# Height function

In [Diophantine geometry](https://www.edgepedia.org/diophantine-geometry), a height function is a function that measures the geometric and arithmetic complexity of rational points on a projective variety defined over a number field. It is a key tool behind the fundamental finiteness results of arithmetic geometry, including the [Mordell–Weil theorem](https://www.edgepedia.org/mordell-weil-theorem), Siegel's theorem, and Faltings' theorem (formerly the Mordell conjecture).<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup>

| Key fact | Detail |
|---|---|
| Purpose | Measures the arithmetic complexity of rational or algebraic points; the general scheme is a measurement of "arithmetic complexity"<sup>[2](https://www.math.columbia.edu/~chaoli/docs/HeightsAndFiniteness.html)</sup> |
| Basic example | For \( (x_0 : \cdots : x_n) \in \mathbb{P}^n(\mathbb{Q}) \) with primitive integer coordinates, \( H = \max\{|x_0|, \ldots, |x_n|\} \)<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup> |
| Northcott property | For any \( T > 0 \), the set \( \{P \in \mathbb{P}^n(\mathbb{Q}) : H(P) \le T\} \) is finite<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup> |
| Weil height machine | The construction \( L \mapsto h_L \) gives a group homomorphism from \( \mathrm{Pic}(X) \) to functions modulo bounded functions, so a Weil height is unique up to a bounded function<sup>[3](https://epubs.siam.org/doi/full/10.1137/25M1804091)</sup> |
| Canonical height | On an abelian variety with symmetric ample \( L \), \( \hat{h}_L(x) = \lim_{n \to \infty} \frac{1}{4^n} h_L(2^n x) \); it is nonnegative and vanishes exactly on torsion points<sup>[3](https://epubs.siam.org/doi/full/10.1137/25M1804091)</sup> |
| Root of unity test | An algebraic number \( x \in \overline{\mathbb{Q}}^{\times} \) satisfies \( h(x) = 0 \) if and only if \( x \) is a root of unity<sup>[2](https://www.math.columbia.edu/~chaoli/docs/HeightsAndFiniteness.html)</sup> |
| Dynamical analog | For a morphism \( \varphi \) of degree \( d \ge 2 \), the Call–Silverman canonical height \( \hat{h}_\varphi(P) = \lim_{n \to \infty} d^{-n} h(\varphi^n(P)) \) satisfies \( \hat{h}_\varphi \circ \varphi = d \, \hat{h}_\varphi \)<sup>[4](https://citeseerx.ist.psu.edu/document?doi=3c9aab7458b4c48a89ddc1d6a6268f81a09fa79b&repid=rep1&type=pdf)</sup> |

## The basic example

For a rational number written in lowest terms as \( p/q \), the height is \( H(p/q) = \max(|p|, |q|) \).<sup>[5](https://arxiv.org/abs/1511.06689)</sup>

The projective version extends this to points \( (x_0 : \cdots : x_n) \in \mathbb{P}^n(\mathbb{Q}) \), where the coordinates are chosen as integers with \( \gcd(x_0, \ldots, x_n) = 1 \), and the height is \( H = \max\{|x_0|, \ldots, |x_n|\} \).<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup> Because the coordinates are homogeneous, this value does not depend on the scaling of the representative.

Over a general number field \( K \), the definition uses all places of the field. For a point \( [x_0 : \cdots : x_n] \in \mathbb{P}^n(k) \), the standard logarithmic height is

\[ h([x_0 : \cdots : x_n]) = \sum_{v \in M_k} \log \max\{|x_0|_v, \ldots, |x_n|_v\}, \]

and this is independent of the choice of coordinates by the product formula \( \prod_{v \in M_k} |x|_v = 1 \) for \( x \in k^{\times} \).<sup>[6](https://link.springer.com/article/10.1007/s00208-023-02564-3)</sup> The absolute (multiplicative) height \( H(x) = H_K(x)^{1/[K:\mathbb{Q}]} \) and the logarithmic height \( h = \log H \) do not depend on the field \( K \) used to define the point.<sup>[2](https://www.math.columbia.edu/~chaoli/docs/HeightsAndFiniteness.html)</sup>

## Northcott's theorem and counting

The defining finiteness feature is the Northcott property: for any positive real \( T \), the set of rational points with \( H(P) \le T \) is finite.<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup> In its general form, Northcott's theorem states the finiteness of the number of points of bounded degree and bounded height in a fixed projective space.<sup>[7](https://www.cambridge.org/core/books/abs/heights-in-diophantine-geometry/weil-heights/934524518B373E7CC59DF0D4D2576AE3)</sup> The bibliographical notes of Bombieri and Gubler record that the general statement appears in a foundational paper by [André Weil](https://www.edgepedia.org/andre-weil), following earlier work of D. G. Northcott.<sup>[8](https://research-solution.com/uplode/books/book-38916.pdf)</sup>

This finiteness allows one to define a counting function for rational points of bounded height and to seek asymptotic formulas as \( T \to \infty \).<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup> Around 1989, [Yuri Manin](https://www.edgepedia.org/yuri-manin) initiated a program to understand the asymptotic behavior of rational points of bounded height on Fano varieties, a line of work in which universal torsors, introduced by Colliot-Thélène and Sansuc, became a useful computational tool.<sup>[9](https://doi.org/10.1007/978-0-8176-8170-8_4)</sup>

## Weil heights on varieties

Given a number field \( F \), a projective variety \( X \) over \( F \), and a line bundle \( L \) on \( X \), one can associate a height function \( H_L : X(F) \to \mathbb{R}_{>0} \); when \( L \) is ample, the height enjoys the Northcott property.<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup> Concretely, one embeds \( X \) into projective space so that the pullback of \( \mathcal{O}(1) \) is a power of \( L \), and composes with the standard height.<sup>[3](https://epubs.siam.org/doi/full/10.1137/25M1804091)</sup>

The resulting height depends on the chosen embedding, but only by a bounded amount. Weil's height machine states that the construction \( L \mapsto h_L \) is a group homomorphism from the [Picard group](https://www.edgepedia.org/picard-group) of \( X \) to real-valued functions on \( X(\overline{K}) \) modulo bounded functions; a representative \( h_L \) of the class is called a Weil height function corresponding to \( L \).<sup>[3](https://epubs.siam.org/doi/full/10.1137/25M1804091)</sup> The same theory associates to every Cartier divisor \( D \) on a projective variety \( V \) a height function \( h_D : V(k) \to \mathbb{R} \) together with local Weil functions \( \lambda_{D,v} \) at each place, satisfying \( \sum_v \lambda_{D,v}(P) = h_D(P) + O(1) \).<sup>[6](https://link.springer.com/article/10.1007/s00208-023-02564-3)</sup> The general theory of local heights is due to [André Néron](https://www.edgepedia.org/andre-neron).<sup>[8](https://research-solution.com/uplode/books/book-38916.pdf)</sup>

## Canonical heights

On abelian varieties, and in particular on elliptic curves, the naive height grows quadratically under multiplication, and a limit process removes the bounded error. For an abelian variety \( A \) over a number field \( K \) with a symmetric ample line bundle \( L \), the Néron–Tate (canonical) height is defined by Tate's limit

\[ \hat{h}_L(x) = \lim_{n \to \infty} \frac{1}{4^n} h_L(2^n x), \]

which converges and is independent of the choice of Weil height \( h_L \).<sup>[3](https://epubs.siam.org/doi/full/10.1137/25M1804091)</sup> On an elliptic curve the equivalent formulation is \( \hat{h}(P) = \lim_{n \to \infty} \frac{1}{n^2} h([n]P) \).<sup>[10](https://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf)</sup>

The canonical height has strong structural properties: \( \hat{h}_L(x) \ge 0 \) with equality exactly for torsion points; it satisfies the parallelogram rule \( \hat{h}_L(x+y) + \hat{h}_L(x-y) = 2(\hat{h}_L(x) + \hat{h}_L(y)) \); and \( \hat{h}_L(mx) = m^2 \hat{h}_L(x) \).<sup>[3](https://epubs.siam.org/doi/full/10.1137/25M1804091)</sup> It induces a positive definite quadratic form on \( E(\mathbb{Q}) \otimes \mathbb{R} \), whose associated lattice determinant is the elliptic regulator.<sup>[10](https://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf)</sup> John Tate described a rapidly convergent series for computing Néron's local canonical heights, which he circulated in an unpublished letter to Serre and which became the standard efficient method of computation.<sup>[4](https://citeseerx.ist.psu.edu/document?doi=3c9aab7458b4c48a89ddc1d6a6268f81a09fa79b&repid=rep1&type=pdf)</sup>

## Role in finiteness theorems

Height functions are the engine of the descent argument in the Mordell–Weil theorem, which says that the group of rational points on an abelian variety over a number field is finitely generated.<sup>[10](https://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf)</sup><sup> • </sup><sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup> The proof reduces the group modulo an integer \( m \), then uses the quadratic behavior of the canonical height to show that representatives of bounded height generate the whole group: if a point \( P \) of smallest height outside the generated span is written \( P = mQ + P_j \), the parallelogram inequality forces \( \hat{h}(Q) < \hat{h}(P) \), a contradiction.<sup>[10](https://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf)</sup>

The same finiteness framework underlies Siegel's theorem and Faltings' theorem, the statement that curves of genus two or more over number fields have only finitely many rational points.<sup>[1](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)</sup><sup> • </sup><sup>[11](https://old.maa.org/press/maa-reviews/heights-in-diophantine-geometry-0)</sup> A distinct invariant, the Faltings height of an abelian variety, is built from the Néron differential and metrized line bundles in the sense of [Arakelov theory](https://www.edgepedia.org/arakelov-theory), and finiteness statements for abelian varieties in bounded height are a key step in the proof of Faltings' theorem.<sup>[2](https://www.math.columbia.edu/~chaoli/docs/HeightsAndFiniteness.html)</sup>

## Heights in arithmetic dynamics

When a morphism \( \varphi : \mathbb{P}^N \to \mathbb{P}^N \) of degree \( d \ge 2 \) is iterated, the Call–Silverman canonical height is

\[ \hat{h}_\varphi(P) = \lim_{n \to \infty} \frac{1}{d^n} h\left(\varphi^n(P)\right), \]

with \( \hat{h}_\varphi = h + O(1) \) and \( \hat{h}_\varphi \circ \varphi = d \, \hat{h}_\varphi \).<sup>[4](https://citeseerx.ist.psu.edu/document?doi=3c9aab7458b4c48a89ddc1d6a6268f81a09fa79b&repid=rep1&type=pdf)</sup> Convergence follows from Tate's telescoping-series argument, since the difference \( |h(\varphi^n(\alpha)) - d\,h(\varphi^{n-1}(\alpha))| \) is bounded and hence divided by \( d^n \) the series converges.<sup>[12](https://people.math.rochester.edu/faculty/ttucker/MSRI.dir/FULL_NOTES.pdf)</sup> More broadly, canonical heights serve as refined measures of the arithmetic complexity of orbits in algebraic dynamical systems, complementing the dynamical degree, which measures geometric complexity of the iterates.<sup>[13](https://arxiv.org/abs/2408.01559)</sup>

## Heights of algebraic numbers

The Weil height of an algebraic number \( z \) measures its arithmetic complexity and is always nonnegative.<sup>[14](https://scholarship.claremont.edu/cmc_fac_pub/85/)</sup> Kronecker's theorem in this language takes a sharp form: \( h(x) = 0 \) if and only if \( x \) is a root of unity. The proof uses Northcott's theorem: if \( h(x) = 0 \) then \( h(x^n) = n\,h(x) = 0 \) for every \( n \), so the bounded-height set \( \{x^n\} \) must be finite, forcing \( x \) to have finite order.<sup>[2](https://www.math.columbia.edu/~chaoli/docs/HeightsAndFiniteness.html)</sup> Lower bounds for heights of non-root-of-unity algebraic numbers, such as those connected with Lehmer's problem and Dobrowolski's theorem, form a continuing line of research in the area.<sup>[15](https://main-site-cimpa-web.apps.ocp.math.cnrs.fr/en/node/7291)</sup>

## References

1. [Height Functions, Chapter 2 of *Birational Geometry and Manin's Conjecture*, Cambridge University Press](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/86576DD02DD8DDE573DE1E070BDFBC10/9781009611633c2_11-17.pdf/height-functions.pdf)
2. [Heights and Finiteness theorems of abelian varieties, Columbia University lecture notes](https://www.math.columbia.edu/~chaoli/docs/HeightsAndFiniteness.html)
3. [Quantitativity in the Mordell Conjecture, ICM 2026 Invited Lectures, SIAM](https://epubs.siam.org/doi/full/10.1137/25M1804091)
4. [Kawaguchi, S. and Silverman, J. H., Canonical heights and the arithmetic dynamics of morphisms](https://citeseerx.ist.psu.edu/document?doi=3c9aab7458b4c48a89ddc1d6a6268f81a09fa79b&repid=rep1&type=pdf)
5. [Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?, arXiv:1511.06689](https://arxiv.org/abs/1511.06689)
6. [Divisibility of polynomials and degeneracy of integral points, Mathematische Annalen (2023)](https://link.springer.com/article/10.1007/s00208-023-02564-3)
7. [Weil Heights, Chapter 2 of *Heights in Diophantine Geometry*, Cambridge University Press](https://www.cambridge.org/core/books/abs/heights-in-diophantine-geometry/weil-heights/934524518B373E7CC59DF0D4D2576AE3)
8. [Bombieri, E. and Gubler, W., *Heights in Diophantine Geometry*, bibliographical notes](https://research-solution.com/uplode/books/book-38916.pdf)
9. [Peyre, E., Counting Points On Varieties Using Universal Torsors, Progress in Mathematics vol. 226, Birkhäuser](https://doi.org/10.1007/978-0-8176-8170-8_4)
10. [Silverman, J. H., An Introduction to the Theory of Elliptic Curves, lecture notes](https://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf)
11. [Heights in Diophantine Geometry, MAA Reviews](https://old.maa.org/press/maa-reviews/heights-in-diophantine-geometry-0)
12. [Tucker, T., Arithmetic Dynamics, MSRI lecture notes](https://people.math.rochester.edu/faculty/ttucker/MSRI.dir/FULL_NOTES.pdf)
13. [Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions, arXiv:2408.01559](https://arxiv.org/abs/2408.01559)
14. [Fukshansky, L., On Heights of Algebraic Numbers, Claremont Colleges](https://scholarship.claremont.edu/cmc_fac_pub/85/)
15. [Lattices, heights and diophantine approximation, CIMPA school](https://main-site-cimpa-web.apps.ocp.math.cnrs.fr/en/node/7291)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Arithmetic geometry › Heights and Diophantine approximation*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license