The Problem

Let F (X, Z) ∈ ℤ[X, Z] be homogeneous of degree 2g + 2 and without multiple factors; then

y^{2} = F (x, 1)

defines a hyperelliptic curve C over ℚ

Problem 1. Find a ‘better’ equation for C (with ‘smaller’ coefficients)!

Since isomorphisms between hyperelliptic curves come from automorphisms of ℙ^{1}, this means the following.

Problem 2. Find a, b, c, d, e ∈ ℚ with ad ≠ bc and e ≠ 0 such that

F_{1}(X, Z) = e^{2} F (aX + bZ, cX + dZ)

has integral coefficients, ‘smaller’ than those of F.

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Two Subproblems

The problem of reducing F splits into two subproblems.

Subproblem 1 (p-adic reduction). Minimise p-adic valuations of the invariants (or of the discriminant) of F.

Subproblem 2 (reduction over ℤ). Reduce F using unimodular transformations (a, b, c, d ∈ ℤ, ad − bc = 1, e = 1).

There exists an algorithm for Subproblem 1, see Q. Liu: Modèles entiers des courbes hyperelliptiques …, Trans. Amer. Math. Soc. 348, 4577–4610 (1996).

We will focus on Subproblem 2 in this talk.

The results we present were obtained in joint work with John Cremona.

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Positive Definite Quadratic Forms

Since Gauss, a positive definite quadratic form

F (X, Z) = a X^{2} + b XZ + c Z^{2}

is called reduced, if its ‘zero’ z(F) ∈ H is in the usual fundamental domain F of SL(2, ℤ) on H. (H is the upper half-plane; F (z(F), 1) = 0)

Since the map F ↦ z(F) is SL(2, ℝ)-covariant:

z(F ∘ γ) = γ^{−1} · z(F) for γ ∈ SL(2, ℝ),

we can reduce F by finding γ ∈ SL(2, ℤ) such that

γ · z(F) ∈ F ;

then F ∘ γ^{−1} is reduced.

Thus, each SL(2, ℤ)-orbit of forms has a unique reduced representative, and the coefficients of the reduced form are the smallest in the orbit (in a precise sense).

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The Basic Idea

We generalise to forms of higher degree. Let ℝ[X, Z]′_{n} denote the set of forms of degree n without multiple factors. Aim. Define an SL(2, ℝ)-covariant map

ℝ[X, Z]′_{n} ⟶ H , F ⟼ z(F)

such that F is ‘small’ when z(F) ∈ F. Then we call F reduced if z(F) ∈ F. We can reduce a given F by the same algorithm as that used for positive definite quadratic forms. We can extend this to complex coefficients:

ℂ[X, Z]′_{n} ⟶ H^{3} , F ⟼ z(F)

should be SL(2, ℂ)-covariant. (H^{3} is upper half-space.)

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Degrees 3 and 4

For geometric reasons, there is a unique such map for forms of degrees 3 and 4. Any set of 3 or 4 distinct points in ℂ has a non-trivial stabiliser in SL(2, ℂ), and this stabiliser has a unique fixed point in H^{3}. z(F) must be the point associated to the roots of F in this way. A formula for real forms F.

z_{0}(F) = z(Q_{0}(F)) , where

Q_{0}(F)(X, Z) = ∑_{α:f(α)=0} \frac{1}{|f′(α)|^{2/(n−2)}}(X − αZ)(X − ᾱZ)

(We use the convention f(x) = F (x, 1), and n = deg F.) Note. This defines a covariant map on real forms of any degree n > 2. So we can use this to define

F is Q_{0}-reduced ⟺ z_{0}(F) ∈ F .

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Higher Degrees — Julia’s Approach

For degrees n ≥ 5, there are many covariant maps F ↦ z(F). How do we find a good one? Gaston Julia (1917) suggests to take

z(F) = z(Q(F)) , where

Q(F)(X, Z) = ∑_{α:f(α)=0} t_{α}(X − αZ)(X − ᾱZ)

with suitable t_{α} > 0. (The possible z(F) ∈ H then form the interior of the convex hull of the roots α.) When z(F) ∈ F, then the coefficients of F can be bounded in terms of

θ = \frac{F (1, 0)^{2}| disc Q(F)|^{n/2}}{∏_{α} t_{α}} .

So we want to choose t_{α} such that θ is minimised.

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Minimising θ

Let

F (X, Z) = a_{0} ∏_{j=1}^{n}(X − α_{j}Z) .

Then

θ = \frac{a_{0}^{2} (∑_{j<k} |α_{j} − α_{k}|^{2}t_{j}t_{k})^{n/2}}{∏_{j=1}^{n} t_{j}} .

We can prove that (for n ≥ 3) this is strictly convex from below on V = {(t_{1}, …, t_{n})|t_{j} > 0, t_{1} … t_{n} = 1} and approaches infinity when one of the t_{j} → ∞.

Conclusion. θ has a unique critical point on V, and it provides a global minimum.

Conclusion. The map F ↦ z(F) = z(Q(F)), where Q(F) is defined using the minimsing t_{j}, is covariant.

We define

F is reduced ⟺ z(F) ∈ F .
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Another Description

Define

F̃(t, u) = \frac{a_{0}^{2} ∏_{j}((t − α_{j})(t − ᾱ_{j}) + u^{2})}{u^{n}}

for t ∈ ℝ, u > 0. If (t, u) is a critical point of F̃, then setting

t_{j} = \frac{u^{2}}{(t − α_{j})(t − ᾱ_{j}) + u^{2}}

gives a critical point of θ and ‘vice versa’. Conclusion. z(F) = t + iu, where (t, u) is the unique critical point of F̃; at this point, F̃ has a global minimum, which is θ (up to a factor of n^{n}). We can use this to find z(F) numerically for any given F.

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A Geometric Interpretation

When t ∈ ℂ, u > 0 and α ∈ ℂ, then

log \frac{|t − α|^{2} + u^{2}}{u}

is a measure for the hyperbolic distance of (t, u) ∈ H^{3} from the boundary point α (up to an additive constant).

So log F̃(t, u) measures the sum of the distances of (t, u) from the roots of F.

Conclusion. z(F) = t + iu, where (t, u) ∈ H^{3} is the point minimising the sum of the distances to the roots of F.

Conclusion. This point (t, u) is characterised by the fact that the sum of the unit tangent vectors pointing towards the roots of F vanishes.

(This is easy to see when the degree is 3 or 4.)

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The Reduction Algorithm

Input: F = a_{0}X^{n} + a_{1}X^{n−1}Z + ⋯ + a_{n}Z^{n} ∈ ℤ[X, Z]′_{n} Output: A reduced form SL(2, ℤ)-equivalent to F.

1. Compute z = z(F).

2. While z ∉ F do

2.1. Let m = Round(Re(z)).

2.2. Set F (X, Z) = F (X + mZ, Z), z = z − m.

2.3. If |z| < 1, then set F (X, Z) = F (Z, −X), z = −1/z.

Remarks.

• We can use this with z_{0}(F) in place of z(F).

(Faster, since z_{0}(F) is computed more easily.)

• Be careful with exit condition to avoid infinite looping.

• Re-compute z(F) from time to time to compensate precision loss.

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Example

C : y^{2} = 19x^{8} − 262x^{7} + 1507x^{6} − 4784x^{5} + 9202x^{4} − 10962x^{3}
+ 7844x^{2} − 3040x + 475

(Taken from H.-J. Weber’s thesis (Essen, 1996).) For the first reduction steps, we use z_{0}(F). The roots of f are

0.42798171, 1.30152156, 1.31947230, 4.31651243, 1.69098301 ± 0.72287100 i, 1.52100984 ± 0.12866975 i .

From the roots, we compute Q_{0}(F) and its root

z = z_{0}(F) = 1.38323301 + 0.31233552 i .

m = 1, so we do a shift and replace f with

f_{1}(x) = f(x + 1)

= 19x^{8} − 110x^{7} + 205x^{6} − 180x^{5} + 47x^{4} + 40x^{3} − 35x^{2} + 10x − 1 and z with z_{1} = z − 1 = 0.38323301 + 0.31233552 i. |z_{1}| < 1, so we invert f_{1} to get

f_{2}(x) = x^{8}f_{1}(−1/x)

= −x^{8} − 10x^{7} − 35x^{6} − 40x^{5} + 47x^{4} + 180x^{3} + 205x^{2} + 110x + 19 and set z_{2} = −1/z_{1} = −1.56792167 + 1.27785869 i.

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Example, continued

Now m = −2, so

f_{3}(x) = f_{2}(x − 2)

= −x^{8} + 6x^{7} − 7x^{6} − 12x^{5} + 27x^{4} − 4x^{3} − 19x^{2} + 10x − 5

and z_{3} = z_{2} + 2 = 0.43207833 + 1.27785869 i ∈ F, so F_{3} is Q_{0}-reduced.

Now we use Julia’s covariant z(F). We find the roots of f_{3} and use some numerical method to compute

z_{4} = z(F_{3}) = 0.64189877 + 1.18525166 i .

This is not in F, and m = 1 in our algorithm. Hence we set

f_{5}(x) = f_{3}(x + 1) = −x^{8} − 2x^{7} + 7x^{6} + 16x^{5} + 2x^{4} − 2x^{3} + 4x^{2} − 5

and z_{5} = z(F_{5}) = z_{4} − 1 = −0.35810123 + 1.18525166 i ∈ F, so F_{5} is reduced.

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