02 / Maths Lab

Arithmetic Heights.

Explore exact heights of curves and tautological cycles on their Jacobians.

Exact formula / Theorem 1.1

One family. A landscape of heights.

Explore the stable Faltings height of the smooth projective curve X_n: y² = xⁿ − 1, equivalently the height of its Jacobian. The formula applies to odd n ≥ 3, with genus g = (n − 1)/2.

Choose n

Stable Faltings height · numerical approximationEnable JavaScript to calculate.

Exact value for the chosen n

This finite gamma expression is exact. The decimal and plotted points are numerical evaluations, displayed to eight decimal places. All logarithms are natural.

Formula & normalization

Writing e_p = ord_p(n), Theorem 1.1 gives

\begin{array}{l} h_{Fal} (X_{n}) = \frac{n}{8} \sum_{p ∣ n} \frac{p^{2 e_{p}} - 1}{p^{2 e_{p} - 1} (p^{2} - 1)} \log p + \frac{n}{8} \log n \\ - \frac{g}{2} \log π + \frac{g}{2 n} \log 2 - \sum_{j = 1}^{g} \log \frac{Γ (\frac{2 j - 1}{2 n})}{Γ (\frac{n + 2 j - 1}{2 n})} . \end{array}

The sum over p runs over prime divisors of n. The exact result above combines the log n term with this prime sum and reduces the rational coefficients.

The normalization is the paper’s: the metric is induced by ⟨η,η′⟩ = (i/2) ∫ η ∧ η̄′. The height is stable, computed after semistable reduction. Γ denotes Euler’s gamma function.

Read the paper on arXiv ↗

Plot every height in an interval

From aTo b

Use integer endpoints 3 ≤ a ≤ b ≤ 10001. Even endpoints are allowed; every odd n in [a,b] is included.

Select a point to inspect its exact height above. With the plot focused, use ← and → to move between points; Home and End select the endpoints. The red point is the chosen n.

The bounds and the leading term

−0.975n < h_Fal(X_n) − (n/8) log n
< (9/64)n log log n − 0.263n.

Corollary 1.2 gives these strict bounds. Toggle them to see the lower bound in teal and the upper bound in dashed red. The second vertical-axis option subtracts the leading term to reveal the remaining variation.

Values for the plotted interval

n	Genus	h_Fal(X_n) ≈

The self-products paper →

Self-products of curves / Theorem 1.1

From a vector to a height.

Let X/K be a smooth projective geometrically connected curve of genus g ≥ 2 with semistable reduction. Fix the canonical degree-one class α = ω/(2g−2), up to torsion. In J = Pic⁰(X), let

Z_m,α = image(X^r → J), (x₁,…,xᵣ) ↦ ∑_j=1^r m_j(x_j−α).

Choose nonzero integers m_j; negative entries are allowed. The calculator gives exact rational coefficients of the curve’s normalized invariants.

Genus g = 3

Number of factors r = 2

1 ≤ r ≤ g. At r = g, the image is the whole Jacobian and its height is zero.

Vector m

Exact Néron–Tate height

Enable JavaScript to calculate the exact expression.

Ω = ω̂²/[K:Q], Φ = φ(X)/[K:Q].

Here φ(X) is the global invariant, including the finite-place and complex-embedding contributions. A numerical height also requires Ω and Φ for your curve.

The formula and its normalization

Write S₂ = ∑ m_j² and P = ∑_j<k m_jm_k. The choice of α makes h_NT(α−ω/(2g−2)) = 0.

h′_ℒ(Z_m,α) = AΩ + BΦ.

For 2 ≤ r < g:

A = \frac{(g - r) [3 g (g - 2) S_{2} - 2 (2 g + 1) P]}{24 g {(g - 1)}^{2} (g - 2)}

B = \frac{(g - r) P}{6 g (g - 1) (g - 2)}

For r = 1: A = m₁²/[8(g−1)] and B = 0. For r = g: A = B = 0. These separate cases also cover genus two without division by g−2.

ℒ is a symmetric line bundle inducing the canonical principal polarization. The normalization is h′_ℒ(Z) = ⟨ℒ̂^r+1|Z⟩ / ([K:Q](r+1)⟨ℒ^r|Z⟩).

Read Theorem 1.1 ↗

Arithmetic Heights.

One family. A landscape of heights.

Exact value for the chosen n

Plot every height in an interval

From a vector to a height.

Comic

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