical polarization of J_{C}. Then we have that

Ψ(h_{E})^{∗} = h_{E} ∘ j_{E}^{−1}. (2)

To verify this, let λ_{E} : J_{E} →^{∼} Ĵ_{E} be the canonical theta-polarization, and let κ_{E} : E →^{∼} ˆ Ê = Ĵ_{E} be the duality isomorphism. Since κ_{E} = −λ_{E} ∘ j_{E}, and j_{D}^{∗} = −λ_{C} (cf. [4], p. 45), we see that Ψ(h_{E})^{∗} = j_{D}^{∗}λ̂_{C} ˆĥ_{E}(j_{E}^{∗})^{−1} ∘ [−1]_{J_{E}} = (−λ_{C})λ̂_{C}κ_{J_{C}}h_{E}κ_{E}^{−1}(λ_{E}) = −h_{E}κ_{E}^{−1}λ_{E} = h_{E} ∘ j_{E}^{−1}, as claimed.

We thus see that the rule E ↦ (equivalence class of Ψ(h_{E})) defines a map Ψ_{C} = Ψ_{C,D} : S(J_{C}) → E(C) = Ẽ(C). Indeed, (2) shows that Ψ(h_{E})^{∗} is a closed immersion, so Ψ(h_{E}) : C → E is an elliptic subcover by Proposition 2.2. Moreover, the equivalence class of Ψ(h_{E}) does not depend on the choice of h_{E} because if h_{E} ∘ α : E ↪ J_{C} is another inclusion with image E, then Ψ(h_{E} ∘ α) = [−1]_{E} ∘ j_{E}^{−1} ∘ α̂ ∘ ĥ_{E} ∘ λ_{C} ∘ j_{D} = (j_{E}^{−1} ∘ α̂ ∘ j_{E}) ∘ Ψ(h_{E}) is equivalent to Ψ(h_{E}).

We now show that Φ_{C} and Ψ_{C} are inverses of each other. For this, suppose first that E ≤ J_{C} is an elliptic subgroup. Then Φ(h_{E})^{∗}J_{E} = Φ(h_{E})^{∗}(j_{E}(E)) = h_{E}(E) = E by (2), and so Φ_{C}(Ψ_{C}(E)) = E.

On the other hand, if f : C → E is a quasi-elliptic subcover, then by the above ∃P_{0} ∈ E(K), so we have an isomorphism j_{E} = j_{P_{0}} : E →^{∼} J_{E} defined by P ↦ cl(P − P_{0}). Then h_{E} := f^{∗} ∘ j_{E} : E ↪ J_{C} is a closed immersion by Proposition 2.2, and f_{1} := Ψ(h_{E}) : C → E is an elliptic subcover, as was shown above. Since

f_{1}^{∗} =^{(2)} h_{E} ∘ j_{E}^{−1} = f^{∗}, it follows that ∃x ∈ E(K) such that f_{1} = T_{x} ∘ f (cf. equations (47) and (48) of [4]) and so f ∼ f_{1}. Thus Ψ_{C}(Φ_{C}(f)) = (class of f), which shows that the maps Φ_{C} and Ψ_{C} are inverses of each other, and so both are bijections.

It thus remains to show that for any C/K, the map Φ_{C} is always injective. For this, let f_{i} : C → E_{i} be two quasi-elliptic subcovers with Φ_{C}(f_{1}) = f_{1}^{∗}J_{E_{1}} = f_{2}^{∗}J_{E_{2}} = Φ_{C}(f_{2}). Then also Φ_{C_{K′}}((f_{1})_{K′}) = Φ_{C_{K′}}((f_{2})_{K′}), for any extension K′/K. Choose K′/K Galois such that C(K′) ≠ ∅. Since Φ_{C_{K′}} is a bijection by what was shown above, we see that (f_{1})_{K′} ∼ (f_{2})_{K′}, so (f_{2})_{K′} = φ′ ∘ (f_{1})_{K′}, for some isomorphism φ′ : (E_{1})_{K′} →^{∼} (E_{2})_{K′}. Now since (f_{1})_{K′} is surjective and hence an epimorphism, it follows that φ′ is uniquely determined by (f_{1})_{K′} and (f_{2})_{K′}, and hence φ′ is invariant under Gal(K′/K). Thus, φ′ = φ_{K′} for some isomorphism φ : E_{1} →^{∼} E_{2}, and then f_{2} = φ ∘ f_{1}. Thus, f_{1} ∼ f_{2}, which means that Φ_{C} is injective.

We observe that the above proof actually yields the following slightly more precise assertion. For this, let

E(C, E) = {f : C → E is an elliptic cover}

denote the set of elliptic subcovers of C/K with target a fixed elliptic curve E/K. Note that the group E(K) of K-rational points of E/K acts on E(C, E) on the left via translations (i.e., via (x, f) = T_{x} ∘ f).

Corollary 2.6. If E/K is an elliptic curve, then the rule f ↦ f^{∗} induces a injection

Φ_{C,E} : E(K)\E(C, E) ↪ I(J_{E}, J_{C}),

where I(J_{E}, J_{C}) denotes the set of injective homomorphisms of J_{E} into J_{C}. Moreover, if C/K has a divisor of degree 1, then Φ_{C,E} is a bijection.

We next observe:

Corollary 2.7. (a) If Φ_{C}(Ẽ(C)) = S(J_{C}), then every finite morphism f : C → E to a curve E/K of genus 1 factors over a quasi-elliptic subcover f_{1} : C → E_{1}. Moreover, the equivalence class of f_{1} is uniquely determined by f.

(b) If Φ_{C}(E(C)) = S(J_{C}), then Ẽ(C) = E(C), i.e., every quasi-elliptic subcover of C is an elliptic subcover. Moreover, a finite morphism f : C → E to an elliptic curve E/K is an elliptic subcover if and only if f does not factor over a non-trivial isogeny.

Proof. (a) Let A = Im(f^{∗}) ≤ J_{C}. Then A is an elliptic subgroup of J_{C} and f^{∗} = h_{A}∘ν, where ν : J_{E} → A is an isogeny and h_{A} : A ↪ J_{C} the inclusion. Since Φ_{C} is a bijection, there exists a quasi-elliptic subcover f_{1} : C → E_{1} such that f_{1}^{∗}J_{E_{1}} = A, and so there is an isomorphism α : J_{E_{1}} →^{∼} A such that f_{1}^{∗} = h_{A} ∘ α. Put ν′ = α^{−1} ∘ ν : J_{E} → J_{E_{1}}. Now choose a Galois extension K′/K such that C(K′) ≠ ∅. Then E_{K′} and (E_{1})_{K′} are elliptic curves and there exists (after choosing origins) an isogeny ν_{1} : (E_{1})_{K′} → E_{K′} such that ν_{1}^{∗} = ν′_{K′}. Now consider f′′ = ν_{1} ∘ (f_{1})_{K′} : C_{K′} → E_{K′}. Since (f′′)^{∗} = (f_{1})_{K′}^{∗} ∘ ν_{1}^{∗} = f_{K′}^{∗}, there exists (as in the above proof) an x ∈ E(K′) such that f_{K′} = T_{x} ∘ f′′. Thus f_{K′} = φ′ ∘ (f_{1})_{K′}, where φ′ = T_{x} ∘ ν_{1} : (E_{1})_{K′} → E_{K′}. Then (as in the above proof) we see that φ′ = φ_{K′} for a unique φ : E_{1} → E and that f = φ ∘ f_{1}. Thus f factors over the quasi-elliptic subcover f_{1}.

To verify the last assertion, note that if f = φ_{1} ∘ f_{2} another such factorization, then f_{2}^{∗}J_{E_{2}} = f^{∗}J_{E} = f_{1}^{∗}J_{E_{1}}, and so f_{1} ∼ f_{2} by the injectivity of Φ_{C}.

(b) The first assertion follows immediately from the injectivity of Φ_{C}. Next, suppose that f : C → E does not factor over an isogeny. By part (a) we have a factorization f = ν ∘ f_{1}, where f_{1} : C → E_{1} is a quasi-elliptic, hence an elliptic subcover of C. Since ν : E_{1} → E is, up to a translation, an isogeny, we have by our hypothesis on f that ν is an isomorphism, and hence f : C → E is an elliptic subcover. Since the converse implication is obvious, this proves the last assertion.

Remark 2.8. If f : C → E is a quasi-elliptic subcover, then F_{E} := f^{∗}κ(E) is a genus 1 subfield of F = κ(C) which is maximal in the sense that F_{E} is not properly contained in any other genus 1 subfield of F. Thus, the map f ↦ f^{∗}κ(E) induces an injection of E(C) into the set Max_{1}(F) of maximal genus 1 subfields of F. Moreover, this map is a bijection whenever Φ_{C} is a bijection, as the above Corollary 2.7 shows.

3 The hyperelliptic case

We now specialize the discussion of quasi-elliptic subcover to the case that C/K hyperelliptic curve of genus g_{C} ≥ 2. Thus, C has a unique involution σ_{C} ∈ Aut(C) such that the quotient curve \overline{C} := C/⟨σ_{C}⟩ has genus 0. (See [7] for various equivalent versions of the hyperellipticity condition.) We let π_{C} : C → \overline{C} = C/⟨σ_{C}⟩ denote the associated hyperelliptic subcover.

Note that if g_{C} = 2, then \overline{C} ≃ ℙ_{K}^{1} because in that case the canonical morphism κ_{C} : C → ℙ_{K}^{g−1} = ℙ_{K}^{1} is surjective. However, for g_{C} > 2, it may happen that \overline{C}(K) = ∅.

The following result, due to Tamme[12], is a basic tool in the study of quasielliptic subcovers of hyperelliptic curves. Note that Kuhn proves only a much weaker statement in [6] (and only for g_{C} = 2 and char(K) ≠ 2).

Proposition 3.1 (Tamme). If f : C → E is a finite morphism to a curve E of genus 1, then there exists a unique involution σ_{f} on E such that

f ∘ σ_{C} = σ_{f} ∘ f. (3)

Moreover, if π_{f} : E → E/⟨σ_{f}⟩ is the associated degree 2 subcover, then \overline{E} := E/⟨σ_{f}⟩ has genus 0 and there is a unique morphism φ_{f} : \overline{C} → \overline{E} such that

φ_{f} ∘ π_{C} = π_{f} ∘ f. (4)

Proof. For convenience of the reader, we present a proof in terms of covers (rather than in terms of function fields as in [12]).

First note that the last assertions are immediate consequences of (3). Indeed, (3) shows that π_{f} ∘ f is σ_{C}-invariant, and so by universal property of the quotient map π_{C}, there is a unique morphism φ_{f} : \overline{C} → \overline{E} satisfying (4). Moreover, by Lüroth we have that g\overline{E} = g\overline{C} = 0.

Next we observe that if σ_{f} exists, then it is uniquely determined by (3) because f is an epimorphism. Thus, it is enough to construct σ_{f} after a suitable Galois baseextension K′/K, for (as in the proof of Proposition 2.3) σ_{f} will be automatically Galois-invariant and hence defined over K.

Thus, we may assume (after replacing K by a suitable K′) that C(K) ≠ ∅. Fix P ∈ C(K). Then f(P) ∈ E(K), and so there exists an isomorphism j = j_{f(P)} : E →^{∼} J_{E} such that j(f(P)) = 0_{J_{E}}. Then j ∘ f(P) = 0_{J_{E}} and so by the Albanese property of (C, j_{P}) there exists a unique homomorphism h = h_{f} : J_{C} → J_{E} such that h ∘ j_{P} = j ∘ f. Next we observe that

j_{P} ∘ σ_{C} = T_{x} ∘ [−1]_{J_{C}} ∘ j_{P} with x = j_{P} (σ_{C}(P)),

as is easy to check. Thus j∘f ∘σ_{C} = h∘j_{P} ∘σ_{C} = h∘T_{x}∘[−1]_{J_{C}} ∘j_{P} = T_{h(x)}∘h∘[−1]_{J_{C}} ∘ j_{P} = T_{h(x)} ∘[−1]_{E} ∘h∘j_{P} = T_{h(x)} ∘[−1]_{E} ∘h∘j_{P} = T_{h(x)} ∘[−1]_{E} ∘j ∘f = j ∘σ_{f} ∘f, where σ_{f} = j^{−1} ∘ T_{h(x)} ∘ [−1]_{E} ∘ j, and so (3) holds because j is an isomorphism. Note that the construction of σ_{f} shows that σ_{f}^{2} = 1 (and σ_{f} ≠ 1), so σ_{f} is an involution.

As usual, the different divisor W_{C} := Diff(π_{C}) of the hyperelliptic cover is called the Weierstrass divisor of the hyperelliptic curve C/K. In [1] and [4], this divisor played a fundamental role in the analysis of elliptic subcovers of a genus 2 curve C, and this turns out to be the case here as well. A key result (which replaces many of the delicate arguments in [1], [4] and [6]) is the following simple fact.

Proposition 3.2. If f : C → E is a finite separable morphism to a curve E/K of genus 1, then there exists an effective divisor D_{f} ∈ Div(E) of degree 2 deg(f)−g_{C} −1 such that

(5) f_{∗}W_{C} = f_{∗}Diff(π_{C}) = deg(f)Diff(π_{f}) − 2D_{f},

where π_{f} is as in Proposition 3.1 and Diff(π_{f}) ∈ Div(E) is its different. Moreover, supp(f_{∗}W_{C}) ⊂ supp(Diff(π_{f})).

To prove this, we will use the following general fact.

Lemma 3.3. Let f_{i} : C → C_{i} and f′_{i} : C_{i} → C′, i = 1, 2 be four separable curve covers such that f′_{1} ∘ f_{1} = f′_{2} ∘ f_{2} and deg(f_{1}) = deg(f′_{2}). If f_{1} and f_{2} are disjoint, i.e., if f_{1}^{∗}κ(C_{1})f_{2}^{∗}κ(C_{2}) = κ(C), then we have that

(6) (f′_{2})^{∗}(f′_{1})_{∗}D_{1} = (f_{2})_{∗}f_{1}^{∗}D_{1}, for all D_{1} ∈ Div(C_{1}).

Moreover, there is an effective divisor D_{2} ∈ Div(C_{2}) such that

(f′_{2})^{∗}(f′_{1})_{∗}Diff(f′_{1}) = (f_{2})_{∗}Diff(f_{2}) + 2D_{2}, (7)

and hence we have

(8) (f_{2})_{∗}Diff(f_{1}) = deg(f_{2})Diff(f′_{2}) − 2D_{2}.

Proof. It clearly suffices to verify (6) in the case that K is algebraically closed. Let f̃: C̃ → C′ be a Galois cover which factors over f := f′_{1} ∘ f_{1} = f′_{2} ∘ f_{2}, so f̃ = f ∘ φ, for some φ : C̃ → C. Put f̃_{i} = f_{i} ∘ φ and write H_{i} = Aut(f̃_{i}), G = Aut(f̃) and H = Aut(φ). By the disjointness and degree conditions (and Galois theory) it follows that H = H_{1} ∩ H_{2} and G = ⟨H_{1}, H_{2}⟩. Moreover, since |G| = d_{1}d_{2}d, where d_{i} = deg(f_{i}) and d = deg(φ) and since |H_{1}H_{2}| = \frac{|H_{1}||H_{2}|}{|H_{1}∩H_{2}|} = \frac{(dd_{1})(dd_{2})}{d} = dd_{1}d_{2}, we have that G = H_{1}H_{2} = H_{2}H_{1}. Thus, for any D̃ ∈ Div(C̃), we obtain

(f̃_{2})^{∗}(f̃_{2})_{∗}(f̃_{1})^{∗}(f̃_{1})_{∗}D̃ = ∑_{h∈H_{2}} h_{∗} ∑_{h′∈H_{1}}(h′)_{∗}D̃ = ∑_{h∈H_{2}} ∑_{h′∈H_{1}}(hh′)_{∗}D̃ = d ∑_{g∈G} g_{∗}D̃ = df̃^{∗}f̃_{∗}D̃.

Taking D̃ = f̃_{1}^{∗}D_{1} yields (f̃_{2})^{∗}(f̃_{2})_{∗}(f̃_{1})^{∗}(dd_{1}D_{1}) = (f̃_{2})^{∗}(f̃_{2})_{∗}(f̃_{1})^{∗}(f̃_{1})_{∗}(D̃) = df̃^{∗}f̃_{∗}D̃ = df̃^{∗}(f′_{1})_{∗}(dd_{1}D_{1}). Thus, since (f̃_{2})^{∗} is injective, it follows from this that (f̃_{2})_{∗}(f̃_{1})^{∗}(D_{1}) = d(f′_{2})^{∗}(f′_{1})_{∗}(D_{1}). This clearly implies (6) because (f̃_{2})_{∗}(f̃_{1})^{∗}(D_{1}) = (f_{2})_{∗}φ_{∗}φ^{∗}(f_{1})^{∗}(D_{1}) = d(f_{2})_{∗}(f_{1})^{∗}(D_{1}).

To verify (7), let U′ = Spec(A) be an open affine subset of C′ which contains all the ramification points of f, and put U_{i} = f_{i}^{−1}(U′) and U = f^{−1}(U′). Then U_{i} = Spec(A_{i}) and U = Spec(B), where A_{i} and B are the integral closures of A in F_{i} = κ(C_{i}) and in F = κ(C), respectively. Since F_{1} and F_{2} are linearly disjoint over F′ = κ(C′) (as we saw above), it follows that F_{1} ⊗_{F′} F_{2} ≃ F_{1}F_{2} = F. Thus, A′ := A_{1} ⊗_{A} A_{2} is an A_{2}-sublattice of B, and so by [10], p. 49, we have that their discriminants are related by the formula

𝔡_{A′/A_{2}} = 𝔡_{B/A_{2}}𝔞^{2},

where 𝔞 = χ(A′, B) ≠ 0 is some A_{2}-ideal. From the definition of the discriminant we see that 𝔡_{A_{1}⊗A_{2}/A_{2}} = 𝔡_{A_{1}/A}A_{2}, and so we obtain by [10], p. 50, the relation

N(𝔇_{A_{1}/A})A_{2} = N(𝔇_{B/A_{2}})𝔞^{2},

where N() denotes the norm and 𝔇_{A_{1}/A} the different. Since this is just the idealtheoretic version of (7), we see that assertion (7) has been proved.

Finally, to verify (8), note first that Diff(f_{1}) = Diff(f_{2}) + f_{2}^{∗}Diff(f′_{2}) − f_{1}^{∗}Diff(f′_{1}) because Diff(f_{1}) + f_{1}^{∗}Diff(f′_{1}) = Diff(f′_{1} ∘ f_{1}) = Diff(f′_{2} ∘ f_{2}) = Diff(f_{2}) + f_{2}^{∗}Diff(f′_{2}). Thus

(f_{2})_{∗}Diff(f_{1}) =^{(6)} (f_{2})_{∗}Diff(f_{2}) + n_{2}Diff(f′_{2}) − (f′_{2})^{∗}(f′_{1})_{∗}Diff(f′_{1})
=^{(7)} (f_{2})_{∗}Diff(f_{2}) + n_{2}Diff(f′_{2}) − ((f_{2})_{∗}Diff(f_{2}) + 2D_{2}),

which proves (8).

Proof of Proposition 3.2. We apply Lemma 3.3 with f_{1} = π_{C}, f′_{1} = φ_{f}, f_{2} = f, f′_{2} = π_{f}, where φ_{f} and π_{f} are as in (4). By Lüroth, κ(E) ⊄ κ(\overline{C}), so π_{C} and f are disjoint. Thus, the hypotheses of Lemma 3.3 hold, and so by (8) there exists a divisor D ≥ 0 such that (5) holds. Since deg(f_{∗}W_{C}) = deg(W_{C}) = 2g_{C} + 2 and deg(Diff(π_{f})) = 4, it follows from (5) that deg(D) = \frac{1}{2}(4 deg(f) − (2g_{C} + 2)) = (2 deg(f) − 1) − g_{C}.

To prove the last assertion, note that if P ∈ supp(f_{∗}(W_{C})), then there exists P′ ∈ supp(W_{C}) such that f(P′) = P. Then σ_{C}(P′) = P′ and so σ_{f}(P) = σ_{f}(f(P′)) = f(σ(P′)) = f(P′) = P, which implies that P ∈ supp(Diff(π_{f})), as claimed.

Remark 3.4. By using the above Proposition 3.2, it is possible to give a quick proof of the “Normalization Lemma” of [4], without recourse to the (more complicated) results of [1] and [6]. More precisely, if char(K) ≠ 2 and if f : C → E is an elliptic subcover of degree n of a genus 2 curve C, then there exists a unique P_{f} ∈ E(K) such that f_{∗}W_{C} = (2, n)Diff(π_{f}) − (−1)^{n}2P_{f}. (9)

Indeed, since P_{f} is uniquely determined by (9), it is enough to verify this over a sufficiently large Galois extension K′/K, and so we may assume that W_{C} = Q_{1} + … + Q_{6} and Diff(π_{f}) = P_{1} + … + P_{4} consist of 6 and 4 distinct rational points, respectively. Then by (5) we see that

v_{P_{i}}(f_{∗}W_{C}) ≡ n (mod 2), for 1 ≤ i ≤ 4.

Thus, if n is odd, then (9) follows immediately because we then have that v_{P_{i}}(f_{∗}W_{C}) ≥ 1 for all i, and so there can be at most one P_{i} with v_{P_{i}}(f_{∗}W_{C}) ≥ 3. Thus f_{∗}W_{C} = 3P_{i} + ∑_{j≠i} P_{j} = 2P_{i} + Diff(π_{f}). If n is even, then a similar analysis shows that (9) holds because we have that v_{P_{i}}(f_{∗}W_{C}) ≤ 3, for all i. (To see this, suppose that v_{P_{i}}(f_{∗}W_{C}) ≥ 4. Then there exist (say) Q_{1}, …, Q_{4} ∈ f^{−1}(P_{i}), and then cl(Q_{k} − Q_{1}) ∈ Ker(f_{∗})[2], for k = 1, …, 4. Since Ker(f_{∗}) ≃ E′ is an elliptic curve, this forces Q_{4} − Q_{1} ∼ Q_{3} − Q_{1} + Q_{2} − Q_{1}. Thus Q_{1} + Q_{4} ∼ Q_{2} + Q_{4}, which is impossible since h^{0}(Q_{1} + Q_{4}) = 1.)

4 Normalized subcovers

In [4], the notion of a normalized subcover was introduced in order to rigidify elliptic subcovers of genus 2 curves (if char(K) ≠ 2). Here we generalize this notion to higher genus (hyperelliptic) curves (and also include the case of characteristic 2).

Definition 4.1. An elliptic cover f : C → E of a hyperelliptic curve C is called pre-normalized if σ_{f} = [−1]_{E}, i.e., if

[−1]_{E} ∘ f = f ∘ σ_{C}. (10)

Moreover, f is called max-normalized if f is pre-normalized and if

(11) v_{0_{E}}(f_{∗}W_{C}) > v_{P} (f_{∗}W_{C}) deg_{i}(P), ∀P ∈ supp(Diff(π_{E})) \ {0_{E}},

where π_{E} : E → E/⟨[−1]_{E}⟩ ≃ ℙ_{K}^{1} is the quotient map, and deg_{i}(P) = [κ(P) : K]_{i} denotes the inseparable degree of P. (Thus, deg_{i}(P) = 1 if K is perfect.) Similarly, f is called min-normalized if f is pre-normalized and if

(12) v_{0_{E}}(f_{∗}W_{C}) < v_{P} (f_{∗}W_{C}) deg_{i}(P), ∀P ∈ supp(Diff(π_{E})) \ {0_{E}}.

Finally, an elliptic cover is called normalized if it is either max-normalized or minnormalized.

We observe the following basic properties of normalized subcovers.

Proposition 4.2. Let f : C → E be an elliptic K-subcover, and let K′/K be an extension field. Then f is pre-normalized or max-normalized or min-normalized if and only if f_{K′} has the corresponding property.

Proof. By properties of faithfully flat base-change, the assertion about pre-normalized subcovers is clear. To prove the rest of the assertions, it suffices to verify them in the case that K′ is algebraically closed. Indeed, suppose that the assertions have been proven in this case, and let K′′ be an algebraic closure of K′. Then from our supposition (applied to K′′/K and to K′′/K′) it follows that f is max-normalized if and only if f_{K′′} = (f_{K′})_{K′′} is max-normalized if and only if f_{K′} is max-normalized. The argument for min-normalized is similar.

Thus, assume that K′ is algebraically closed, and let β = β_{K′/K} : C_{K′} → C be the base-change map. If D_{K′} = β^{∗}D is the pullback of a divisor D ∈ Div(E), then

(13)  v_{P′}(D_{K′}) = v_{β(P)}(D) deg_{i}(β(P′)), for all P′ ∈ C(K′),
(14)  supp(D_{K′}) = β^{−1}(supp(D)).

Thus, since β is surjective and since β^{−1}(0_{E}) = 0_{E_{K′}}, it is clear from the definitions (and the fact that (f_{∗}W_{C})_{K′} = (f_{K′})_{∗}W_{C_{K′}}) that (11) or (12) holds for f if and only if the corresponding property holds for f_{K′}.

Proposition 4.3. Let f_{i} : C → E be two elliptic subcovers with f_{1}^{∗} = f_{2}^{∗}.

(a) If f_{1} and f_{2} are both pre-normalized, then there exists a unique x ∈ E[2](K) such that f_{1} = T_{x} ∘ f_{2}.

(b) If f_{1} and f_{2} are both max-normalized (or both are min-normalized), then f_{1} = f_{2}.

Proof. (a) Since f_{1}^{∗} = f_{2}^{∗}, there exists x ∈ E(K) such that f_{1} = T_{x} ∘ f_{2}; cf. Corollary 2.6. (Note that T_{x} and hence x is unique because f_{2} is an epimorphism.) Then [−1]_{E} ∘ T_{x} ∘ f_{2} = [−1]_{E} ∘ f_{1} = f_{1} ∘ σ_{C} = T_{x} ∘ f_{2} ∘ σ_{C} = T_{x} ∘ [−1]_{E} ∘ f_{2}, and so [−1]_{E} ∘ T_{x} = T_{x} ∘ [−1] because f_{2} is an epimorphism. Thus T_{−x} = T_{x}, so 2x = 0 or x ∈ E[2](K).

(b) By part (a) there exists x ∈ E[2](K) such that f_{1} = T_{x} ∘ f_{2}. Note that x ∈ supp(Diff(π_{E})). Thus, if x ≠ 0_{E}, then v_{0_{E}}((f_{2})_{∗}W_{C}) > v_{x}((f_{2})_{∗}W_{C}). Now (f_{1})_{∗}W_{C} = (T_{x})_{∗}(f_{2})_{∗}W_{C}, so v_{0_{E}}((f_{1})_{∗}W_{C}) = v_{x}((f_{2})_{∗}W_{C}) < v_{0_{E}}((f_{2})_{∗}W_{C}) = v_{x}((f_{1})_{∗}W_{C}). Since this contradicts the hypothesis that f_{1} is max-normalized, we must have x = 0, and so f_{1} = f_{2}. The proof for min-normalized maps is similar.

Thus, max/min-normalized subcovers are uniquely determined by their induced maps on the Jacobians. This fact allows us to descend max/min-normalized subcovers as follows.

Corollary 4.4. Let E ≤ J_{C} be an elliptic subgroup with embedding h : E ↪ J_{C}. If there exists a Galois extension K′/K and a normalized cover f′ : C_{K′} → E_{K′} such that (f′)^{∗} ∘ j_{E_{K′}} = h_{K′}, then there exists a normalized cover f : C → E such that f^{∗} ∘ j_{E} = h.

Proof. By Galois descent, it is enough to verify that (f′)^{σ} = f′, for all σ ∈ Gal(K′/K). Now ((f′)^{σ})^{∗} = (f′)^{∗} because ((f′)^{σ})^{∗} ∘ j_{E_{K′}} = ((f′)^{∗} ∘ j_{E_{K′}})^{σ} = (h_{K′})^{σ} = h_{K′} = (f′)^{∗}∘j_{E_{K′}} (and j_{E_{K′}} is an isomorphism). Thus, the assertion follows from Proposition 4.3(b) once we have shown that (f′)^{σ} is also max/min-normalized.

For this, note first that (f′)^{σ} is pre-normalized because [−1]_{E_{K′}} ∘(f′)^{σ} = ([−1]_{E_{K′}} ∘ f′)^{σ} = (f′ ∘ σ_{C_{K′}})^{σ} = (f′)^{σ} ∘ σ_{C_{K′}}. Now since (W_{C_{K′}})^{σ} = W_{C_{K′}}, we see that (f′)_{∗}^{σ}(W_{C_{K′}}) = ((f′)_{∗}(W_{C_{K′}}))^{σ}. Thus, if f′ is max-normalized, i.e., if v_{0_{E_{K′}}} ((f′)_{∗}W_{C_{K′}}) > v_{P} ((f′)_{∗}W_{C_{K′}}) deg_{i}(P), for all P ∈ supp(Diff(π_{E_{K′}}))\{0_{E_{K′}}}, then also v_{0_{E_{K′}}} (((f′)_{∗}W_{C_{K′}})^{σ}) > v_{P} (((f′)_{∗}W_{C_{K′}})^{σ}) deg_{i}(P), for all P ∈ supp(Diff(π_{E_{K′}})) \ {0_{E_{K′}}}, because 0_{E_{K′}}^{σ} = 0_{E_{K′}} and supp(Diff(π_{E_{K′}}))^{σ} = supp(Diff(π_{E_{K′}})). Thus (f′)^{σ} is also max-normalized, and so the assertion follows. The proof for f′ min-normalized is analogous.

We now turn to investigate the existence of normalized subcovers (inside a given equivalence class of subcovers). For this, we first consider the existence of prenormalized subcovers.

Proposition 4.5. Let f : C → E be an elliptic subcover. Then the following conditions are equivalent:

(i) ∃x ∈ E(K) such that T_{x} ∘ f is pre-normalized.

(ii) ∃x ∈ E(K) such that σ_{f} = T_{x}^{−1} ∘ [−1]_{E} ∘ T_{x}.

(iii) supp(Diff(π_{f})) ∩ E(K) ≠ ∅.

Proof. (ii) ⇒ (i): By (ii) and (10) we have that [−1]_{E}∘T_{x}∘f = (T_{x}∘σ_{f} ∘T_{x}^{−1})∘T_{x}∘f = T_{x} ∘ f ∘ σ_{C}, so T_{x} ∘ f is pre-normalized.

(i) ⇒ (iii): By (i) and (10) we have [−1]_{E} ∘ T_{x} ∘ f = T_{x} ∘ f ∘ σ_{C} = T_{x} ∘ σ_{f} ∘ f, and so [−1]_{E} ∘ T_{x} = T_{x} ∘ σ_{f} because f is an epimorphism. Thus σ_{f}(−x) = T_{x}^{−1}([−1]_{E}(T_{x}(−x))) = T_{−x}(0_{E}) = −x, and so −x ∈ Diff(π_{f}) ∩ E(K).

(iii) ⇒ (ii): Let P ∈ supp(Diff(π_{f})) ∩ E(K), so σ_{f}(P) = P. Put σ_{1} = T_{P}^{−1}σ_{f} ∘ T_{P}. Then σ_{1}(0_{E}) = T_{P}^{−1}(σ_{f}(P)) = 0_{E}, so σ_{1} ∈ Aut(E). Now since σ_{f}^{∗} = [−1]_{J_{E}} (as is clear from the construction of σ_{f} in the proof of Proposition 3.1), we thus have that σ_{1}^{∗} = σ_{f}^{∗} = [−1]_{J_{E}} = [−1]_{E}^{∗}, and so σ_{1} = [−1]_{E}. Thus (ii) holds with x = −P.

Remark 4.6. Note that unless E(K) is 2-divisible, not every involution σ of E satisfies condition (ii) of Proposition 4.5. Indeed, for any y ∈ E(K), the element σ_{y} = T_{y} ∘ [−1]_{E} is clearly an involution of E, but σ_{y} = T_{x}^{−1} ∘ [−1]_{E} ∘ T_{x} if and only if y = −2x. Thus, the conditions of the Proposition 4.5 are not automatically satisfied.

Corollary 4.7. If f : C → E be an elliptic subcover, then there exists a y ∈ E(K) such that T_{y} ∘ f is max-normalized (respectively, min-normalized) if and only if there exists an x ∈ supp(Diff(π_{f})) ∩ E(K) such that

(15) v_{x}(f_{∗}W_{C}) > v_{P} (f_{∗}W_{C}) deg_{i}(P), respectively, v_{x}(f_{∗}W_{C}) < v_{P} (f_{∗}W_{C}) deg_{i}(P), for all P ∈ supp(Diff(π_{f})) \ {x}.

Proof. If T_{y}∘f is max-normalized, then by (the proof of) Proposition 4.5 we know that x := −y ∈ supp(Diff(π_{f}))∩E(K) and that σ_{f} = T_{y}^{−1} ∘[−1]_{E} ∘T_{y}. Thus π_{E} ∘T_{y} ∘σ_{f} = π_{E} ∘[−1]_{E} ∘T_{y} = π_{E} ∘T_{y} and so, by the universal property of quotients, there exists an isomorphism α : ℙ^{1} →^{∼} ℙ^{1} such that α ∘ π_{f} = π_{E} ∘ T_{y}. Thus Diff(π_{f}) = Diff(π_{E} ∘ T_{y}) = T_{y}^{∗}Diff(π_{E}), and so supp(Diff(π_{f})) = T_{y}^{−1}(supp(Diff(π_{E}))) = T_{x}(supp(Diff(π_{E}))). Moreover, since (T_{y}f)_{∗}W_{C} = (T_{x}^{−1})_{∗}f_{∗}W_{C} = T_{x}^{∗}f_{∗}W_{C}, we have that

v_{Q}((T_{y}f)_{∗}W_{C}) deg_{i}(Q) = v_{T_{x}(Q)}(f_{∗}W_{C}) deg_{i}(T_{x}(Q)), ∀ prime divisors Q on E.

Thus, taking Q = x and Q = P ∈ supp(Diff(π_{f}))\{x} = T_{x}(supp(Diff(π_{E}))\{0_{E}}), it follows from the fact that T_{y}f is max-normalized that v_{x}(f_{∗}W_{C}) = v_{0_{E}}((T_{y}f)_{∗}W_{C}) > v_{T_{x}^{−1}P} ((T_{y}f)_{∗}W_{C}) = v_{P} (f_{∗}W_{C}), and so the first inequality of (15) holds. Similarly, if T_{y}f is min-normalized, then the second inequality of (15) holds.

Conversely, suppose that there exists x ∈ supp(Diff(π_{f})) ∩ E(K) such that the first inequality of (15) holds for all P ∈ supp(Diff(π_{f})) \ {x}. Then by Proposition 4.5 we know that if y = −x, then T_{y} ∘ f is pre-normalized and the previous discussion shows that T_{y} ∘ f is max-normalized. Similarly, if the second equality holds, then T_{y} ∘ f is min-normalized.

We now come to the main result of this section.

Theorem 4.8. Suppose that g_{C} is even. If f : C → E is an elliptic subcover, then there exists an x ∈ E(K) such that T_{x} ∘ f is normalized.

Proof. We first note that it sufficies to verify this fact after Galois base change K′/K. Indeed, suppose that there exists x ∈ E(K′) such that f′ := T_{x} ∘ f_{K′} is normalized. Then (f′)^{∗} = (f_{K′})^{∗}, so (f′)^{∗} ∘ j_{E_{K′}} = (f^{∗} ∘ j_{E})_{K′}, and hence by Corollary 4.4 there is a normalized cover f_{1} : C → E such that (f_{1})_{K′} = f′. Now (f_{1}^{∗})_{K′} = (f′)^{∗} = f_{K′}^{∗} and so by Galois decent f_{1}^{∗} = f^{∗}. Thus, by the equivalences (47) and (48) of [4] it follows that there exists x ∈ E(K) such that f_{1} = T_{x} ∘ f, which proves the assertion.

Thus, we may assume that all prime divisors appearing in W_{C} and in Diff(π_{f}) are purely inseparable because this is the case after a suitable Galois extension. We now distinguish cases. Case 1: char(K) ≠ 2. Since π_{f} and π_{C} are tame extensions of degree 2, we see that Diff((π_{f})\overline{K}) and Diff((π_{C})\overline{K}) are reduced and so all prime divisors of Diff(π_{f}) and of Diff(π_{C}) are separable. Thus, by our hypothesis on K we have that Diff(π_{f}) = ∑_{i=1}^{4} P_{i} where the P_{i} ∈ E(K) are distinct. Thus by Proposition 3.2 we see that

f_{∗}W_{C} = ∑_{i=1}^{4} n_{i}P_{i}

for some integers n_{i} ≥ 0 with ∑ n_{i} = deg f_{∗}W_{C} = 2g_{C} + 2. By renumbering the P_{i}’s if necessary, we may assume that n_{1} ≥ n_{2} ≥ n_{3} ≥ n_{4}. Now suppose that it could happen that n_{1} = n_{2} and n_{3} = n_{4}. But then, since n_{i} ≡ deg(f) (mod 2) by (5), it follows that n_{1} + n_{3} ≡ 2 deg(f) ≡ 0 (mod 2) and so 2g_{C} + 2 = 2n_{1} + 2n_{3} ≡ 0 (mod 4), which contradicts the fact that g_{C} ≡ 0 (mod 2).

Thus we must have that either n_{1} > n_{2} or that n_{4} < n_{3}. In the former case we see that the first condition of (15) holds for x = P_{1} and in the latter case we see that the second condition of (15) holds for x = P_{4}. Thus, by Corollary 4.7 there exists x ∈ E(K) such that T_{x} ∘ f is normalized. Case 2: char(K) = 2, E supersingular. Since here E[2](\overline{K}) = \{0_{E\overline{K}}\}, it follows that Diff(π_{E\overline{K}}) = 4(0_{E\overline{K}}), which implies that Diff((π_{f})\overline{K}) = 4\overline{P}, for some \overline{P} ∈ E(\overline{K}). (Indeed, since E(\overline{K}) is 2-divisible, there exists \overline{P} ∈ E(\overline{K}) such that T_{−\overline{P}} ∘ f\overline{K} is pre-normalized (cf. Remark 4.6 and Proposition 4.5) and then Diff((π_{f})\overline{K}) = T_{−\overline{P}}^{∗} (Diff(π_{E\overline{K}})) = 4\overline{P} by the proof of Corollary 4.7.)

It therefore follows that Diff(π_{f}) = eP, where P is a prime divisor of E and e deg(P) = e deg_{i}(P) = 4. Thus, by Proposition 3.2 we see that f_{∗}W_{C} = \frac{2g_{C}+2}{deg(P)}P, and so, since g_{C} +1 is odd, it follows that deg(P)|2. Now if deg(P) = 2 then Diff(π_{f}) = 2P and f_{∗}W_{C} = (g_{C} +1)P, so by (5) we obtain that (g_{C} +1)P = deg(f)(2P)−2D, which is impossible since g_{C} + 1 is odd. Thus deg(P) = 1, so P ∈ E(K). Since condition (15) holds vacuously here (because supp(Diff(π_{f})) = {P }), we see by Corollary 4.7 that T_{−P} ∘ f is both max-normalized and min-normalized. Case 3: char(K) = 2, E ordinary. Since here E[2](\overline{K}) = \{0_{E\overline{K}}, \overline{P}_{1}\}, it follows that Diff(π_{E\overline{K}}) = 2(0_{E\overline{K}}) + 2\overline{P}_{1}. By the same argument as in Case 2 we see thus see that Diff((π_{f})\overline{K}) = 2\overline{P} ′_{0} + 2\overline{P} ′_{1}, for some (distinct) \overline{P} ′_{0}, \overline{P} ′_{1} ∈ E(\overline{K}). Thus Diff(π_{f}) = e_{0}P_{0} + e_{1}P_{1}, where P_{0}, P_{‘} are distinct prime divisors of E with e_{k} deg(P_{k}) = e_{k} deg_{i}(P_{k}) = 2, for k = 0, 1.

Suppose first that deg(P_{k}) = 2, for k = 0 and k = 1. Then Diff(π_{f}) = P_{0} + P_{1} and so f_{∗}W_{C} = n_{0}P_{0} + n_{1}P_{1}. Comparing degrees yields n_{0} + n_{1} = g_{C} + 1. Put m_{k} = v_{P_{k}}(D), where D is as in (5). Then by (5) we have that n_{k} = deg(f) − 2m_{k}, for k = 0, 1, and so g_{C} +1 = n_{0}+n_{1} = 2 deg(f)−2(m_{0}+m_{1}), which is impossible since g_{C} is even. Thus, there exists at least one P_{k} with deg(P_{k}) = 1, say (after renumbering) deg(P_{0}) = 1. Thus Diff(π_{f}) = 2P_{0} + e_{1}P_{1} and P_{0} ∈ supp(Diff(π_{f})) ∩ E(K). Write, as before, f_{∗}W_{C} = n_{0}P_{0} + n_{1}P_{1}, so n_{0} + n_{1} deg(P_{1}) = 2g_{C} + 2.

We now claim that n_{0} ≠ n_{1} deg(P_{1}). Indeed, if this is the case, then n_{0} = n_{1} deg(P_{1}) = g_{C} + 1. But then from (5) we obtain that g_{C} + 1 = n_{0} = deg(f)(2) − 2v_{P_{0}}(D), which is impossible. Thus n_{0} ≠ n_{1} deg(P_{1}), and so either the first or second inequality of (15) holds, and so by Corollary 4.7 there exists x = −P_{0} such that T_{x} ∘f is either max-normalized or min-normalized.

In the case of genus 2 one can say more. The following result generalizes Proposition 2.2 of [4] to the case of characteristic 2, and also gives a different proof of it^{2} when char(K) ≠ 2.

2There is an unfortunate typo in formula (4) of [4] which was corrected in [5].

Theorem 4.9. Let C be a curve of genus g_{C} = 2, and let f : C → E be an elliptic subcover of degree N. Then there exists a unique P_{f} ∈ E(K) such that

f_{∗}W_{C} = (2, N)Diff(π_{f}) − (−1)^{N}(2P_{f}). (16)

Thus, if N is odd, then there exists a unique x ∈ E(K) such that T_{x} ∘ f is maxnormalized, and if N is even, then there exists a unique x ∈ E(K) such that T_{x} ∘ f is min-normalized.

Proof. First note that the last assertions follow immediately from (16) and Corollary 4.7 by taking x = −P_{f}. Thus, it is enough to verify (16).

Now if char(K) ≠ 2, then (16) was established in Remark 3.4. Thus, assume that char(K) = 2. If E is supersingular, then by the proof of Theorem 4.8 we see that there exists an x ∈ E(K) such that T_{x}∘f is normalized. Thus, since Diff(π_{E}) = 4(0_{E}), we see that Diff(π_{f}) = 4P_{f} for a unique P_{f} ∈ E(K) and that f_{∗}W_{C} = 6P_{f}. Thus,

(16) holds in this case because 6P_{f} = 4P_{f} + 2P_{f} = 2(4P_{f}) − 2P_{f}.

Now suppose that E is ordinary, so |E[2](\overline{K})| = 2. We first verify (16) in the case that K = \overline{K} is algebraically closed. Clearly, (16) follows from the following more precise statement. Claim: If K = \overline{K}, then Diff(f) = 2Q_{1} with Q_{1} ∈ C(K). If P_{1} = f(Q), then Diff(π_{f}) = 2P_{1} + 2P_{2} with P_{2} ∈ E(K), and P_{2} ≠ P_{1}. Furthermore,

(17) #(f^{−1}(P_{2}) ∩ supp(W_{C})) = (2, N) and #(f^{−1}(P_{1}) ∩ supp(W_{C})) = w − (2, N),

where w = #supp(W_{C}), and hence

(f_{∗}W_{C}) = (2, N)Diff(π_{f}) − (−1)^{N}(2P_{1}). (18)

To verify this, note first that since deg(Diff(f)) = 2, we have e_{Q} ≤ 3 for all Q ∈ supp(Diff(f)). Moreover, if e_{Q} = 3, for one Q, then Diff(f) = 2Q and if e_{Q} = 2, for one Q, then due to wild ramification we have 2Q ≤ Diff(f) and so Diff(f) = 2Q in this case as well. This proves the first statement of the claim.

Next we observe that since Diff(f) is σ_{C}-invariant, so is Q_{1}, and hence Q_{1} ∈ supp(W_{C}). Thus P_{1} ∈ supp(Diff(π_{f})). Now the Deuring-Shafarevich formula (cf. [8]) applied to π_{f} shows that #supp(Diff(π_{f})) = γ_{E} + 1 = 2, where γ_{E} is the Hasse-Witt rank (or 2-rank) of E. Thus, Diff(π_{f}) = 2P_{1} + 2P_{2} for some P_{2} ≠ P_{1}, which proves the second statement of the claim.

To verify (17), note first that second equality of (17) is clearly a consequence of the first equality. To prove it, we first observe that since f^{−1}(P_{2}) is a ⟨σ_{C}⟩-set of cardinality N, we have that #(f^{−1}(P_{2}) ∩ supp(W_{C})) ≡ N (mod 2). Now clearly w ≤ 3 (since deg(W_{C}) = 6 and all points in W_{C} have multiplicity ≥ 2), so the first equality follows once we have shown that 1 ≤ #(f^{−1}(P_{2}) ∩ supp(W_{C})) ≤ 2 or, equivalently, that supp(W_{C}) = {Q_{1}, … Q_{w}} cannot lie completely in a single fibre of f. Indeed, if this were the case, then cl(Q_{i} − Q_{j}) ∈ Ker(f_{∗})[2](K), ∀i, j. Now since E′ := Ker(f_{∗}) is an elliptic curve and J_{C} ∼ E × E′, we see that |Ker(f_{∗})[2](K)| = 2^{γ_{C}−1} < 2^{γ_{C}} = |J_{C}[2](K)|, where γ_{C} denotes the 2-rank of J_{C}. We thus obtain a contradiction, because the classes cl(Q_{i} − Q_{j}) generate J_{C}[2](K). This proves (17).

We now deduce (18) from (17). Consider first the case that w = 3. Then W_{C} = 2Q_{1} + 2Q_{2} + 2Q_{3}, so from (17) it follows that f_{∗}W_{C} = 2((3 − (2, N))P_{1} + (2, N)P_{2}) = (2, N)Diff(π_{f}) − (−1)^{N}(2P_{1}), which proves (18) in this case. Next, suppose that w < 3. Then w = 2 (because otherwise supp(W_{C}) would lie completely in a fibre), so supp(W_{C}) = {Q_{1}, Q_{2}} with Q_{2} ∈ f^{−1}(P_{2}). Since e_{Q_{2}}(f) = 1, it follows that v_{Q_{2}}(Diff(π_{f} ∘ f)) = v_{f(Q_{2})}(Diff(π_{f})) = 2, and so v_{Q_{2}}(W_{C}) ≤ v_{Q_{2}}(Diff(φ_{f} ∘ π_{C})) = v_{Q_{2}}(Diff(π_{f} ∘ f)) = 2. Thus, W_{C} = 4P + 2P_{2}, and hence f_{∗}W_{C} = 4Q_{1} + 2Q_{2} = Diff(π_{f}) + 2Q_{1}. This proves (18) also when w = 2 because in that case we must have that N is odd. Indeed, in that case J_{C} ∼ E × E′ has 2-rank w − 1 = 1 (by Deuring/Shafarevich applied to π_{C}) and then E′ is supersingular (whereas E is ordinary). Thus, N has to be odd, for otherwise we would have an induced isomorphism E[2] ≃ E′[2] which is impossible; cf. [3], Theorem 3.4. Thus, (18) holds in all cases.

We now consider the case of an arbitrary ground field K. Since by the claim Diff(f)\overline{K} = 2\overline{Q}_{1} with \overline{Q}_{1} ∈ C(\overline{K}), it follows that either Diff(f) = 2Q_{1} with Q_{1} ∈ C(K) or that Diff(f) = Q_{1} where deg(Q_{1}) = deg_{i}(Q_{1}) = 2. Since in the latter case that κ(Q_{2})/κ(f(Q_{2})) must be inseparable of degree ≥ 2, it follows that in both cases P_{1} := f(Q_{2}) ∈ E(K) is K-rational. Thus, the right hand side of (18) is a divisor which is defined over K, and so it follows from (18) (applied to f\overline{K}) that this equation already holds over K. In particular, (16) holds with P_{f} = P_{1}.

We can use the above results to prove the results stated in the introduction.

Proof of Theorem 1.1. By Proposition 2.3 it is enough to show that Φ_{C} is surjective. Thus, let E ≤ J_{C} be an elliptic subgroup with embedding h : E → J_{C}. Choose a Galois extension K′/K such that C(K′) ≠ ∅. Then by (the proof of) Proposition 2.3 we know that there is an elliptic subcover f′ : C_{K′} → E_{K′} such that (f′)^{∗} ∘j_{E_{K′}} = h_{K′}. By Theorem 4.8 we know that there exists an x ∈ E(K′) such that f′′ := T_{x} ∘ f′ is normalized. Since (f′′)^{∗} = (f′)^{∗}, it follows from Corollary 4.4 that there is a (normalized) subcover f : C → E such that f′′ = f_{K′}, and thus f^{∗}J_{E} = h(E), so Φ_{C} is surjective.

Proof of Corollaries 1.2, 1.3 and 1.4. In view of Theorem 1.1, these follow directly from Corollary 2.7(b) and Remark 2.8.

Acknowledgements I would like to thank the referee for the useful comments and in particular for posing the question about counterexamples to the extension of the main theorem to curves of odd genus. In addition, I gratefully acknowledge receipt of funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).

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