Heights on Projective Space

Definition. Let P = (x_{0} : x_{1} : ⋯: x_{N}) ∈ ℙ^{N}(ℚ). Then its non-logarithmic height is

H(P) = ∏_{v} max{|x_{0}|_{v}, |x_{1}|_{v}, …, |x_{N}|_{v}} ,

and its (logarithmic) height is

h(P) = log H(P) = ∑_{v} log max{|x_{0}|_{v}, |x_{1}|_{v}, …, |x_{N}|_{v}} .

v runs through the places of ℚ, and | · |_{v} is the normalised absolute value (|x|_{p} = p^{−v_{p}(x)} and |x|_{∞} = |x|).

Product formula ⟹ H, h well-defined.

If we choose the coordinates x_{j} for P to be relatively prime integers, then

H(P) = max{|x_{0}|, |x_{1}|, …, |x_{N}|} .

Fact. For every B ∈ ℝ, there are only finitely many P ∈ ℙ^{N}(ℚ) with h(P) ≤ B.

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Heights on Abelian Varieties

Let A be an abelian variety over ℚ, and fix a (suitable) morphism ϕ : A ⟶ ℙ^{N}.

Definition. For P ∈ A(ℚ), we define the non-logarithmic naive height H(P) and the naive height h(P) by

H(P) = H(ϕ(P)) and h(P) = h(ϕ(P)) = log H(P) .

Definition. The (canonical) height of P ∈ A(ℚ) is

ĥ(P) = lim_{n→∞} \frac{h(nP)}{n^{2}} = lim_{m→∞} 4^{−m} h(2^{m}P) .

Definition. We define the global height constant of A to be

γ = sup_{P ∈A(ℚ)}(h(P) − ĥ(P)) < ∞ .

This means we have h(P) ≤ ĥ(P) + γ for all P ∈ A(ℚ).

Fact. ĥ induces a positive definite quadratic form on A(ℚ) ⊗_{ℤ} ℝ.

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What are Heights Good for?

Let T = A(ℚ)_{tors} be the torsion subgroup of A(ℚ). Let Λ = A(ℚ)/T ≅ ℤ^{r} (where r is the Mordell-Weil rank). Then (Λ, ĥ) is a lattice. We can use this additional structure in order to find A(ℚ) (given a finite index subgroup). We need

(1) A bound for the global height constant γ;

(2) An algorithm that computes ĥ(P) for P ∈ A(ℚ);

(3) A method for enumerating all P ∈ A(ℚ) with H(P) ≤ some bound.

We will deal with (1) and (2) in the special case that A is the Jacobian of a genus two curve.

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Jacobians of Genus 2 Curves

Let C be a curve of genus two over ℚ. Then C has an affine model given by an equation of the form

Y^{2} = F (X, 1)

where F (X, Z) ∈ ℤ[X, Z] is homogeneous of degree 6 and without multiple (non-constant) factors.

Let J be the Jacobian of C; it is an abelian surface over ℚ.

We need a map ϕ : J ⟶ ℙ^{N} to define a height on J.

On an elliptic curve E, we can use the x-coordinate E ⟶ ℙ^{1}. In analogy to this case, we identify P and −P on J to obtain the Kummer surface K. It is a hypersurface in ℙ^{3}. We then take

ϕ : J ⟶ K ⊂ ℙ^{3} .

We use a specific embedding K ⊂ ℙ^{3} that depends on F.

We define H, h, ĥ on J(ℚ) via ϕ.

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Local Height Constants I

We cannot bound γ directly, but we can use the usual local-global trick.

The duplication map P ↦ 2P on J induces a morphism δ : K ⟶ K. It can be defined by four homogeneous polynomials (δ_{1}, δ_{2}, δ_{3}, δ_{4}) of degree four with integral coefficients.

Let p be a prime, and let v_{p} be the normalised additive p-adic valuation. For x = (x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℚ_{p}^{4} define v_{p}(x) = min_{j} v_{p}(x_{j}).

Definition. We define ε_{p} : K(ℚ_{p}) ⟶ ℤ_{≥0} by

ε_{p}(x) = v_{p}(δ(x)) − 4 v_{p}(x) .

We lift to a map ε_{p} : J(ℚ_{p}) ⟶ ℤ_{≥0} via J ⟶ K.

Definition. Similarly, we define ε_{∞} : K(ℝ) ⟶ ℝ by

ε_{∞}(x) = 4 max_{j} log |x_{j}| − max_{j} log |δ_{j}(x)|

and lift it to a map ε_{∞} : J(ℝ) ⟶ ℝ.

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Local Height Constants II

Definition. Let v be a place of ℚ. The local height constant at v is

γ_{v} = max_{P ∈J(ℚ_{v})} ε_{v}(P) ∈ ℝ_{≥0} .

We have

h(2P) = 4 h(P) − ε_{∞}(P) − ∑_{p} ε_{p}(P) log p

and hence

ĥ(P) = h(P) − ∑_{n=0}^{∞} 4^{−(n+1)}ε_{∞}(2^{n}P) − ∑_{p} ∑_{n=0}^{∞} 4^{−(n+1)}ε_{p}(2^{n}P) log p .

Lemma.

γ ≤ \frac{1}{3}(γ_{∞} + ∑_{p} γ_{p} log p) .

• The sum is finite.

• We can bound the γ_{v}.

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Quick Bounds

Using representation theory, we can find 10 homogeneous polynomials z_{ν} of degree two over ℚ̄ such that

4\sqrt{disc F} x_{j}^{2} = ∑_{ν=1}^{10} a_{jν}z_{ν}
z_{ν}^{2} = ∑_{i=1}^{4} b_{νi}δ_{i} + b_{ν0}δ_{0}

where the a_{jν} and the b_{νi} are integral and δ_{0} = 0 defines the Kummer surface.

Proposition.

γ_{p} ≤ v_{p}(2^{4} disc F)

and

γ_{∞} ≤ 2 log max_{j}(∑_{ν=1}^{10} |a_{jν}|\sqrt{∑_{i=1}^{4} |b_{νi}|}) − log(2^{4}| disc F |) .

• Refinements are possible.

• Get bounds on max_{P ∈K(ℚ̄_{v})} ε_{v}(P).

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Canonical Height Algorithm (Sketch)

Theorem. Let p be a prime, P, Q ∈ J(ℚ_{p}). Then

ε_{p}(P) = 0 ⟹ ε_{p}(P + Q) = ε_{p}(Q) .

Corollary. Define U_{p} = {P ∈ J(ℚ_{p}) | ε_{p}(P) = 0}. Then

(1) U_{p} ⊂ J(ℚ_{p}) is a subgroup of finite index.

(2) ε_{p} factors through J(ℚ_{p})/U_{p}.

• We can find ∑_{p} ∑_{n} 4^{−(n+1)}ε_{p}(2^{n}P) log p in a finite number of steps.

• We can compute ĥ(P) to any desired accuracy. (Use floating-point arithmetic for ε_{∞}(2^{n}P).)

Recall that

ĥ(P) = h(P) − ∑_{n=0}^{∞} 4^{−(n+1)}ε_{∞}(2^{n}P) − ∑_{p} ∑_{n=0}^{∞} 4^{−(n+1)}ε_{p}(2^{n}P) log p .

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Better Bounds for γ_{p}

There are precise conditions for when a point on K(ℚ_{p}) has ε_{p} > 0 and conditions that have to be satisfied for the point to lift to J(ℚ_{p}).

• We can use a p-adic search procedure to find γ_{p} exactly.

We also get a general result.

Theorem. Let p be an odd prime and suppose that the reduction mod p of F has zeroes of multiplicities (1, 1, …, 1, k) in ℙ^{1}(𝔽̄_{p}), with 1 ≤ k ≤ 5.

If v_{p}(disc F) = k − 1, then γ_{p} = 0.

This is always the case (with k = 2) when v_{p}(disc F) = 1.

In practice, this means that large bad primes do not make trouble.

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How to Find A(ℚ)

We assume we have got

(1) A bound for γ;

(2) An algorithm that computes ĥ on A(ℚ);

(3) An algorithm that enumerates points on A(ℚ)

of naive height ≤ B;

(4) Generators of a finite index subgroup U ⊂ A(ℚ)

We have discussed (1) and (2) for Jacobians of genus two curves.

(3) is OK for Jacobians of genus two curves when B ≤ 9 or so.

(4) gives us a finite index sublattice Λ′ ⊂ Λ where Λ = A(ℚ)/A(ℚ)_{tors} is sitting in the euclidean vector space V = A(ℚ) ⊗ ℝ (with norm induced by ĥ).

Lemma. Let ρ = max_{x∈V} dist(x, Λ′) be the covering radius of Λ′. Then

A(ℚ) = ⟨U ∪ {P ∈ A(ℚ) | ĥ(P) ≤ ρ^{2}}⟩ .

Corollary.

A(ℚ) = ⟨U ∪ {P ∈ A(ℚ) | H(P) ≤ ⌊exp(ρ^{2} + γ)⌋}⟩ .

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Example 1

C : Y^{2} = X^{5} + 16 X^{4} − 274 X^{3} + 817 X^{2} + 178 X + 1

J = Jacobian of C

• Found by Ed Schaefer who proved that

• Torsion subgroup of J(ℚ) is trivial and

rank J(ℚ) = 7

• He also has generators of odd index subgroup U of J(ℚ)

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Example 1

C : Y^{2} = X^{5} + 16 X^{4} − 274 X^{3} + 817 X^{2} + 178 X + 1 J = Jacobian of C

• disc F = 191^{2} · 941^{4}

• Mod 941, F factors as Z(X − 185 Z)^{5} ⟹ γ_{941} = 0 • Find γ_{2} = 4, γ_{191} = 2, γ_{∞} ≤ 4.2640

• Hence γ ≤ \frac{1}{3}(γ_{∞} + γ_{2} log 2 + γ_{191} log 191) ≤ 5.8470

• Get bound on covering radius of lattice (U, ĥ):

ρ^{2} ≤ 4.29021

• Therefore J(ℚ) generated by U and all P ∈ J(ℚ) with

H(P) ≤ ⌊exp(ρ^{2} + γ)⌋ ≤ 25 266

• Find all 683 such points and check they are in U already

• Conclusion:  J(ℚ) = U

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Example 1

C : Y^{2} = X^{5} + 16 X^{4} − 274 X^{3} + 817 X^{2} + 178 X + 1 J = Jacobian of C Generators of J(ℚ) (Reduced Basis)

generator height
(2 + \sqrt{10}, 21 − 5\sqrt{10}) + conj. − 2 · ∞  1.647351
(0, 1) − ∞  2.017800
(4, 37) − ∞  2.990177
(−2, 73) − ∞  3.292326
(\frac{21+\sqrt{329}}{8}, \frac{965+9\sqrt{329}}{64}) + conj. − 2 · ∞  3.379621
(\frac{1}{2}(15 + \sqrt{129}), 355 + 26\sqrt{129}) + conj. − 2 · ∞ 3.692315
(\frac{35+\sqrt{1189}}{9}, \frac{11492+307\sqrt{1189}}{243}) + conj. − 2 · ∞  3.695602

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Example 2

C : Y^{2} = X^{6} − 56 X^{5} + 176 X^{4} + 74 X^{3} − 81 X^{2} − 282 X + 169 J = Jacobian of C

• Found by Colin Stahlke

• Small coeffs and #C(ℚ) ≥ 126

• Divisor classes [P − Q] with P, Q ∈ C(ℚ) generate subgroup U ⊂ J(ℚ) of rank 12

• 2-descent gives rank J(ℚ) ≤ 12, hence

rank J(ℚ) = 12

• Torsion subgroup of J(ℚ) is trivial

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Example 2

C : Y^{2} = X^{6} − 56 X^{5} + 176 X^{4} + 74 X^{3} − 81 X^{2} − 282 X + 169

J = Jacobian of C

• disc F = −2^{9} · 27605791 · 12261635838401, hence γ_{v} = 0 for v ∉ {2, ∞}

• Find γ_{2} = 5, γ_{∞} ≤ 6.6482

• Hence γ ≤ \frac{1}{3}γ_{∞} + \frac{1}{3}γ_{2} log 2 ≤ 3.3713

• Get bound on covering radius of lattice (U, ĥ):

ρ^{2} ≤ 5.3031

• Therefore J(ℚ) generated by U and all P ∈ J(ℚ) with

H(P) ≤ ⌊exp(ρ^{2} + γ)⌋ ≤ 5851

• Find all 1347 such points and check they are in U already

• Conclusion:  J(ℚ) = U

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Example 2

C : Y^{2} = X^{6} − 56 X^{5} + 176 X^{4} + 74 X^{3} − 81 X^{2} − 282 X + 169 J = Jacobian of C Generators of J(ℚ) (Reduced Basis)

generator height
(1, 1) − ∞_{+}  3.6069846330
(1, 1) − ∞_{−}  3.6248921651
(0, 13) − ∞_{−}  3.9724077310
(−1, 23) − ∞_{+}  4.4794963440
(2, 31) − ∞_{+}  4.5306263681
(\frac{1}{2}, \frac{41}{8}) − ∞_{−}  4.6541675885
(1, 1) + (−2, 67) − ∞_{+} − ∞_{−}  4.7683627239
(\frac{1}{3}, \frac{227}{27}) − ∞_{+}  4.9309921342
(\frac{3}{5}, −\frac{283}{125}) − ∞_{−}  5.1654708604
(\frac{4}{3}, \frac{331}{27}) − ∞_{−}  5.2192990555
(1, 1) + (−4, −319) − ∞_{+} − ∞_{−} 5.2817228322
(\frac{5}{3}, \frac{589}{27}) − ∞_{+}  5.3847930886
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