Richelot isogenies

Suppose that the embedding corresponding to v is real and view C as em-bedded into the real projective plane usingv, given by an equation satisfying H = 0. Moreover suppose that all the roots of F(X,1) are real. Bost and Mestre show in [11] how to construct a sequence of mapsφ_n−1:C_n−→C_n−1 on genus 2 curves C_n :Y² =f_n(X), where C₀ =C and f₀(X) =F(X,1), that deform C into a singular curve with 3 nodes. The roots of the poly-nomialsf_n converge quadratically to the x-coordinates of these points and this approach provides a quick method for the computation of the period matrixτv.

The maps φ_n induce isogenies on the corresponding Jacobians, which are called Richelot isogenies. See [20, Chapter 9] for a more general account of these classical maps. Flynn has found explicit formulas for the induced maps on the Kummer surface in [42], so it seems quite promising to use these formulas for an analog of the isogeny method of Bost and Mestre described in Section 2.4, since we know from Proposition 3.24 how the canonical local height changes under isogenies.

Suppose thatP₀ ∈J(R) lies on the connected component of the identity (otherwise there might be no real preimage ofP₀ underφ₀) and (P_n)_nis the sequence of points defined byP_n−1=φ_n(P_n). We supposeκ_i(P_n)6= 0 for all n, so that we can use ˆλ_i,v, since we need a canonical local height associated with some divisor – the sequence of canonical local heights on Kummer coordinates does not converge, unless we use a consistent normalization.

This is problematic, but what is worse is that the sequence of canonical local heights (ˆλ_Di,v(P_n))_nonly converges linearly with convergence factor 2, coming from Proposition 3.24 and the fact that the roots of the f_n con-verge quadratically. So this is slower than the series approach which has convergence factor 4; moreover finding the preimage of a set of Kummer co-ordinate using Flynn’s formulas involves computing inverses of 4×4 matrices

and several square roots and is thus slower in general than an application ofδ.

Jacobian threefolds

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4.1 Embedding the Kummer variety

Let l denote a field of characteristic char(l) 6= 2. In order to generalize the results from the previous chapter we need to find an embedding of the Kummer variety associated to the Jacobian of a smooth projective curve of genus 3 into projective space of dimension 2^g −1 = 7. Since not all those curves are hyperelliptic, we first restrict to those which are. This is reasonable because all genus 2 curves are hyperelliptic and so we expect that the results for Jacobian surfaces generalize more easily to Jacobians of other hyperelliptic curves.

Since the genus is odd there is another complication that we have not encountered so far. Recall that in Sections 3.1 and 3.3 we constructed the Kummer surface by finding an embedding using the fact that generic points of the Jacobian J of a smooth projective genus 2 curve C are represented by divisors of the form

(P₁) + (P₂)−2(∞) or

(P₁) + (P₂)−((∞⁺) + (∞⁻)),

where P₁, P₂ ∈ C, depending on whether there exists a unique point at infinity or not. Therefore generic points on the Jacobian can be represented using unordered pairs of points on the curve.

In the present situation, we have an analogous result if C is a smooth projective genus 3 curve over l with Jacobian J and an l-rational Weier-strass point, which we can assume to be at infinity. Then we can find a representative of the form

(P₁) + (P₂) + (P₃)−3(∞), (4.1) whereP1, P2, P3 ∈C and this representation is unique for generic points on J. Here generic means that if we have a representation (4.1), then P_i 6=∞ and we haveP_i6=P_j⁻ for all distincti, j∈ {1,2,3}, whereQ7→Q⁻ denotes the hyperelliptic involution on C.

However, in the complementary case we have to consider representatives of the form

(P₁) + (P₂) + (P₃) + (P₄)−2((∞⁺) + (∞⁻)),

and these are not unique, even for generic points. In this more general situation there exists a different approach due to Stoll, see [99], which also contains some ideas for the embedding of the Kummer threefold in the non-hyperelliptic case.

Therefore we first restrict to curves having anl-rational Weierstrass point at infinity. Consider an affine equation of the form

Y² =F(X,1) (4.2)

where F(X, Z) =

f₀Z⁸+f₁XZ⁷+f₂X²Z⁶+f₃X³Z⁵+f₄X⁴Z⁴+f₅X⁵Z³+f₆X⁶Z²+f₇X⁷Z (4.3) is a binary octic form inl[X, Z] without multiple factors with deg(F(X,1)) = 7. LetCdenote the hyperelliptic curve of genus 3 given by the smooth pro-jective model of (4.2). Let J be the Jacobian of C. An embedding of the Kummer variety K =J/{±1} of J can be given by a basis of L(2Θ);

such a basis has been found by Stubbs in [100] and we reproduce it here.

Suppose we have a generic point, where for the remainder of this section generic means thatP is represented by an unordered triple of affine points (x₁, y₁),(x₂, y₂),(x₃, y₃)∈C, where allx_i are pairwise distinct. An embed-ding of the Kummer threefold is given by

κ(P) = (κ1(P), . . . , κ8(P)), where the functionsκ₁, . . . , κ₈ are given by

κ₁ = 1,

κ₂ = x₁+x₂+x₃, κ₃ = x₁x₂+x₁x₃+x₂x₃, κ4 = x1x2x3,

κ₅ = b²₀−f₇κ³₂+f₇κ₃κ₂−f₆κ²₂+ 3f₇κ₄+ 2f₆κ₃,

κ₆ = κ₂b²₀+ 2b₀b₁−f₇κ⁴₂+ 3f₇κ₃κ²₂−f₆κ³₂−f₇κ²₃−f₇κ₄κ₂+ 2f₆κ₃κ₂

−f5κ²₂+ 2f₅κ₃,

κ₇ = b²₁−κ₃b²₀+f₇κ₃κ³₂−2f₇κ²₃κ₂+f₆κ₃κ²₂+f₇κ₄κ₃−f₆κ²₃+f₅κ₃κ₂

−3f5κ4,

κ₈ = κ₂b²₁+ 2κ₃b₀b₁+κ₄b²₀+f₇κ²₃κ²₂−f₇κ³₂κ₄+f₇κ₂κ₃κ₄−f₇κ³₃ +f₆κ²₃κ₂−f₆κ₄κ²₂+f₅κ²₃−f₅κ₄κ₂.

Here we have

b₀ = (x₁y₂−x₂y₁−x₃y₂+x₃y₁−x₁y₃+x₂y₃)/d, b1 = (x²₃y2−x²₃y1+x²₂y1+y3x²₁−y2x²₁−y3x²₂)/d,

d = (x₁−x₂)(x₁−x₃)(x₂−x₃).

We also provide formulas for the values of κ(P) when P is not of the form considered above, because apparently they do not exist in the literature. For this we first notice that in the generic case b₀ and b₁ satisfy that

B(X, Z) =b₂Z⁴+b₁XZ³+b₀X²Z² (4.4)

for someb₂∈l, where the Mumford representation of P is (A(X, Z), Y −B(X, Z)),

see the discussion in Section 5.2.2. We can use this observation for our generalization ofκ. If P still has a unique representative of the form (4.1), where now at least two of thexi are equal, then the formulas for κ remain valid ifb₀ and b₁ are not given by the explicit formulas above, but rather as coefficients ofB(X, Z) as in (4.4).

In order to find formulas whenP has a unique representative of the form ((x₁, y₁)) + ((x₂, y₂))−2(∞)

we first assume that P is generic and satisfies x₁x₂x₃ 6= 0 and write the κ_i in terms of (w₁, z₁),(w₂, z₂),(w₃, z₃), where z_j = 1/x_j and w_j = y_j/x_j. We then multiply by the common denominator and setw₃ = 0. Assuming x₁ 6=x₂ we find

κ₁ = 0, κ₂ = 1, κ₃ = x₁+x₂, κ₄ = x₁x₂,

κ5 = f5+ 2f6κ3+f7κ²₃+ 2κ4f7, κ₆ = f₄+f₅κ₃−κ₄f₇κ₃,

κ₇ = −f4κ₃−3κ₄f₅+κ²₄f₇,

κ₈ = (f₃κ³₃+f₁κ₃+f₂κ²₃+ 2f₀−2y₁y₂+κ₄f₄κ²₃−3κ₄f₃κ₃−2κ₄f₂ +κ²₄f₅κ₃−2κ²₄f₄+κ³₄f₇κ₃+ 2κ³₄f₆)/(x₁−x₂)².

For the casex₁ =x₂ it suffices to use the sameκ₁, . . . , κ₇ and

κ₈=b²₁−(κ₄−κ²₃)(−2κ4f₇κ₃−κ₄f₆+f₇κ³₃+κ²₃f₆+f₅κ₃+f₄), whereb₁ is as in (4.4).

Now consider points represented by ((x₁, y₁))−(∞).

We first look at quotients of the formκ_i(P)/κ₅(P), whereP is again assumed generic, and then take the limit (x2, y2)→(x3,−y3).The result is

κ(P) = (0,0,0,0,1,−x1, x²₁, x³₁).

A similar argument shows that we have

κ(0) = (0,0,0,0,0,0,0,1).