Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves

In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $E_1\times E_2$ without complex multiplication are integers. By studying elliptic curves on $E_1\times E_2$ we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $E_1\times E_2$ are integers if and only if the minimal degree of an isogeny $E_1\to E_2$ equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $E_1\times E_2$.


Introduction
For an ample line bundle L on a smooth projective variety X, the Seshadri constant of L at a point x ∈ X is by definition the real number ε(L, x) = inf L · C mult x (C) C irreducible curve through x .
On abelian varieties, ε(L, x) is independent of the chosen point x. Moreover, one knows by [3] that on abelian surfaces Seshadri constants are always rational numbers.
In the present paper we focus on the question of when all Seshadri constants ε(L) on a given abelian surface are integers. This question was first approached in [2], where it was shown that integrality of Seshadri constants on an abelian surface X is equivalent to requiring that for every ample line bundle L on X, either √ L 2 is an integer or ε(L) is computed by an elliptic curve. From work of Bauer and Schulz [4] one knows that the latter condition is always satisfied when X is a self-product E × E of an elliptic curve without complex multiplication. In the present paper we study the more general situation that X is a product E 1 × E 2 of two isogenous elliptic curves without complex multiplication. One might hope that the Seshadri constants on these surfaces behave in a similar way as those on a self-product, since the surfaces E 1 × E 1 and E 1 × E 2 are isogenous. As our result will show, however, Seshadri constants exhibit quite unexpected behaviour under isogenies.
A first indication of this phenomenon follows from work of Kani [11], who studied the question of when E 1 × E 2 is the Jacobian of a smooth genus 2 curve, i.e., when Keywords: abelian surface, elliptic curve, Seshadri constant, isogeny, principal polarization. Mathematics Subject Classification (2010): 14C20, 14H40, 14H52, 14J99, 14K02, 14K99, 11H55. . E 1 × E 2 carries an irreducible principal polarization. His result states that E 1 × E 2 is not a Jacobian if and only if the minimal degree d of an isogeny E 1 → E 2 satisfies d ∈ {1, 2, 4, 6, 10, 12, 18, 22, 28, 30, 42, 58, 60, 70, 78, 102, 130, 190, 210, 330, 462} and for at most one more unknown value d > 462. Since the Seshadri constant of an irreducible principal polarization is 4 3 by [15], it follows that on E 1 × E 2 non-integer Seshadri constants occur in any event when d is not contained in the list above. Our result shows that non-integer Seshadri constants are even more frequent: Theorem 1 All Seshadri constants on E 1 × E 2 are integers if and only if the minimal degree d of an isogeny E 1 → E 2 equals 1 or 2.
In particular, the non-integer Seshadri constants predicted by the theorem whenever d 3 must come from polarizations other than principal polarizations whenever d is contained in the list above.
Our proof is based on the idea to study the intersection numbers of line bundles with elliptic curves on E 1 × E 2 , which allows us to rephrase the problem into a purely numerical question in terms of quadratic forms. We will solve that numerical problem by using reduction theory of binary quadratic forms. This approach also enables us to characterize irreducible principal polarizations on E 1 × E 2 in terms of quadratic forms. Recall that two polarizations L 1 and L 2 are equivalent if there exists an automorphism ψ of X such that ψ * L 1 is algebraically equivalent to L 2 , and denote the set of isomorphism classes of principal polarizations on X = E 1 × E 2 by P (X). We will see in Sect. 4 that there is a bijection between P (X) and a set of certain quadratic forms: We can characterize irreducible principal polarizations in these terms: Theorem 2 A matrix M as above corresponds to a class of reducible principal polarizations if and only if B = 0.
In particular, there exists an irreducible principal polarization on E 1 × E 2 if and only if d can be written as As an application we can use Thm. 2 to give an alternative proof of Kani's result mentioned above. Throughout this paper we will work over the complex numbers and on the abelian surface X = E 1 × E 2 , and we denote by ϕ : E 1 → E 2 an isogeny of smallest degree d.

Preliminaries
Let E 1 and E 2 be two isogenous elliptic curves without complex multiplication. Throughout this paper we will work on the abelian surface X := E 1 × E 2 and we fix an isogeny ϕ : E 1 → E 2 of smallest degree d. On the product surface X, denote by F 1 = {0} × E 2 and F 2 = E 1 × {0} the fibers of the projections and by ∆ the graph of the isogeny ϕ. The classes of these three elliptic curves on X form a basis of the Néron-Severi group NS(X) (see [16,Thm. 22] or [5,Thm. 11.5.1]).
Note that the determinant of the intersection matrix coincides with the discriminant of the Néron-Severi group on X (see [1,Chp. 3]). Thus implies that (F 1 , F 2 , ∆) is a basis of NS(X).
Proof. We argue as in [2,Prop. 2.1]. All three curves are elliptic, so we have F 2 1 = F 2 2 = ∆ 2 = 0. As each curve intersects the other ones transversely, it is enough to count the number of intersection points. So we have since these curves intersect only in the origin. For F 2 and ∆ one has and this shows that we need to count the number of solutions x ∈ E 1 of the equation ϕ(x) = 0. But this number is equal to the degree of the isogeny ϕ, so we get As a first application, we will compute Seshadri constants for line bundles in the cone generated by (F 1 , F 2 , ∆): with non-negative coefficients a i . Then one has Proof. Let D be the divisor a 1 F 1 + a 2 F 2 + a 3 ∆, and let C be any irreducible curve C passing through 0, which is not a component of D. As D is effective, we have This implies that ε(L) is computed by one of the curves F 1 , F 2 or ∆. Their intersection numbers with L are given by the intersection matrix in Prop. 1.1 and this yields the assertion. Remark 1.3 Let ϕ : E 2 → E 1 be the isogeny corresponding to ϕ such that the maps ϕ • ϕ and ϕ • ϕ are the multiplication by d. Then the previous arguments can be used to see that the triple (F 1 , F 2 , ∆) with ∆ := {x ∈ X | x 1 = ϕ(x 2 )} also forms a basis of NS(X). Then, we can analogously formulate Prop. 1.2 for line bundles in the cone generated by (F 1 , F 2 , ∆). In general the resulting cones will not coincide since it is possible to get negative coefficients by changing bases: In fact, the cones coincide if and only if d = 1, that is, if E 1 and E 2 are isomorphic.
For our purpose it will be useful to change the basis (F 1 , F 2 , ∆) of the Néron-Severi group by choosing an element which is orthogonal to F 1 and F 2 . We define ∇ := ∆ − dF 1 − F 2 , then the triple (F 1 , F 2 , ∇) forms a basis of NS(X) and the intersection matrix is given by  Then L is ample if and only if the two inequalities are satisfied.
Proof. By the Nakai-Moishezon criterion for abelian varieties [5,Cor. 4.3.3] a line bundle L is ample if and only if both L 2 = 2(a 1 a 2 − da 2 3 ) and the intersection of L with any fixed ample line bundle L 0 are positive. By choosing the ample line bundle L 0 := O E×E (F 1 + F 2 ) the intersection of L with L 0 is given by a 1 + a 2 . Now assume that L 2 is positive, then a 1 and a 2 are either both positive or both negative. Thus, the sum a 1 + a 2 is positive if and only if a 1 is positive.

Numerical classes of elliptic curves on X
In this section we will determine the intersection number of line bundles with elliptic curves. To this end, we need to know all elliptic curves on E 1 × E 2 . In [9] Hayashida and Nishi described the elliptic curves on E n and with the same argument it immediately follows: By the previous lemma we know that for every elliptic curve N on X, there are integers (a, b) = (0, 0) such that N is a translate of Note that for any multiple (λa, λb) we have N a,b = N λa,λb = m λ (N a,b ), where m λ describes the multiplication with λ, which is a map of degree λ 2 that maps N a,b surjective onto itself. Next, we want to describe the intersection numbers of elliptic curves with line bundles. For this, we have to determine the numerical class of the elliptic curves N a,b . As N a,b and F 1 intersect transversely for a = 0, we have where σ a,b : E 1 → N a,b is the map x → (ax, bϕ(x)). But the numerator is equal to the degree of the map a : E 1 → E 1 , x → ax, which has degree a 2 . If a = 0, then N a,b = F 1 and, therefore, N a,b · F 1 = 0 = a 2 deg(σ a,b ) . With the same arguments we obtain .
The coefficients of the numerical representation of N a,b are then given by and, thus, we have To fully understand the intersection numbers we have to determine the degree of the map σ a,b . For this, we use the following criteria to identify elliptic curves on abelian varieties (see [11,Prop. 2

.1, Prop. 2.3]):
Proposition 2.2 ( [11]) Let A be an abelian surface and C 1 , C 2 ⊂ A be two irreducible curves on A. Then C 1 · C 2 0. Furthermore, C 1 · C 2 = 0 if and only if C 1 is an elliptic curve and C 2 a translate of C 1 . With these criteria we can calculate the degree of σ a,b and, thus, the numerical class of elliptic curves. Lemma 2.4 Let N ⊂ X be an elliptic curve and let σ a,b : and, therefore, the numerical class of N is given by Proof. By our previous arguments the numerical class of N a,b is given by Since all the coefficients of the representation must be integers, it follows that So, there exists a positive integer λ such that If we show that λ = 1, then the assertion follows. By using the previous equation in the expression for N a,b we get where N is an element of NS(X), since all the coefficients are integers. Next, we make use of Prop. 2.3 to show that N is a positive multiple of an elliptic curve. Since N a,b is an elliptic curve by assumption, we can apply Prop. 2.3 to N a,b and by using N ≡ 1 λ N a,b we see that Thus, by Prop. 2.3 we have N ≡ mE for an elliptic curve E and m 1. So, we obtain N a,b ≡ mλE.
Since N a,b · E = 1 mλ N 2 a,b = 0, it follows from Prop 2.2 that E is a translate of N a,b and, therefore, their numerical classes coincide. But since m and λ are positive integers, it follows that m = λ = 1. So, we have deg(σ a,b ) = gcd(a 2 , b 2 d, ab) as claimed.
The natural question might arise whether it is possible to find for any given pair (a, b) another pair (a ′ , b ′ ) such that the elliptic curves N a,b and N a ′ ,b ′ coincide and such that the degree of the map σ a ′ ,b ′ : E 1 → N a ′ ,b ′ is 1. While this is apparently true for d = 1 by dividing a and b by gcd(a, b), it is not true for d > 1, since it would imply that there exists a map of degree 1, i.e. an isomorphism, of E 1 to N 0,1 ∼ = E 2 . Now we have available the required tools to calculate the intersection numbers of elliptic curves with line bundles: Proposition 2.5 Let L be a line bundle with L ≡ a 1 F 1 + a 2 F 2 + a 3 ∇ and N a,b an elliptic curve on X. Then the intersection number of L and N a,b is given by Proof. This follows by explicitly calculating the intersection of the numerical class of N a,b given in Lemma 2.4 with the numerical class of L.
Remark 2. 6 We denote the matrix associated with L ≡ a 1 F 1 +a 2 F 2 +a 3 ∇ ∈ NS(X) through Prop. 2.5 by Note that the map L → M L is an isomorphism between the abelian groups It turns out that this representation enables an ampleness criterion for line bundles on X: (i) L is ample.
(ii) M L is positive definite.
(iii) The intersection L · E is positive for every elliptic curve E on X.
Proof. This is an immediate consequence of Prop. 1.4 and Prop. 2.5.

Integrality of Seshadri constants on X
In this section we will provide a complete answer, in terms of d, to the question whether there exists a non-integer Seshadri constant on X or not. For this, we first develop a criterion to check if all Seshadri constants are computed by elliptic curves.
Obviously, if all Seshadri constants are computed by elliptic curves, then all Seshadri constants are integers by definition. Our starting point is: The following are equivalent: (i) All Seshadri constants on X are computed by elliptic curves.
(ii) Every ample line bundle L on X has a weakly submaximal elliptic curve, i.e., there exists an elliptic curve E on X such that L · N √ L 2 .
(iii) For every positive definite matrix of the form Proof. The implication (i) ⇒ (ii) follows from the definition of Seshadri constants and the upper bound ε(L) √ L 2 . The implication (ii) ⇒ (i) is a result from Schulz. The argument can be found in the proof of [4,Thm. 4.5].
The equivalence of (ii) ⇐⇒ (iii) is a consequence of Prop. 2.5 and the fact that it is enough to consider coprime pairs (a, b), since we have L·N a,b = L·N λa,λb for any multiple λ ∈ N. Moreover, if a and b are coprime, then gcd(a 2 , b 2 d, ab) = gcd(a, d).
In Prop. 3.1 we have translated the question whether all Seshadri constants are computed by elliptic curves into a purely numerical problem. To progress further, we will use results for binary quadratic forms and apply them to our setting. We start by briefly collecting the relevant language and facts. Recall that a (binary) quadratic form Q : Z × Z → Z is given by where A, B and C are integers. We will denote Q by the triple (A, B, C). A quadratic form Q is called primitive if gcd(A, B, C) = 1. Two quadratic forms Q and P are called (properly) equivalent if there exists a matrix S ∈ GL 2 (Z) such that P (x, y) = Q(S(x, y)) (and, respectively det(S) = 1). It is more common to use proper equivalence with quadratic forms, since the classes of primitive quadratic forms of a fixed determinant form a group. The crucial ingredient are so-called reduced forms and their properties:  It follows that any positive definite quadratic form is equivalent to a unique form (A, B, C) with 0 B A C. This will be used in the last section. Now, we turn our attention to the two relevant properties which we obtain by using reduced forms: .
With this, we can refine Prop. 3.1 in terms of reduced positive definite forms.
Proposition 3.5 The following is equivalent: (i) All Seshadri constants on X are computed by elliptic curves.
(ii) For every reduced positive definite quadratic form Q = (A, B, C) and every Proof. To begin with, we will determine how a base change affects the inequality given in Prop. 3.1. Assume that we have a positive definite matrix M and for S = α β γ δ ∈ SL 2 (Z) we set R := S T M S. Then, by changing coordinates a coprime pair (x, y) corresponds to the coprime pair (v, w) := S −1 (x, y) T and it follows that Thus, the change of basis leads to the inequality stated in the Proposition. The implication (ii) ⇒ (i) follows from the fact that every positive definite matrix appearing in Prop. 3.1 is by Thm. 3.3 equivalent to a unique reduced form. Thus, if the inequality holds for all reduced quadratic forms and all S ∈ SL 2 (Z), then it already holds for every positive definite form.
The implication (i) ⇒ (ii) is less apparent, since Prop. 3.1 gives a statement on positive definite quadratic forms given by  The Theorem implies that ε * (L) = ε(L) and hence ε(L) can be effectively computed by Prop. 3.6.
Proof. We will show that Prop. 3.5 applies in this situation. First note that, since the inequality in Prop. 3.5 does not depend on γ and δ, it is enough to consider coprime pairs (α, β). Let Q = (A, B, C) be a reduced positive definite quadratic form. We have to show that there exists a coprime pair (a, b) such that Q(a, b) gcd(aα + bβ, d) First, assume that d = 1. Then, by Prop. 3.4 (i) it follows that we have for (1, 0) the inequality and, thus, we have found a coprime pair. Next, we treat the case d = 2. We claim that at least one of the pairs (1, 0), (0, 1), (1, ±1) satisfies the inequality ( * ). We will discuss three cases dependent on the divisibility of α and β by 2. Note that α and β cannot be both divisible by 2 at the same time, since they are coprime.
For a more thorough discussion of idoneal numbers and the existence of the additional number d * we refer to [12] and [13].
We will now show that there in fact exists an ample line bundle with a noninteger Seshadri constant for every d 3. (Such line bundles cannot be principal polarizations in the cases listed in Thm. 3.8.) For this, we will use Thm. 2 from [2]. So, the assertion follows, if we show that there exists a line bundle L such that ε(L) is not computed by an elliptic curve and that √ L 2 is not an integer.
Theorem 3.9 If d 3, then there exists a line bundle on X, which does not have a weakly submaximal elliptic curve.
Proof. We will use Prop. 3.5. So, the issue is to exhibit a reduced positive definite form Q = (A, B, C) and a matrix S = α β γ δ ∈ SL 2 (Z) such that for every coprime pair (a, b) the inequality is satisfied. We will show that the quadratic form Q = (2, 1, d) and the matrix satisfy for every coprime pair (a, b) the even stronger inequality The idea is as follows: We begin by exhibiting two properties a coprime pair (a, b) must satisfy, if it contradicts the inequality ( * ). Then we show that no coprime pair can satisfy both properties at the same time.
First, we observe that it is enough to consider coprime pairs (a, b) with Q(a, b) < 2d , because any coprime pair (a, b) with Q(a, b) 2d will satisfy the inequality ( * ) since the denominator is at most d. Secondly, we claim that it is enough to consider coprime pairs (a, b) = (1, 0) such that To this end, we know by Prop. 3.4 that the two smallest non-zero integers represented by Q with coprime pairs are Q(1, 0) = 2 and Q(0, 1) = d. Since the pair (1, 0) satisfies the inequality ( * ), we may assume that (a, b) = (1, 0) and, thus, Q(a, b) is at least d. If such a coprime pair has gcd(a + b⌈ d 2 ⌉, d) = q < d, then we conclude as for any divisor q of d with q = d the quotient d q is at least 2. Consequently, we are left with coprime pairs (a, b) = (1, 0) such that gcd(a + b⌈ d 2 ⌉, d) = d. Next, we will create a set of pairs which will contain all coprime pairs (a, b) = (1, 0) such that Q(a, b) < 2d. We claim that if |b| > 1, then Q(a, b) is at least 2d. For this, we consider for a fixed b the function x → Q(x, b), which has the minimum in −b 4 . It follows that we have for every x ∈ R and for |b| 2 we obtain Q(x, b) 2d. Thus, we only have to consider the cases b = 1 and b = −1. Now, we calculate the range of a depending on b ∈ {−1, 1} by estimating the possible values of a such that Q(a, b) = 2a 2 +ab+db 2 < 2d holds. We conclude by solving these two quadratic inequalities that It follows that all coprime pairs (a, b) = (1, 0) with Q(a, b) < 2d are contained in (the possibly bigger set): The crucial point is that if d 4, then we can derive the following bounds for a + b⌈ d 2 ⌉: If b = 1, then (This is also the moment where the argument hinges on the choice of the matrix S.) As a consequence, the sum a + b⌈ d 2 ⌉ always lies in the interval (0, d) or in (−d, 0) and, therefore, is not divisible by d. So, we conclude that every coprime pair (a, b) satisfies ( * ).
So far we have found line bundles L whose Seshadri constants are not computed by elliptic curves. This concludes the first step of the proof of the converse of Thm. 3.7. It still might be possible that the Seshadri constants are all integers if ε(L) = √ L 2 ∈ Z (see [2, Ex. 1.1]). To exclude this, we have to show that L 2 is not a square number. With that we can deduce the even stronger statement: Proof. We will apply Thm. 2 from [2]. Thus, the claim follows, if we show that there exists a line bundle L such that ε(L) is not computed by an elliptic curve and that √ L 2 is not an integer. To find such a line bundle, we will use the binary quadratic form Q = (2, 1, d) and matrix S = 1 d 2 0 1 ∈ SL 2 (Z) given in Thm. 3.9 and apply the arguments from the proof of Prop. 3.5. So, we consider the scaled quadratic form Q ′ = 2dQ = (4d, 2d, 2d 2 ) together with the matrix S. By using the change of basis S we get the positive definite matrix which by Rmk. 2.6 corresponds to the class of the ample line bundle which has no weakly submaximal elliptic curve by Thm. 3.9. Thus, it is left to show that L 2 is not a perfect square. By using Cor. 2.7 we have for the self-intersection If d is odd, then 16d 2 − 2d ≡ 2 modulo 4 and hence L 2 is not a perfect square. If d is even, then we write d = 2 n d ′ for an odd integer d ′ 1 and n 1. If n is even, then L 2 = 2 n (2 n+4 d ′2 − 2d ′ ). Since 2 n is a square number, it is enough to show that 2 n+4 d ′2 − 2d ′ can not be a perfect square. This, however, can not happen since it is 2 modulo 4.
Lastly, if n is odd, then L 2 = 2 n+1 (2 n+3 d ′2 − d ′ ). Suppose to the contrary that L 2 is a perfect square. Then there exists an integer r such that 2 n+3 d ′2 − d ′ = r 2 . We claim that d ′ already is a perfect square. Let p be a prime number such that p 2n−1 is a divisor of d ′ . It follows that p 2n−1 must be a divisor of r 2 and since r 2 is a perfect square it is divisible by p 2n . Thus, d ′ = 2 n+3 d ′2 −r 2 is divisible by p 2n as well and, therefore, d ′ must be an odd square number. Then, however, 2 n+3 d ′2 − d ′ ≡ 3 modulo 4 and hence L 2 cannot be a square number, which is a contradiction.

Remark 3.11
Note that the application of Thm. 2 from [2] does not yield an explicit line bundle with a fractional Seshadri constant. Thus, the line bundles given in the proof do not necessarily have a non-integer Seshadri constant themselves, but they imply their existence. For such an example see [2,Prop. 2.8].

Classification of Principal Polarizations on X
In this section we will characterize the isomorphism classes of reducible and irreducible principal polarization. We give a brief summary of notation, which can be found in [14]. Two principal polarizations L 1 and L 2 are equivalent if there exists an automorphism ψ of X such that ψ * L 1 ≡ L 2 . We denote the isomorphism classes of principal polarizations on X by Recall that we have the equality dL 2 2 = det(M L ) and, thus, by Rmk. 2.6 a principal polarization L is given by a positive definite matrix M L = a 1 −da 3 −da 3 da 2 with determinant d. Furthermore, since the self-intersection is 2, it follows that the matrix M L is primitive, that is gcd(a 1 , da 2 , da 3 ) = 1. Thus, the quadratic form M L is equivalent to a unique reduced form Q = (A, 2B, C) with 0 B A C and gcd(A, B, C) = 1; we will call such forms principally reduced forms of determinant d. Using ( * ), we now characterize the principally reduced forms of reducible principal polarizations L, i.e., the polarizations such that L ≡ E ′ 1 + E ′ 2 with elliptic curves In other words, irreducible principal polarizations correspond to principally reduced forms with B = 0.
Proof. First, assume that B = 0. Let S = α β γ δ ∈ GL 2 (Z) be a base change such that Q corresponds to a principal polarization L. It is enough to find an elliptic curve such that the intersection with L is 1. It follows for that d divides m 2,2 = β 2 A + δ 2 C and m 1,2 = αβA + δγC. By using AC = d and the coprime properties of α, β, γ and δ resulting from the fact that the determinant of S is 1, we can deduce that C is a divisor of β and A is a divisor of δ. Therefore, we see that gcd(β, d) = C. Hence, we obtain Thus, the elliptic curve N −β,δ which corresponds to S −1 (0, 1) has intersection number 1 with L. Thus, L is reducible. Let L be a reducible principal polarization. The issue is to find a base change S ∈ GL 2 (Z) such that S T M L S is a diagonal matrix. Since L is reducible, there exists two elliptic curves E ′ 1 and E ′ 2 on X such that L ≡ E ′ 1 + E ′ 2 . By Lemma 2.1 there are coprime pairs (α, β) and (γ, δ) such that E ′ 1 ≡ N α,β and E ′ 2 ≡ N γ,δ with deg(σ α,β ) = gcd(α, d) and deg(σ γ,δ ) = gcd(γ, d). We will show that the matrix S := α γ β δ has the required properties.
For that we first claim that α and γ are coprime. Let p be any prime number and we factorize α = p r α ′ , γ = p s γ ′ and d = p t d ′ . By Lemma 2.4 the numerical classes are given by: If p is a common factor of α and γ, then 1 r, s. Without loss of generality we assume that 1 r s. Then we deduce by explicitly calculating the intersection number N α,β · N γ,δ = 1 that For all t 0 both factors are integers and, moreover, the left factor is at least p. Thus, this equation represents a factorization of 1, but this is impossible and, therefore, α and γ are coprime.
Next, we will show that S has determinant ±1. The map is an isogeny and we have for the degree where the last equality comes from the fact that α and γ are coprime. Thus, the degree of σ α,β × σ γ,δ is at most d. On the other hand, the minimal degree of any isogeny E 1 × E 1 → E 1 × E 2 is at least d and, hence, the gcd(αγ, d) equals d. From the equation given by the intersection N α,β · N γ,δ = 1 we then deduce that Lastly, by using explicit calculations we see that S T M L S = gcd(α, d) 0 0 gcd(γ, d) .
Note that if gcd(α, d) > gcd(γ, d), then we swap the roles E ′ 1 and E ′ 2 . This completes the proof.
To conclude, we briefly return to Kani's result in Thm. 3.8. His proof is based on studying the refined Humbert invariant Case 2.1: If A = 2B, then 4d ′ + B 2 = 2BC and it follows that B is even. Thus, there exists an odd number B ′ and n 1 such that B = 2 n B ′ . Since we have 4d ′ + 2 2n B ′2 = 2 n+1 B ′ C, it follows that n = 1, because otherwise d ′ would be divisible by 2. But then C has to be an even number, since we have d ′ + B ′ = B ′ C. This, however, is impossible, because then we would have gcd(A, B, C) 2.
Case 2.2: If A = C, then we argue as in case 1.2 and deduce that 4d ′ = (C − B)(C + B) is divisible by 8 which is a contradiction since d ′ is odd.