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Height function

In arithmetic geometry and number theory, a height function is a real-valued invariant defined on the set of rational points (or more generally, points over a number field) of an algebraic variety, quantifying the arithmetic complexity of these points by measuring the size of their coordinates in a minimal integral representation.^[1] These functions, first systematically developed in the mid-20th century, enable the study of infinite sets of solutions to Diophantine equations by bounding their growth and proving finiteness results for subsets of bounded height.^[2] The foundational height, known as the (absolute, multiplicative) Weil height, is defined for a point P = [x_0 : \dots : x_n] in projective space \mathbb{P}^n(\overline{\mathbb{Q}}) over the algebraic closure of the rationals, where the x_i are algebraic numbers; it takes the form H(P) = \prod_v \max_i \{ \|x_i\|_v \} ^{1/[K:\mathbb{Q}]}, with the product over all places v of a number field K containing the coordinates, normalized absolute values \| \cdot \|_v, and the exponent ensuring independence from the field extension degree.^[1] The logarithmic height h(P) = \log H(P) is often used for its additive properties under field operations, such as h(\alpha \beta) \leq h(\alpha) + h(\beta) + O(1) for algebraic numbers \alpha, \beta.^[2] For rational points in \mathbb{P}^1(\mathbb{Q}), this simplifies to H(p/q) = \max\{|p|, |q|\} for coprime integers p, q.^[3] A key property of height functions is Northcott's theorem, which states that for any number field K and bound B > 0, there are only finitely many points in \mathbb{P}^n(K) with height at most B, reflecting the finite number of algebraic points of bounded degree and size.^[1] This finiteness underpins many applications, including estimates for the number of rational points of bounded height, such as N(B) \sim c B^2 for points in \mathbb{P}^1(\mathbb{Q}), where c = 12/\pi^2.^[1] On abelian varieties, such as elliptic curves, canonical heights refine the naive height by incorporating the group law; for a point P, the canonical height \hat{h}(P) = \lim_{n \to \infty} n^{-2} h(P) satisfies \hat{h}(P) = n^2 \hat{h}(P) and vanishes exactly on torsion points, facilitating the proof of the Mordell-Weil theorem that the rational points form a finitely generated abelian group.^[2] These tools extend to broader Diophantine problems, providing lower bounds on heights to resolve equations like Lehmer's conjecture on the minimal height of non-cyclotomic algebraic integers.^[1]

Fundamentals

Significance

Height functions serve as fundamental tools in Diophantine geometry for quantifying the arithmetic complexity of algebraic numbers, rational points on algebraic varieties, and related objects, typically measured on a logarithmic scale that reflects their "size" in terms of minimal defining polynomials or coordinates.^[1] This logarithmic nature allows heights to capture intrinsic properties independent of representation, enabling comparisons across different number fields and embeddings.^[1] A primary application lies in establishing finiteness results for infinite sets, exemplified by Northcott's theorem, which asserts that for a fixed number field K and projective space \mathbb{P}^N(K), there are only finitely many points of bounded height and degree.^[1] This principle extends to varieties, implying only finitely many rational points below any height bound, as crucially utilized in Faltings' 1983 proof of the Mordell conjecture, which shows that curves of genus at least 2 over \mathbb{Q} have finitely many rational points by bounding their heights. Such theorems transform qualitative statements about finiteness into quantitative bounds on solution sets. Heights also connect to effective Diophantine approximation, providing tools to bound solutions to equations like norm form equations or unit equations through inequalities involving local heights and subspace theorems, such as Schmidt's subspace theorem, which refines approximation properties via height controls.^[1] In general, heights increase with the "size" of points in projective embeddings—such as the naive height growing with the maximum absolute value of coordinates—ensuring that sublevel sets remain finite and facilitating algorithmic searches for rational points on varieties.^[1]

History

The concept of height functions in Diophantine geometry traces its origins to the late 19th century, when Georg Cantor introduced an inhomogeneous height for algebraic numbers in 1874 to demonstrate that the cardinality of the real algebraic numbers is smaller than that of the real numbers.^[4] This early notion quantified the complexity of algebraic numbers based on their minimal polynomials. In 1903, Émile Borel extended similar ideas to systems of rational numbers, laying groundwork for measuring arithmetic complexity in Diophantine approximations.^[4] The systematic development of height functions began in the mid-20th century, building on André Weil's 1928 work on the Mordell-Weil theorem, which presaged the use of heights to bound rational points on elliptic curves without explicitly defining them. In 1949, Douglas G. Northcott formalized the Northcott-Weil height and proved a finiteness theorem stating that only finitely many algebraic points of bounded degree and height exist on projective space.^[5] André Weil further advanced the theory in 1951 by integrating heights into arithmetic geometry on algebraic varieties.^[4] During the 1950s and 1960s, key contributions included Northcott's finiteness results and the introduction of the Néron-Tate height on elliptic curves and abelian varieties, with André Néron conjecturing its existence in a 1958 address and John Tate developing its properties in subsequent works around 1960.^[6] In the 1970s, Suren Arakelov introduced metrics on arithmetic surfaces in 1974, enabling global heights that incorporate archimedean and non-archimedean places for intersection theory.^[7] A landmark application came in 1983, when Gerd Faltings proved Mordell's conjecture using heights on abelian varieties to establish the finiteness of rational points on curves of genus greater than 1. Post-1980s developments extended height functions to higher-dimensional varieties and arithmetic dynamics, with applications to equidistribution and finiteness theorems on projective spaces and dynamical systems over number fields.^[8]

Height Functions in Diophantine Geometry

Naive Height

The naive height provides a basic measure of the arithmetic complexity of rational numbers and algebraic integers. For a rational number \alpha = p/q expressed in lowest terms with p, q \in \mathbb{Z} and q > 0, the absolute naive height is defined as H(\alpha) = \max\{|p|, |q|\}, while the logarithmic naive height is h(\alpha) = \log H(\alpha).^[1] This definition captures the "size" of the rational in terms of its numerator and denominator after reduction.^[3] For an algebraic integer \alpha in a number field K/\mathbb{Q} of degree d = [K : \mathbb{Q}], the absolute naive height extends naturally via the product formula over all places v of K: H(\alpha) = \prod_{v} \max(1, |\alpha|_{v})^{[K_v : \mathbb{Q}_{v_0}] / d}, where v_0 is the place of \mathbb{Q} below v.^[9] The corresponding logarithmic height is h(\alpha) = \log H(\alpha). For finite places, the contribution is often trivial for integers, leaving the height largely determined by archimedean places.^[2] Key properties include multiplicativity of the logarithmic height: h(\alpha \beta) = h(\alpha) + h(\beta) for algebraic integers \alpha, \beta.^[2] These ensure that only finitely many algebraic integers of bounded degree and height exist, a consequence of Northcott's theorem in this naive setting.^[1] A representative example arises with units in real quadratic fields K = \mathbb{Q}(\sqrt{d}) for square-free d > 0. The fundamental unit \varepsilon > 1 satisfies h(\varepsilon) = \frac{1}{2} \log \varepsilon, since the two real embeddings yield contributions \max(1, \varepsilon)^{1/2} and \max(1, |\varepsilon'|)^{1/2} = 1^{1/2} (with conjugate \varepsilon' = \pm 1/\varepsilon < 1), and finite places contribute 1; here \log \varepsilon equals the regulator R_K.^[2] This height relates to continued fractions, as \varepsilon is the convergent at the period length k of the continued fraction expansion of \sqrt{d}, with k \leq R_K / \log((1 + \sqrt{5})/2).^[10] For instance, in K = \mathbb{Q}(\sqrt{5}), \varepsilon = (1 + \sqrt{5})/2 gives h(\varepsilon) \approx 0.241, reflecting the short period k=1 of [\sqrt{5}] = [2; \overline{4}].^[1]

Weil Height

The Weil height provides a measure of the arithmetic complexity of rational points on projective varieties, extending the naive height to more general geometric settings. For a projective variety X defined over \mathbb{Q} and an ample line bundle L on X, the Weil height h_L(P) of a point P \in X(\mathbb{Q}) is defined as the sum over all places v of the local heights \lambda_v(L, P), where the local heights \lambda_v are constructed using valuations at each place v.^[1] This definition relies on embedding X into projective space via sections of L, allowing the reduction to heights on \mathbb{P}^n.^[2] The absolute multiplicative height is then given by H_L(P) = \exp(h_L(P)), which captures the exponential growth associated with the point's coordinates in projective embeddings.^[1] Key properties include near-independence from the choice of model: if two ample line bundles L and L' are linearly equivalent, then h_L(P) = h_{L'}(P) + O(1), where the error term is bounded independently of P.^[2] Additionally, the height exhibits functoriality under rational morphisms \phi: X \to Y: h_{\phi^* M}(P) = h_M(\phi(P)) + O(1) for an ample line bundle M on Y.^[1] In the special case of projective space \mathbb{P}^n over \mathbb{Q}, with the tautological line bundle \mathcal{O}(1), the Weil height of a point [x_0 : \dots : x_n] \in \mathbb{P}^n(\mathbb{Q}) simplifies to h([x_0 : \dots : x_n]) = \sum_v \log \max_i |x_i|_v, where the x_i are integers with \gcd(x_0, \dots, x_n) = 1 and the sum is over all places v of \mathbb{Q}, normalized appropriately.^[2] This coincides with the naive height up to a constant when restricted to \mathbb{P}^1. A fundamental consequence is Northcott's theorem, which states that there are only finitely many points in X(\overline{\mathbb{Q}}) of bounded degree over \mathbb{Q} and bounded Weil height.^[1] This finiteness property underpins many applications in Diophantine geometry, ensuring controlled growth in the set of points of interest.^[2]

Néron–Tate Height

The Néron–Tate height \hat{h}, also known as the canonical height, is a quadratic form defined on the Mordell-Weil group E(\mathbb{Q}) of rational points on an elliptic curve E over \mathbb{Q}. It refines the Weil height h by providing a translation-invariant measure that vanishes precisely on the torsion subgroup. The height is given by the limit formula

\hat{h}(P) = \lim_{n \to \infty} \frac{1}{4^n} h(2^n P)

for P \in E(\mathbb{Q}), where $2^n P denotes the n-fold doubling of P on E, and the limit exists and is independent of the choice of model for E.^[11] This height satisfies key quadratic properties, including \hat{h}(mP) = m^2 \hat{h}(P) for integers m, and it induces a positive semi-definite bilinear pairing on E(\mathbb{Q}) via

\langle P, Q \rangle = \frac{1}{2} \bigl( \hat{h}(P+Q) - \hat{h}(P) - \hat{h}(Q) \bigr).

The pairing is alternating, non-degenerate on the free part of E(\mathbb{Q}), and defines the Néron–Tate height as its associated quadratic form. Moreover, \hat{h}(P) = 0 if and only if P is a torsion point, and \hat{h}(P) > 0 otherwise, ensuring the height bounds the growth of points under the group law.^[11] The Néron–Tate height plays a central role in the arithmetic of elliptic curves, particularly in the study of the regulator of E(\mathbb{Q}), which is the absolute value of the determinant of the Gram matrix formed by the height pairing on a basis of the free part of the Mordell-Weil group. This regulator appears in the conjectural formula for the leading term of the L-function L(E,s) at s=1 under the Birch and Swinnerton-Dyer conjecture, linking the algebraic rank of E(\mathbb{Q}) to analytic data and facilitating computations of ranks via descent methods and partial BSD verifications.^[11] For the family of elliptic curves y^2 = x^3 + k with k \in \mathbb{Q}, explicit formulas for the Néron–Tate height can be derived using integrals involving the Weierstrass \wp-function and periods of the curve, particularly when the curve has complex multiplication. These formulas express \hat{h}(P) in terms of logarithmic heights adjusted by archimedean contributions from elliptic integrals over the lattice associated to E.^[11]

Arakelov Height

The Arakelov height represents a refinement of classical height functions in arithmetic geometry, extending the algebraic Weil height to incorporate contributions from archimedean places through differential geometric structures on the complex fibers of arithmetic varieties. This extension equips projective varieties over the ring of integers of a number field with Hermitian metrics, enabling a unified treatment of finite and infinite places in Diophantine problems. Originally developed to facilitate intersection theory on arithmetic surfaces, the Arakelov height provides tools for bounding the complexity of rational points and subvarieties, bridging algebraic and analytic aspects of number theory. For a point P on an arithmetic curve X over \mathcal{O}_K, equipped with a Hermitian line bundle \overline{\mathcal{L}} = (\mathcal{L}, \|\cdot\|), the Arakelov height is defined as

h_{\mathrm{Ark}}(P) = h_{\mathrm{alg}}(P) + \frac{1}{2} \int_{X(\mathbb{C})} \omega \log \|s(P)\|^2,

where h_{\mathrm{alg}}(P) denotes the algebraic component derived from the Weil height on the finite places, s is a local section of \mathcal{L}, and \omega is the Fubini–Study Kähler form inducing the Hermitian metric on the complex fiber X(\mathbb{C}). This formula ensures the height measures both the "size" of P at non-archimedean places via logarithmic norms and its geometric position at archimedean places through the integral, which captures the contribution of the metric to the point's embedding. The choice of the Fubini–Study metric standardizes the archimedean term, making the height independent of coordinates up to bounded error.^[12]^[13] On arithmetic surfaces, which are proper flat models of curves over \Spec \mathcal{O}_K, Arakelov intersection theory extends classical intersection multiplicities by incorporating Green's functions to handle the archimedean components. For divisors D_1, D_2 on such a surface \mathcal{X}, the intersection number is given by (D_1 \cdot D_2)_{\mathrm{Ark}} = \sum_{v \nmid \infty} \log N_v(\mathcal{O}_{\mathcal{X}} (D_1) \cdot D_2) + \iint_{\mathcal{X}(\mathbb{C})} g_{D_1} dd^c g_{D_2} \wedge \omega, where g_D is the Green's function associated to D satisfying dd^c g_D = \omega - [D] on the complex fiber, and \omega is a smooth positive (1,1)-form representing the first Chern class of the metric. This construction allows for a Riemann-Roch theorem in the arithmetic setting, relating degrees and genera while controlling the growth of heights for integral points. For non-archimedean places, the theory uses the standard logarithmic valuations, ensuring compatibility with the finite part of the height.^[12] Key properties of the Arakelov height include its Hermiticity, arising from the sesquilinear nature of the Hermitian metrics, which ensures that the associated intersection pairing on arithmetic cycles is positive semi-definite and compatible with complex conjugation on the archimedean fibers. This Hermiticity underpins the positivity of arithmetic degrees and facilitates equidistribution results for sequences of points of bounded height. The height is also closely tied to the Arakelov degree, defined for a Hermitian line bundle \overline{\mathcal{L}} on an arithmetic variety as \widehat{\deg}(\overline{\mathcal{L}}) = \deg_{\mathrm{fin}}(\mathcal{L}) + \frac{1}{2} \int_{X(\mathbb{C})} c_1(\overline{\mathcal{L}}) \wedge \omega^{ \dim X - 1}, where the finite degree sums local contributions over non-archimedean places. This degree measures the "arithmetic size" of the bundle and relates the height of points to intersection numbers via pullback formulas.^[13]^[14] Applications of Arakelov heights have been pivotal in addressing conjectures in Diophantine geometry, notably providing effective bounds toward Szpiro's conjecture on the discriminants of elliptic curves. By estimating Arakelov intersection numbers on the arithmetic surfaces arising from modular curves, the theory yields uniform bounds on the ratio of the conductor to the discriminant, supporting ineffective proofs and partial effective results for elliptic curves over number fields. Similarly, Arakelov heights contribute to effective versions of the Shafarevich theorem, bounding the heights of isomorphism classes of elliptic curves with bounded conductor and enabling algorithms to compute finiteness results for integral points on modular curves. These applications leverage the positivity and continuity properties of the heights to control the distribution of special points.^[15]^[16]^[17]^[18]

Faltings Height

The Faltings height provides a fundamental invariant for measuring the arithmetic size of abelian varieties over number fields, playing a pivotal role in finiteness results in Diophantine geometry. Introduced by Gerd Faltings in his seminal work on finiteness theorems, it extends classical height functions to higher-dimensional objects by incorporating Arakelov-theoretic metrics on line bundles associated to the variety. For an abelian variety A of dimension g defined over a number field K, the Faltings height relies on the line bundle \omega_A of invariant differentials, which is the determinant of the sheaf H^0(A, \Omega^1_{A/K}). This bundle is metrized at finite places using the Néron model of A and at infinite places via the natural Hermitian structure on the associated complex tori.^[19] The precise definition of the Faltings height h_F(A) is given by the normalized Arakelov degree of the metrized line bundle \overline{\omega}_A:

h_F(A) = \frac{1}{[K : \mathbb{Q}]} \left( \deg(\omega_A) - \frac{1}{2} \sum_{v \mid \infty} \log \|\omega_A\|_v \right),

where \deg(\omega_A) is the usual degree on the finite part derived from the model over the ring of integers of K, and \|\omega_A\|_v denotes the norm of the fiber at the infinite place v, computed as the exponential of half the integral of the logarithm of the pointwise norm over the complex fiber with respect to the invariant volume form. This formulation ensures the height is independent of the choice of model, provided it is semi-stable, and it remains invariant under base field extensions within the same isogeny class after normalization. The metrics employed are of Arakelov style, emphasizing global arithmetic intersection theory.^[19] Faltings extended this notion to semi-stable vector bundles on abelian varieties, defining the height of such a bundle E as the minimal Faltings height over all possible determinant line bundles arising from modifications of E that preserve semi-stability. This minimal height captures the "simplest" arithmetic embedding of the bundle and is crucial for bounding the complexity of families of bundles in higher dimensions.^[19] A key application lies in Faltings' finiteness theorem, which asserts that for fixed dimension g and a bound C > 0, there are only finitely many isomorphism classes of abelian varieties over K (up to isogeny) with h_F(A) \leq C. Bounding the Faltings height thus implies finiteness of isomorphism classes, resolving the Shafarevich conjecture for abelian varieties and providing the arithmetic foundation for the Mordell conjecture in higher dimensions. This result relies on the height's ability to control the geometry of the special fiber in semi-stable models.^[19] The Faltings height exhibits strong stability properties under isogenies: for an isogeny \phi: A \to B of degree d, the difference satisfies |h_F(A) - h_F(B)| \leq \frac{1}{g} \log d, ensuring that heights within an isogeny class differ by at most a logarithmic factor in the isogeny degree. Additionally, the height is intimately related to minimal models of the abelian variety; it is computed using the semi-stable or Néron minimal model, which minimizes the contribution from bad reduction and aligns the infinite metrics with the principal polarization. These properties make the Faltings height a robust tool for studying moduli spaces of abelian varieties.^[19]

Height Functions in Commutative Algebra

Height of Polynomials

In commutative algebra and number theory, the height of a polynomial provides a measure of the arithmetic complexity based on the size of its coefficients. For a polynomial P(x) = a_n x^n + \cdots + a_1 x + a_0 with integer coefficients a_i \in \mathbb{Z}, the house height (or naive height) is defined as H(P) = \max_{0 \leq i \leq n} |a_i|. The corresponding logarithmic height is h(P) = \log H(P). This definition extends naturally to polynomials over the rationals by clearing denominators and adjusting for content, ensuring the height reflects the maximal absolute coefficient size after normalization. The height function exhibits useful properties under arithmetic operations. For scalar multiplication, H(c P) = |c| H(P) for c \in \mathbb{Z}. Under polynomial multiplication, the height is nearly submultiplicative: if f, g \in \mathbb{Z} with \deg f \leq \deg g, then H(f g) \leq (1 + \deg f) H(f) H(g), arising from the convolution structure of coefficients in the product. For substitution, the height remains invariant under certain linear changes; specifically, H(\pm P(\pm x^k)) = H(P) for integer k \geq 1, and H((x - c) P(x)) = H(P) if |c| \leq 1 and c \in \mathbb{Z}. These properties facilitate estimates in algebraic manipulations and highlight the height's role in controlling coefficient growth.^[20] Applications of polynomial heights include irreducibility criteria and bounds on roots. In irreducibility studies, the height helps quantify the scarcity of reducible polynomials: for fixed degree d, the number of reducible monic polynomials in \mathbb{Z} with height at most T is asymptotically o(T^d) as T \to \infty, implying most low-height polynomials are irreducible. This supports effective irreducibility tests, such as variants of Eisenstein's criterion adapted to coefficient bounds. For bounding roots, consider a monic polynomial P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0 with height H(P); any root r satisfies |r| \leq 1 + \max(1, H(P)), with sharper bounds like |r| < 1 + (H(P))^{1/(n-1)} for complex roots outside the unit disk. These estimates are crucial for locating roots in Diophantine approximations and effective versions of theorems like Roth's.^[21]^[22] A representative example is the height of cyclotomic polynomials \Phi_n(x), which generate the factors of x^n - 1. For n < 105, H(\Phi_n) = 1, as all coefficients are $0, \pm 1; the first exception is \Phi_{105}(x), with H(\Phi_{105}) = 2. In general, the height A(n) = H(\Phi_n) grows slowly, bounded above by \exp(c (\log n \log \log n)^{1/2}) for some constant c > 0, reflecting the structured yet bounded coefficient growth in these irreducible polynomials. The height of polynomials connects to the distribution of their roots via the Mahler measure, with details covered in the subsequent section on the relation to Mahler measure.^[23]

Relation to Mahler Measure

The Mahler measure of a polynomial P(z) = a_n \prod_{i=1}^n (z - \zeta_i) \in \mathbb{C} of degree n is defined as

M(P) = \exp\left( \int_0^1 \log |P(e^{2\pi i \theta})| \, d\theta \right),

which, by Jensen's formula, equals |a_n| \prod_{i=1}^n \max(1, |\zeta_i|).^[24] This quantity provides a root-sensitive refinement of the naive height H(P), typically defined as the maximum absolute value of the coefficients of P, by incorporating the geometric distribution of the roots relative to the unit circle rather than solely bounding coefficients.^[24] Specifically, for an algebraic number \alpha of degree n with minimal polynomial P, the (absolute logarithmic) Weil height satisfies h(\alpha) = \frac{1}{n} \log M(P), linking the Mahler measure directly to heights in Diophantine geometry.^[24] highlighting how the Mahler measure captures the "average" growth of roots outside the unit disk in a way that scales with the degree.^[24] A key property of the Mahler measure is its multiplicativity: for polynomials P and Q with no common roots, M(PQ) = M(P) M(Q), which facilitates its use in factoring and product formulas for heights of composite algebraic numbers.^[24] This multiplicativity extends naturally to dynamical systems on tori, where the Mahler measure of a Laurent polynomial f \in \mathbb{Z}[u_1^{\pm 1}, \dots, u_d^{\pm 1}] equals the topological entropy of the associated \mathbb{Z}^d-action \alpha_f on the d-torus, providing a bridge between algebraic geometry and ergodic theory. In this context, the measure quantifies the exponential growth rate of periodic points and mixing properties, with zero entropy corresponding to polynomials defining finite orbits. Applications of the Mahler measure include Lehmer's problem, which conjectures the existence of a constant c > 1 such that for any non-constant, non-cyclotomic polynomial P \in \mathbb{Z}, either M(P) = 1 or M(P) \geq c.^[24] The smallest known value greater than 1 is M(L) \approx 1.17628 for Lehmer's polynomial L(z) = z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1, and partial results bound the measure from below using effective versions of Dobrowolski's theorem, though the conjecture remains open.^[24] These investigations connect to zeta functions via special values, such as evaluations at algebraic points yielding L-series related to the measure of minimal polynomials.^[24]

Automorphic Forms

In the theory of automorphic forms, particularly cohomological ones, the height of an automorphic form \pi is defined as the L^2 norm of an integrally normalized representative of its cohomology class in the minimal cohomological degree on the associated locally symmetric space Y_K.^[25] This height depends on an invariant metric on the global symmetric space and provides a measure of the arithmetic complexity of the form. For cusp forms on \mathrm{GL}_2, this often corresponds to the Petersson norm \|f\|^2 = \int_{\Gamma \backslash \mathfrak{H}} |f(z)|^2 y^k \frac{dx dy}{y^2}, where k is the weight, which quantifies the L^2 size and ensures square-integrability on the quotient \Gamma \backslash \mathfrak{H}. Cusp forms satisfy exponential decay as y = \Im(z) \to \infty, with |f(z)| \ll e^{-c y} for some c > 0, due to the vanishing constant term in their Fourier expansion at the cusps. This decay aligns with the cuspidality condition and the automorphy factor f(\gamma z) = j(\gamma, z)^k f(z) for \gamma \in \mathrm{SL}_2(\mathbb{Z}). Forms of bounded height in this sense contribute to the unitarity of representations and the spectral theory on arithmetic quotients. They form an orthonormal basis for the cuspidal spectrum under the Petersson inner product, facilitating the decomposition of the Laplacian and applications of the Selberg trace formula. Bounded height conditions aid in subconvexity bounds for associated L-functions L(s, f) and their analytic continuation, linking spectral parameters to arithmetic invariants. The Maass-Selberg relations provide inner product formulas for truncated Eisenstein series, ensuring meromorphic continuation of L-functions. For the Eisenstein series E(z, s) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \Im(\gamma z)^s, the functional equation \Xi(s, f) = C(s) \Xi(1 - s, \tilde{f}) relates to the scattering matrix C(s), bounding residues and informing the continuous spectrum's role in arithmetic invariants. In the context of motives, the height of the automorphic form relates to the arithmetic height of the associated motive via conjectures involving L(1, \mathrm{Ad} \pi).^[25]

Arithmetic Dynamics

In arithmetic dynamics, height functions are adapted to study the behavior of points under iteration of algebraic morphisms. For a morphism f: \mathbb{P}^n \to \mathbb{P}^n of degree d \geq 2 defined over a number field K, the dynamical height of a point P \in \mathbb{P}^n(\overline{K}) is defined as h_f(P) = h(P) + O(1), where h is the absolute Weil height and the implied constant depends only on f, n, and the places of K. This notion captures the arithmetic growth of orbits under f. The associated canonical height is given by

\hat{h}_f(P) = \lim_{n \to \infty} d^{-n} h(f^{\circ n}(P)),

where f^{\circ n} denotes the n-th iterate of f; this limit exists and is independent of the choice of Weil height used in the definition. The canonical height \hat{h}_f satisfies key functional and algebraic properties that mirror those of the Néron–Tate height on abelian varieties. In particular, it obeys the equation \hat{h}_f(f(P)) = d \cdot \hat{h}_f(P), making it a homogeneous function with respect to the dynamics, and \hat{h}_f(P) \geq 0 for all P, with equality if and only if P is preperiodic (i.e., its forward orbit under f is finite). These properties enable the classification of preperiodic points via vanishing heights and facilitate equidistribution results for sequences of iterates. For instance, in the case of rational maps on \mathbb{P}^1, the zero set of \hat{h}_f precisely identifies the preperiodic points, providing a tool analogous to torsion points in elliptic curve arithmetic.^[26] Applications of canonical heights abound in proving finiteness results for periodic and preperiodic points. Silverman established theorems showing that, for morphisms with good reduction outside a finite set of primes, the number of rational periodic points of bounded period is finite, leveraging height bounds to control orbit growth. More broadly, these heights underpin analogs of classical conjectures, such as the dynamical Bogomolov conjecture, which posits that there exists a constant \epsilon_f > 0 depending only on f such that \hat{h}_f(P) \geq \epsilon_f or \hat{h}_f(P) = 0 for all P \in \mathbb{P}^n(\overline{K}); this has been proven in special cases, including split rational maps on \mathbb{P}^1 \times \mathbb{P}^1. Silverman's specialization theorems further imply finiteness of periodic points for specializations of families of morphisms with sufficiently good reduction.^[26]^[27] Recent developments extend height functions to non-archimedean settings and uniform bounds across families. In p-adic dynamics, canonical heights are defined using p-adic valuations, enabling analogs of Northcott's theorem for points of bounded p-adic height under iteration; for example, attracting cycles in p-adic polynomial dynamics yield explicit height bounds for post-critically finite maps. Ingram and collaborators have advanced uniform boundedness results post-2010, including lower bounds on canonical heights for quadratic polynomials and rational functions over number fields, supporting the dynamical Northcott conjecture that the set of points with \hat{h}_f(P) \leq T and bounded degree is finite for fixed T. These works culminate in proofs of the geometric dynamical Northcott property for polarized endomorphisms over function fields, with implications for uniform estimates in arithmetic families.^[28]^[29]

References

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Jan 21, 2006 · A height function on X(K) is a function. H : X(K) −→ R whose value H(P) measures the arithmetic complexity of the point P. For example, in ...
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Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem by Weil (1929) and Faltings (1983) respectively.
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Northcott's theorem on heights I. A general estimate
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[PDF] class number problems for real quadratic fields with small ...
Version of 23 Feb 2021. CLASS NUMBER PROBLEMS FOR REAL QUADRATIC FIELDS. WITH SMALL FUNDAMENTAL UNIT. MARK WATKINS. Abstract. Chowla conjectured that ...Missing: naive | Show results with:naive
[12]
[PDF] Joseph H. Silverman - The Arithmetic of Elliptic Curves
The past two decades have witnessed tremendous progress in the study of elliptic curves. Among the many highlights are the proof by Merel [170] of uniform bound ...
[13]
[PDF] ARAKELOV GEOMETRY
subsequent treatment of heights in Arakelov geometry. Our basic reference is FULTON. (1998), supplemented by MATSUMURA (1980); THORUP (1990) and ...
[14]
[PDF] Arakelov geometry, heights, equidistribution, and the Bogomolov ...
The advent of arithmetic intersection theory with Arakelov (1974) and, above all, its extension in any dimension by Gillet & Soulé (1990) (“Arakelov geom- etry”) ...Missing: 1970s | Show results with:1970s
[15]
[PDF] Heights in Arakelov Geometry - Universiteit Leiden
In diophantine geometry heights are used to control the number of rational points, they are used for finiteness statements or describing distributions of ...
[16]
On Szpiro's Discriminant Conjecture - Oxford Academic
The proofs use the theory of logarithmic forms and Arakelov theory for arithmetic surfaces. Issue Section: Articles · Download all slides. Advertisement ...<|separator|>
[17]
A Survey of the Hodge-Arakelov Theory of Elliptic Curves II
Theorem 4.3). Thus, development (i) brings us closer to the goal of applying Hodge-. Arakelov theory to proving Szpiro 's conjecture (for elliptic curves over.
[18]
The effective Shafarevich conjecture for abelian varieties of ${\text {GL}
May 19, 2021 · The proof combines Faltings' method with Serre's modularity conjecture, isogeny estimates and results from Arakelov theory.<|separator|>
[19]
An effective Shafarevich theorem for elliptic curves - ResearchGate
Aug 9, 2025 · ... Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as ...
[20]
Finiteness Theorems for Abelian Varieties over Number Fields
Finiteness Theorems for Abelian Varieties over Number Fields. Chapter. pp 9–26; Cite this chapter. Download book PDF.
[21]
[PDF] MAXIMAL HEIGHT OF DIVISORS OF xn − 1 - Dartmouth Mathematics
Apr 20, 2005 · Abstract. The size of the coefficients of cyclotomic polynomials is a problem that has been well-studied. This paper investigates.<|control11|><|separator|>
[22]
On the number of reducible polynomials of bounded naive height
Jan 19, 2014 · We prove an asymptotical formula for the number of reducible integer polynomials of degree d and of naive height at most T when T → ∞ .
[23]
ON THE DISTANCE BETWEEN ROOTS OF INTEGER POLYNOMIALS
In this paper we denote by H(P) the naive height of an integer polynomial P(X), that is, the maximum of the absolute values of its coefficients. In ...
[24]
[PDF] A Survey on Coefficients of Cyclotomic Polynomials - arXiv
Dec 15, 2021 · Upper bounds for the height of ternary cyclotomic polynomials have been studied by many authors. Bang (1895) [17] proved that A(pqr) ≤ p − 1.<|control11|><|separator|>
[25]
[PDF] the mahler measure of algebraic numbers: a survey
In [101] Mahler called M(P) the measure of the polynomial P, apparently. to distinguish it from its (naıve) height. This was first referred to as Mahler's.
[26]
[PDF] Automorphic forms on GL(2) Hervé Jacquet and Robert P. Langlands
Since a grössencharakter is an automorphic form on GL(1) one is tempted to ask if the Euler prod- ucts associated to automorphic forms on GL(2) play a role in ...
[27]
[PDF] AUTOMORPHIC FORMS Contents 1. Representations of adelic ...
Our objective now is to present a classical automorphic form (which is, in partic- ular, a function on Γ0(N)\GL2(R)0) as an automorphic form on GL2(A). We will.<|control11|><|separator|>
[28]
[PDF] Lecture Notes on Arithmetic Dynamics - Arizona Winter School
Feb 8, 2010 · Tate used the telescoping sum trick as in Theorem 17, while. N้ron constructed the canonical height as a sum of local heights. See [3] for the ...
[29]
On the dynamical Bogomolov conjecture for families of split rational ...
We prove that Zhang's dynamical Bogomolov conjecture holds uniformly along 1-parameter families of rational split maps and curves.Missing: Morioka | Show results with:Morioka
[30]
[PDF] Attracting Cycles in p-adic Dynamics and Height Bounds for Post ...
ATTRACTING CYCLES IN p-ADIC DYNAMICS AND HEIGHT BOUNDS FOR. POST-CRITICALLY FINITE MAPS. ROBERT BENEDETTO, PATRICK INGRAM, RAFE JONES, AND ALON LEVY. Abstract ...
[31]
The Geometric Dynamical Northcott and Bogomolov Properties - arXiv
Dec 17, 2019 · We establish the dynamical Northcott property for polarized endomorphisms of a projective variety over a function field \mathbf{K} of characteristic zero.