21.09.03 · number-theory / arakelov-faltings

Heights and the Néron–Tate canonical height

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Anchor (Master): Néron 1965 *Annals of Mathematics* 82, 249-331 (originator paper — *Quasi-fonctions et hauteurs sur les variétés abéliennes*: the local decomposition of the canonical height into Néron local height functions, one per place); Tate (unpublished, ~1962, exposed in Lang *Elliptic Curves: Diophantine Analysis* and Manin-Panchishkin) — the limit construction $\hat h(P) = \lim_n h(nP)/n^2$ producing the canonical height by telescoping the parallelogram defect; Weil 1929 *Acta Mathematica* 52, 281-315 (the Mordell-Weil theorem over number fields, the descent in which heights first appear) and Weil 1951 *J. Math. Pures Appl.* 29 (the height machine attaching functorial heights to divisor classes); Northcott 1949 *Proc. Cambridge Phil. Soc.* 45 (the finiteness property); Silverman 2009 *The Arithmetic of Elliptic Curves* GTM 106 Ch. VIII; Bombieri-Gubler 2006 *Heights in Diophantine Geometry* Ch. 1-9; Lang 1983 *Fundamentals of Diophantine Geometry* (Springer) Ch. 4-5; Manin-Panchishkin 2005 *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed.) Ch. 4 §3, Ch. 6 §3

Intuition Beginner

A rational number like $\frac{3}{4}$ feels small, while $\frac{1000001}{999998}$ feels large and complicated, even though both are close to the same real value. A height is a way to make this feeling precise: a number that measures the arithmetic complexity of a point, roughly the number of digits needed to write it down. The fraction $\frac{a}{b}$ in lowest terms has height about $max (∣ a ∣, ∣ b ∣)$ , so $\frac{3}{4}$ has small height and $\frac{1000001}{999998}$ has large height.

The single most useful fact about heights is a finiteness statement. There are only finitely many rational numbers whose height is below any fixed bound, because there are only finitely many fractions $\frac{a}{b}$ with $∣ a ∣$ and $∣ b ∣$ both small. This is the engine behind many theorems: if you can show the rational points you care about all have bounded height, then there can be only finitely many of them.

On an elliptic curve, points can be added together by the group law. One wants a height that interacts well with this addition, so that doubling a point multiplies its height in a predictable way. The naive height almost does this, but with an annoying bounded error. The canonical height is a polished version that removes the error completely: it behaves like a perfect squared-length, a precise gauge in which adding points and measuring their size fit together exactly.

Visual Beginner

Picture the rational points of an elliptic curve as dots, each tagged with a number measuring its complexity. Small dots (the simple points like a torsion point) sit near the centre; large dots (points reached by adding a generator to itself many times) march outward, their tags growing like the square of how many steps they took. A dashed circle of fixed radius encloses only finitely many dots: this is the Northcott finiteness property drawn as a picture.

The key visual idea: the canonical height turns the group $E (K)$ into something like a lattice of dots in space, where the height is the squared distance from the origin, and only finitely many lattice points fit inside any fixed ball.

Worked example Beginner

Compute heights of a few rational points and watch the canonical height emerge.

Step 1. The logarithmic height of a rational number $\frac{a}{b}$ in lowest terms is $h (\frac{a}{b}) = lo g max (∣ a ∣, ∣ b ∣)$ . So $h (\frac{3}{4}) = lo g 4 \approx 1.386$ , and a torsion-like simple value such as $\frac{0}{1} = 0$ has height $lo g 1 = 0$ . Bigger fractions have bigger heights: $h (\frac{1000001}{999998}) = lo g 1000001 \approx 13.8$ .

Step 2. On a curve, doubling a point roughly squares the size of its coordinates. So the naive height of $2 P$ is about four times the naive height of $P$ : the doubling formula on an elliptic curve is a degree-four map on coordinates, and a degree-four map multiplies logarithmic height by about $4$ . In symbols, $h (2 P) \approx 4 h (P)$ , but with a bounded error that depends on the point.

Step 3. Tate's idea removes the error. Form the sequence $\frac{h ( 2 P )}{4}, \frac{h ( 4 P )}{16}, \frac{h ( 8 P )}{64}, \dots$ , dividing the height of $2^{n} P$ by $4^{n}$ . Each step corrects a little of the bounded error, and the numbers settle down to a limit. That limit is the canonical height $\hat{h} (P)$ .

Step 4. The canonical height squares under doubling with no error at all: $\hat{h} (2 P) = 4 \hat{h} (P)$ , and more generally $\hat{h} (n P) = n^{2} \hat{h} (P)$ . The "squared-length" behaviour is now exact.

Step 5. The finiteness pay-off. Since only finitely many points have height below any bound, and since a point of height zero turns out to be a torsion point (one of finite order), the canonical height sorts the points of the curve cleanly: a finite torsion core at height zero, and everything else spreading outward with quadratically growing height.

What this tells us: the naive height already measures complexity and already gives finiteness, but it wobbles under the group law. The canonical height is the wobble-free version, a perfect quadratic gauge in which "small means few" and "doubling means times four" both hold on the nose.

Check your understanding Beginner

Exercise (easy, multiple choice).

Why are there only finitely many rational numbers of height below a fixed bound $B$ ?

A. Because the rationals are countable B. Because a bounded height forces both numerator and denominator to be bounded, and there are finitely many such fractions C. Because every rational number is a root of unity D. Because heights are always irrational

Hint

The logarithmic height of $\frac{a}{b}$ in lowest terms is $lo g max (∣ a ∣, ∣ b ∣)$ . A bound on the height is a bound on $max (∣ a ∣, ∣ b ∣)$ .

Answer

B. A bound $h (\frac{a}{b}) \leq B$ means $max (∣ a ∣, ∣ b ∣) \leq e^{B}$ , so both numerator and denominator are bounded integers, and only finitely many fractions satisfy that. Option A is true but does not give finiteness of a bounded set. Option C is false. Option D is false: heights are typically logarithms of integers and can be rational or zero.

Formal definition Intermediate+

Let $K$ be a number field with set of places $M_{K}$ , each place $v$ normalised so that the product formula holds: $\prod_{v \in M_{K}} ∣ x ∣_{v} = 1$ for every $x \in K^{\times}$ , where the normalisation absorbs the local degrees $n_{v} = [K_{v} : Q_{v}]$ into $∣ \cdot ∣_{v}$ .

Multiplicative Weil height. For a point $P = [x_{0} : \dots : x_{n}] \in P^{n} (K)$ , the multiplicative height is $$ H(P) = \prod_{v \in M_K} \max_{0 \le i \le n} |x_i|_v. $$ The product formula makes $H (P)$ independent of the choice of homogeneous coordinates: scaling all $x_{i}$ by a common $λ \in K^{\times}$ multiplies the product by $\prod_{v} ∣ λ ∣_{v} = 1$ . The logarithmic (Weil) height is $$ h(P) = \log H(P) = \sum_{v \in M_K} \log \max_i |x_i|_v \ge 0. $$ The absolute height is obtained by normalising for the field of definition, $H (P)^{1/ [K : Q]}$ , so that $h$ is well-defined on $P^{n} (\overline{Q})$ independently of the number field through which $P$ is presented. The additive form is $h$ ; the multiplicative form is $H = e^{h}$ .

Northcott finiteness. For every pair of bounds $B, d$ , the set $$ { P \in \mathbb{P}^n(\overline{\mathbb{Q}}) : h(P) \le B,\ [\mathbb{Q}(P) : \mathbb{Q}] \le d } $$ is finite. Kronecker's theorem is the boundary case: an algebraic number $α$ has $h (α) = 0$ if and only if $α = 0$ or $α$ is a root of unity.

Weil height machine. For a smooth projective variety $X / K$ and a divisor class $D \in Pic (X)$ , there is a height function $h_{X, D} : X (\overline{Q}) \to R$ , well-defined up to a bounded function $O (1)$ , characterised by: (i) normalisation $h_{P^{n}, O (1)} = h + O (1)$ ; (ii) additivity $h_{X, D + E} = h_{X, D} + h_{X, E} + O (1)$ ; (iii) functoriality $h_{X, f^{*} D} = h_{Y, D} \circ f + O (1)$ for a morphism $f : X \to Y$ . The machine is a homomorphism $Pic (X) \to {functions X (\overline{Q}) \to R} / O (1)$ .

Néron–Tate canonical height. Let $A / K$ be an abelian variety (the elliptic-curve case $A = E$ is the main one) and $D$ a symmetric ample divisor class, so $[n]^{*} D \sim n^{2} D$ in $Pic (A) \otimes Q$ . The canonical height is the limit $$ \hat h(P) = \hat h_D(P) = \lim_{n \to \infty} \frac{h_{A,D}(nP)}{n^2}, $$ a genuine real-valued function (no $O (1)$ ambiguity) satisfying $\hat{h} = h_{A, D} + O (1)$ . On an elliptic curve with $D = (O)$ doubled to a degree-two class one usually writes $\hat{h} (P) = lim_{n} h (2^{n} P) / 4^{n}$ using the $x$ -coordinate height. The canonical height is a positive-definite quadratic form on $E (K) \otimes R$ : it satisfies the parallelogram law $$ \hat h(P+Q) + \hat h(P-Q) = 2\hat h(P) + 2\hat h(Q), $$ the scaling $\hat{h} (n P) = n^{2} \hat{h} (P)$ , and $\hat{h} (P) = 0 ⟺ P \in E (K)_{tors}$ .

Height pairing and regulator. The associated bilinear form is $$ \langle P, Q \rangle = \tfrac{1}{2}\big(\hat h(P+Q) - \hat h(P) - \hat h(Q)\big), $$ positive-definite on $E (K) / E (K)_{tors}$ . If $P_{1}, \dots, P_{r}$ generate the free part of $E (K)$ of rank $r$ , the regulator is $$ R_E = \det\big( \langle P_i, P_j \rangle \big)_{1 \le i, j \le r} > 0, $$ the covolume of the Mordell-Weil lattice, which enters the Birch-Swinnerton-Dyer formula.

Counterexamples to common slips

The naive height $h_{A, D}$ is not a quadratic form: $h (n P) = n^{2} h (P) + O (1)$ , with the error genuinely present. Only the canonical limit kills the error. Treating $h$ itself as quadratic produces false exact identities.
$\hat{h} (P) = 0$ characterises torsion, not the identity point alone. A nonzero torsion point of order $5$ also has canonical height zero. The vanishing locus of $\hat{h}$ is the whole torsion subgroup.
The canonical height depends on the chosen symmetric divisor class $D$ through a positive scalar: $\hat{h}_{D}$ and $\hat{h}_{D^{'}}$ differ by the ratio of their classes when both are symmetric ample. Positive-definiteness and the torsion characterisation are independent of this normalisation, but the numerical regulator is normalised by a fixed choice (the class of the origin for an elliptic curve).

Key theorem with proof Intermediate+

Theorem (existence and quadraticity of the canonical height; Néron 1965, Tate c. 1962 ^{[source pending]}). Let $E / K$ be an elliptic curve and $h$ the logarithmic Weil height attached to the degree-two divisor class of the origin. For every $P \in E (\overline{K})$ the limit $$ \hat h(P) = \lim_{n \to \infty} \frac{h(2^n P)}{4^n} $$ exists, satisfies $\hat{h} = h + O (1)$ , and defines a quadratic form on $E (\overline{K})$ : the parallelogram law $\hat{h} (P + Q) + \hat{h} (P - Q) = 2 \hat{h} (P) + 2 \hat{h} (Q)$ holds, $\hat{h} (n P) = n^{2} \hat{h} (P)$ , and $\hat{h} \geq 0$ with $\hat{h} (P) = 0$ if and only if $P$ is a torsion point.

Proof. The duplication formula on $E$ realises $[2] : E \to E$ as a morphism of degree four on the divisor class of the origin, so by the functoriality of the height machine there is a constant $C$ with $$ |h(2Q) - 4 h(Q)| \le C \qquad \text{for all } Q \in E(\overline{K}). $$ Apply this with $Q = 2^{n} P$ and divide by $4^{n + 1}$ : $$ \left| \frac{h(2^{n+1}P)}{4^{n+1}} - \frac{h(2^n P)}{4^n} \right| \le \frac{C}{4^{n+1}}. $$ The partial sums of $\sum_{n} C / 4^{n + 1}$ converge, so the sequence $a_{n} = h (2^{n} P) / 4^{n}$ is Cauchy and the limit $\hat{h} (P)$ exists. Telescoping the same bound from $n = 0$ gives $∣ \hat{h} (P) - h (P) ∣ \leq C /3$ , so $\hat{h} = h + O (1)$ .

For the scaling relation, the height machine gives $h (2 Q) = 4 h (Q) + O (1)$ , hence $h (2 \cdot 2^{n} P) / 4^{n} = 4 h (2^{n} P) / 4^{n} + O (4^{- n})$ ; passing to the limit yields $\hat{h} (2 P) = 4 \hat{h} (P)$ , and the analogous degree- $n^{2}$ bound for $[n]$ gives $\hat{h} (n P) = n^{2} \hat{h} (P)$ .

For the parallelogram law, the theorem of the square on $E$ supplies the divisor-class identity $[P + Q] + [P - Q] \sim 2 [P] + 2 [Q] - 2 [O]$ on the abelian variety, which through the height machine reads $h (P + Q) + h (P - Q) = 2 h (P) + 2 h (Q) + O (1)$ . Replacing each point by $2^{n}$ times itself, dividing by $4^{n}$ , and passing to the limit kills the $O (1)$ and yields the exact parallelogram law for $\hat{h}$ . A function satisfying the parallelogram law is a quadratic form, so $\hat{h}$ is quadratic.

Non-negativity follows because $\hat{h} = h + O (1)$ and $h \geq 0$ on a set bounded below after the scaling normalisation; more precisely, if $\hat{h} (P) < 0$ then $\hat{h} (n P) = n^{2} \hat{h} (P) \to - \infty$ , contradicting $\hat{h} \geq h - C /3$ bounded below, so $\hat{h} \geq 0$ . If $\hat{h} (P) = 0$ then all multiples $n P$ have $h (n P) = \hat{h} (n P) + O (1) = O (1)$ bounded, so by Northcott finiteness the set ${n P : n \in Z}$ is finite, forcing $P$ to have finite order. Conversely a torsion point has a finite orbit of bounded height, so its canonical height vanishes. $□$

Bridge. The canonical height builds toward the entire arithmetic of the Mordell-Weil group: it is exactly the positive-definite quadratic form that turns $E (K) / E (K)_{tors}$ into a lattice, and the finiteness it encodes is the foundational reason the descent argument terminates. This is exactly the input that powers the Birch-Swinnerton-Dyer regulator $R_{E} = det ⟨ P_{i}, P_{j} ⟩$ , where the height pairing measures the covolume of that lattice; the canonical height generalises the naive Weil height by removing its bounded defect, and putting these together one sees that the central insight — that arithmetic complexity is a squared-length — is dual to the geometric fact that $[n]^{*} D \sim n^{2} D$ on an abelian variety. The bridge is that this same quadratic form reappears as an arithmetic self-intersection number: the canonical height is the Arakelov height developed in 21.09.01, and it appears again in 21.09.02 where heights drive Faltings's proof of the Mordell conjecture, and in 21.06.01 where the regulator enters BSD.

Exercises Intermediate+

Exercise 5 (medium, symbolic).

Write the regulator $R_{E}$ of an elliptic curve of Mordell-Weil rank $r$ with free generators $P_{1}, \dots, P_{r}$ , and state why it is positive.

Hint

The regulator is the Gram determinant of the height pairing on the generators; the pairing is positive-definite.

Answer

R_{E} = det (⟨ P_{i}, P_{j} ⟩)_{1 \leq i, j \leq r} .

It is the Gram determinant of the height pairing on a basis of $E (K) / E (K)_{tors}$ . Because $⟨ \cdot, \cdot ⟩$ is positive-definite on the free part (the canonical height vanishes only on torsion), its Gram matrix on an independent set is positive-definite, hence has strictly positive determinant. Geometrically $R_{E}$ is the squared covolume of the Mordell-Weil lattice. Rubric: full credit for the determinant formula plus the positive-definiteness justification.

Exercise 6 (hard, short-answer).

Explain the role of the height function in the proof of the Mordell-Weil theorem (the descent argument), and why both the weak Mordell-Weil theorem and the height machine are needed.

Hint

The weak Mordell-Weil theorem gives finiteness of $E (K) / m E (K)$ ; the height supplies the descent that turns that into finite generation of $E (K)$ itself.

Answer

The Mordell-Weil theorem ( $E (K)$ finitely generated) is proved by descent. The weak Mordell-Weil theorem gives that $E (K) / m E (K)$ is finite for some $m \geq 2$ ; choose coset representatives $Q_{1}, \dots, Q_{t}$ . The height machine supplies two facts: $h (m P) = m^{2} h (P) + O (1)$ , and Northcott finiteness. Given any $P$ , write $P = m P^{'} + Q_{i_{1}}$ , then descend $P^{'}$ likewise; the quadratic growth of height under multiplication by $m$ means each descent step strictly decreases the height of the representative (once it exceeds a threshold), because $h (P^{'}) \approx h (P) / m^{2} + O (1) < h (P)$ for large $h (P)$ . After finitely many steps the height drops below a fixed bound, and by Northcott only finitely many points lie below that bound. Thus $E (K)$ is generated by the $Q_{i}$ together with the finite set of low-height points. The weak theorem alone gives no generators; the height alone cannot start the descent — the combination is what closes the argument. Rubric: full credit for identifying weak Mordell-Weil + the height-decrease descent + Northcott as the three ingredients.

Advanced results Master

Theorem (Néron's local decomposition; Néron 1965 ^{[source pending]}). Let $A / K$ be an abelian variety with symmetric ample $D$ and canonical height $\hat{h}_{D}$ . There exist Néron local height functions $λ_{v} : (A ∖ ∣ D ∣) (K_{v}) \to R$ , one for each place $v$ of $K$ , each a quasi-function bounded by an intersection-theoretic local term and continuous in the $v$ -adic topology, such that for every $P \in (A ∖ ∣ D ∣) (K)$ $$ \hat h_D(P) = \sum_{v \in M_K} \lambda_v(P). $$ The local heights are determined up to normalisation by their transformation under the group law and their logarithmic singularity along $D$ ; at archimedean places $λ_{v}$ is given by a Green's function on the complex torus $A (C)$ , and at non-archimedean places by an intersection number on the Néron model.

This decomposition is the precise bridge to Arakelov geometry. The archimedean local heights are Green's-function integrals on $A_{σ} (C)$ , and the non-archimedean local heights are local intersection multiplicities on the Néron model over $O_{K, v}$ ; their sum reconstitutes the global canonical height. In the language of 21.09.01, $\hat{h}_{D} (P)$ is the arithmetic intersection number $(P) \cdot D$ of the section $P$ against the hermitian line bundle attached to $D$ , so the canonical height is an Arakelov self-intersection.

Theorem (positive-definiteness over $R$ ; Néron-Tate). The canonical height extends by the quadratic-form property to a positive-definite real quadratic form on the finite-dimensional real vector space $E (K) \otimes R$ . Consequently $E (K) / E (K)_{tors}$ is a lattice of full rank in $E (K) \otimes R$ with respect to the inner product $⟨ \cdot, \cdot ⟩$ , and the regulator $R_{E} = det ⟨ P_{i}, P_{j} ⟩$ is the squared covolume of this lattice, independent of the choice of basis up to sign.

The positive-definiteness is the analytic heart of the theory. It implies the finiteness of points of bounded canonical height directly (a ball in a Euclidean lattice contains finitely many lattice points), recovering Northcott in the geometric language, and it makes the regulator a well-defined real invariant of the curve. The regulator is one of the arithmetic factors in the conjectural Birch-Swinnerton-Dyer formula for the leading Taylor coefficient of $L (E, s)$ at $s = 1$ .

Theorem (lower bounds and Lehmer-type problems; Bombieri-Gubler 2006 ^{[source pending]}). For a fixed elliptic curve $E / K$ , the canonical heights of non-torsion points are bounded below by a positive constant depending on $E$ (a result of the Northcott property), and the elliptic Lehmer conjecture predicts a uniform lower bound $\hat{h} (P) ≫ 1/ [Q (P) : Q]$ scaling with the degree of the field of definition. The classical Lehmer problem is the analogous statement for the Weil height of algebraic numbers (the multiplicative-group case), and the two are unified by the height machine viewed across distinct group-variety structures: each is a statement that the height cannot be too small unless the point is special (torsion, or a root of unity).

Synthesis. The foundational reason heights exist is that arithmetic complexity must be a single real number that grows predictably under the operations of the geometry, and this is exactly what the Weil height machine delivers: a homomorphism from divisor classes to functions-up-to- $O (1)$ . Putting these together, the naive Weil height already encodes Northcott finiteness and Kronecker's height-zero characterisation, the canonical height generalises it by removing the $O (1)$ defect through Tate's telescoping limit, and the resulting quadratic form is dual to the geometric relation $[n]^{*} D \sim n^{2} D$ on an abelian variety — the central insight that turns the group law into a squared-length. The bridge to the rest of the arc is Néron's local decomposition: the global canonical height splits as a sum of local heights, archimedean Green's functions plus non-archimedean intersection numbers, and this is exactly the Arakelov self-intersection of 21.09.01, so the canonical height is simultaneously a Diophantine measuring stick, a positive-definite lattice inner product, and an arithmetic-intersection number. This same object appears again in 21.06.01 as the BSD regulator and in 21.09.02 as the height driving Faltings's finiteness, and putting these together one sees a single quadratic form organising the descent that proves Mordell-Weil, the lattice geometry that gives the regulator, and the arithmetic intersection theory that proves Mordell.

Full proof set Master

Proposition (well-definedness of the Weil height under change of coordinates). The multiplicative height $H (P) = \prod_{v} max_{i} ∣ x_{i} ∣_{v}$ on $P^{n} (K)$ is independent of the choice of homogeneous coordinates representing $P$ .

Proof. Two coordinate vectors for the same projective point differ by a common scalar $λ \in K^{\times}$ : if $P = [x_{0} : \dots : x_{n}] = [λ x_{0} : \dots : λ x_{n}]$ , then at each place $v$ , $$ \max_i |\lambda x_i|_v = |\lambda|_v \max_i |x_i|_v. $$ Taking the product over all places, $$ \prod_v \max_i |\lambda x_i|_v = \Big(\prod_v |\lambda|_v\Big) \prod_v \max_i |x_i|_v = \prod_v \max_i |x_i|_v, $$ where $\prod_{v} ∣ λ ∣_{v} = 1$ by the product formula for the number field $K$ . Hence $H (P)$ is unchanged, and so is $h (P) = lo g H (P)$ . $□$

Proposition (the canonical height is well-defined and independent of the limit base). The limit $\hat{h} (P) = lim_{n} h (2^{n} P) / 4^{n}$ exists for all $P$ , and equals $lim_{n} h (n P) / n^{2}$ along the full sequence of positive integers; in particular the value does not depend on using base $2$ .

Proof. Existence and the Cauchy estimate were established in the Key Theorem: the bound $∣ h (2 Q) - 4 h (Q) ∣ \leq C$ gives $∣ a_{n + 1} - a_{n} ∣ \leq C / 4^{n + 1}$ for $a_{n} = h (2^{n} P) / 4^{n}$ , a convergent geometric tail. For independence of base, set $\hat{h} (P) = lim_{n} a_{n}$ and use $\hat{h} = h + O (1)$ together with the relation $\hat{h} (m P) = m^{2} \hat{h} (P)$ , which was proved from the degree- $m^{2}$ height-machine bound $∣ h (m Q) - m^{2} h (Q) ∣ \leq C_{m}$ . Then for any $n$ , $$ \frac{h(nP)}{n^2} = \frac{\hat h(nP) + O(1)}{n^2} = \frac{n^2 \hat h(P) + O(1)}{n^2} = \hat h(P) + O(n^{-2}) \to \hat h(P), $$ so the limit along all integers exists and equals the base- $2$ limit. The canonical height is therefore intrinsic. $□$

Proposition (the parallelogram law characterises $\hat{h}$ as quadratic). A function $q : G \to R$ on an abelian group $G$ satisfying $q (P + Q) + q (P - Q) = 2 q (P) + 2 q (Q)$ for all $P, Q$ and $q (0) = 0$ is a quadratic form: $B (P, Q) = \frac{1}{2} (q (P + Q) - q (P) - q (Q))$ is bi-additive and $q (P) = B (P, P)$ .

Proof. Setting $Q = 0$ in the parallelogram law gives $q (P) + q (P) = 2 q (P) + 2 q (0)$ , consistent with $q (0) = 0$ . Setting $P = Q$ gives $q (2 P) = 4 q (P)$ , and inductively $q (n P) = n^{2} q (P)$ . Symmetry of $B$ is immediate. For additivity in the first slot, expand $B (P_{1} + P_{2}, Q) - B (P_{1}, Q) - B (P_{2}, Q)$ using the definition and the parallelogram law repeatedly; the standard polarisation identity shows this difference is zero, so $B$ is bi-additive. Finally $B (P, P) = \frac{1}{2} (q (2 P) - 2 q (P)) = \frac{1}{2} (4 q (P) - 2 q (P)) = q (P)$ . Applying this with $q = \hat{h}$ , whose parallelogram law was proved via the theorem of the square, shows $\hat{h}$ is a quadratic form with associated pairing $⟨ P, Q ⟩ = B (P, Q)$ . $□$

Proposition (positivity of the regulator). If $P_{1}, \dots, P_{r}$ are $Z$ -independent in $E (K) / E (K)_{tors}$ , the Gram matrix $G = (⟨ P_{i}, P_{j} ⟩)$ is positive-definite, so $R_{E} = det G > 0$ .

Proof. The pairing $⟨ \cdot, \cdot ⟩$ is symmetric and bilinear on $E (K) \otimes R$ , and $⟨ P, P ⟩ = \hat{h} (P) \geq 0$ with equality only for torsion points. For independent $P_{1}, \dots, P_{r}$ and any nonzero real vector $(c_{1}, \dots, c_{r})$ , the combination $v = \sum_{i} c_{i} P_{i}$ in $E (K) \otimes R$ is nonzero (by independence), so $\sum_{i, j} c_{i} c_{j} ⟨ P_{i}, P_{j} ⟩ = ⟨ v, v ⟩ = \hat{h} (v) > 0$ . Thus $G$ is positive-definite, and a positive-definite symmetric matrix has strictly positive determinant, giving $R_{E} > 0$ . The determinant is the squared covolume of the lattice $E (K) / E (K)_{tors}$ in the inner-product space $(E (K) \otimes R, ⟨ \cdot, \cdot ⟩)$ . $□$

Connections Master

Elliptic curves and Mordell-Weil 04.04.03. The canonical height is the quadratic form that makes the descent proof of the Mordell-Weil theorem work: the height-decrease under multiplication by $m$ combined with Northcott finiteness turns the finiteness of $E (K) / m E (K)$ into finite generation of $E (K)$ . The height pairing then organises the free part $E (K) / E (K)_{tors}$ into a Euclidean lattice, and $\hat{h} (P) = 0 ⟺ P$ torsion pins down exactly which points collapse to the origin of that lattice.
Abelian varieties 04.10.16. On a general abelian variety $A / K$ with symmetric ample $D$ , the relation $[n]^{*} D \sim n^{2} D$ is the geometric source of the quadratic scaling $\hat{h}_{D} (n P) = n^{2} \hat{h}_{D} (P)$ . The theorem of the square supplies the parallelogram law, so the canonical height is a positive-semidefinite quadratic form on $A (K)$ , positive-definite modulo torsion. The elliptic-curve case is the dimension-one specialisation, and the Néron-Tate height on a Jacobian is the same construction applied to the theta divisor.
Arakelov geometry and arithmetic surfaces 21.09.01. Néron's local decomposition $\hat{h} (P) = \sum_{v} λ_{v} (P)$ identifies the canonical height with an Arakelov arithmetic intersection number: archimedean local heights are Green's-function integrals on $A_{σ} (C)$ and non-archimedean local heights are intersection multiplicities on the Néron model. The canonical height is therefore the arithmetic self-intersection $(P) \cdot D$ in the framework of that unit, making heights a special case of arithmetic intersection theory.
Modularity and the BSD conjecture 21.06.01. The regulator $R_{E} = det ⟨ P_{i}, P_{j} ⟩$ built from the height pairing is one of the arithmetic factors in the Birch-Swinnerton-Dyer formula for the leading coefficient of $L (E, s)$ at $s = 1$ . The Gross-Zagier theorem computes the canonical height of a Heegner point in terms of the derivative $L^{'} (E, 1)$ , directly linking the height pairing developed here to the analytic side of BSD.
Faltings / Mordell theorem 21.09.02. Heights are the bookkeeping that drives Faltings's and Vojta's proofs of the Mordell conjecture: the Faltings height of an abelian variety and the canonical height of points on a curve in its Jacobian are the invariants bounded in the finiteness arguments. The Northcott property of the height — finitely many points of bounded height and degree — is the abstract finiteness principle that those proofs ultimately exploit.

Historical & philosophical context Master

The height function entered number theory as the bookkeeping device in André Weil's 1929 thesis, where he proved the Mordell-Weil theorem over arbitrary number fields by an infinite descent that needed a numerical measure of the "size" of a rational point to guarantee termination ^{[Weil 1929]}. Weil's height $h$ measured the number of digits of a point's coordinates, and the decisive property was finiteness: only finitely many points have bounded height. Mordell had proved the genus-one case over $Q$ in 1922, and the descent already implicitly used heights; Weil made the height a systematic tool and later, around 1950, abstracted it into the height machine that attaches a functorial height to every divisor class on a projective variety, well-defined up to a bounded error.

The finiteness half was crystallised by D. G. Northcott in 1949 ^{[Northcott 1949]}, whose inequality showed that algebraic points of bounded height and bounded degree are finite in number — the pigeonhole principle on which the whole subject rests. Kronecker's much older theorem, that an algebraic number of height zero is a root of unity, is the boundary case marking the height-zero locus. The qualitative naive height, however, wobbled under the group law on an abelian variety: doubling a point only multiplied the height by four up to a bounded error, so the height was not a clean quadratic form.

The decisive refinement came around 1962 from John Tate, who observed that the bounded defect $h (2 P) - 4 h (P)$ could be telescoped away by averaging: the limit $\hat{h} (P) = lim_{n} h (2^{n} P) / 4^{n}$ converges and is an exact quadratic form. Independently and more structurally, André Néron's 1965 Annals paper Quasi-fonctions et hauteurs sur les variétés abéliennes ^{[Néron 1965]} constructed the canonical height through a decomposition into local height functions, one per place, each a quasi-function controlled by local intersection theory. The two constructions agree: Tate's global limit and Néron's local sum are two faces of one object. Néron's local viewpoint proved prophetic, for it is precisely the decomposition that, two decades later, would be recognised as an Arakelov arithmetic intersection number, fusing the Diophantine theory of heights with the arithmetic intersection theory of the present chapter. The canonical height now sits at the centre of the subject: it is the positive-definite quadratic form whose determinant is the Birch-Swinnerton-Dyer regulator, the invariant Faltings bounded to prove Mordell, and the arithmetic self-intersection that closes the Arakelov circle.

Bibliography Master

@article{Neron1965,
  author  = {N{\'e}ron, Andr{\'e}},
  title   = {Quasi-fonctions et hauteurs sur les vari{\'e}t{\'e}s ab{\'e}liennes},
  journal = {Annals of Mathematics},
  volume  = {82},
  year    = {1965},
  pages   = {249--331}
}

@article{Weil1929,
  author  = {Weil, Andr{\'e}},
  title   = {L'arithm{\'e}tique sur les courbes alg{\'e}briques},
  journal = {Acta Mathematica},
  volume  = {52},
  year    = {1929},
  pages   = {281--315}
}

@article{Northcott1949,
  author  = {Northcott, D. G.},
  title   = {An inequality in the theory of arithmetic on algebraic varieties},
  journal = {Proceedings of the Cambridge Philosophical Society},
  volume  = {45},
  year    = {1949},
  pages   = {502--509}
}

@book{Silverman2009,
  author    = {Silverman, Joseph H.},
  title     = {The Arithmetic of Elliptic Curves},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {106},
  edition   = {2},
  year      = {2009}
}

@book{BombieriGubler2006,
  author    = {Bombieri, Enrico and Gubler, Walter},
  title     = {Heights in Diophantine Geometry},
  publisher = {Cambridge University Press},
  series    = {New Mathematical Monographs},
  volume    = {4},
  year      = {2006}
}

@book{HindrySilverman2000,
  author    = {Hindry, Marc and Silverman, Joseph H.},
  title     = {Diophantine Geometry: An Introduction},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {201},
  year      = {2000}
}

@book{Lang1983,
  author    = {Lang, Serge},
  title     = {Fundamentals of Diophantine Geometry},
  publisher = {Springer-Verlag},
  year      = {1983}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yuri I. and Panchishkin, Alexei A.},
  title     = {Introduction to Modern Number Theory},
  publisher = {Springer-Verlag},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  edition   = {2},
  year      = {2005}
}

Prerequisites

04.04.03
04.10.16
21.09.01

Tier anchors

beginner: Silverman *The Arithmetic of Elliptic Curves* (Springer GTM 106, 2nd ed. 2009) Ch. VIII §§4-5 informal opening on measuring the size of a rational point; Bombieri-Gubler *Heights in Diophantine Geometry* (Cambridge 2006) Ch. 1 informal preface on the naive height
intermediate: Silverman *The Arithmetic of Elliptic Curves* (Springer GTM 106, 2nd ed. 2009) Ch. VIII (heights, the canonical height, the descent theorem); Bombieri-Gubler *Heights in Diophantine Geometry* (Cambridge 2006) Ch. 1-2 (Weil height machine, Northcott, functoriality); Hindry-Silverman *Diophantine Geometry* (Springer GTM 201, 2000) Part B
master: Néron 1965 *Annals of Mathematics* 82, 249-331 (originator paper — *Quasi-fonctions et hauteurs sur les variétés abéliennes*: the local decomposition of the canonical height into Néron local height functions, one per place); Tate (unpublished, ~1962, exposed in Lang *Elliptic Curves: Diophantine Analysis* and Manin-Panchishkin) — the limit construction $\hat h(P) = \lim_n h(nP)/n^2$ producing the canonical height by telescoping the parallelogram defect; Weil 1929 *Acta Mathematica* 52, 281-315 (the Mordell-Weil theorem over number fields, the descent in which heights first appear) and Weil 1951 *J. Math. Pures Appl.* 29 (the height machine attaching functorial heights to divisor classes); Northcott 1949 *Proc. Cambridge Phil. Soc.* 45 (the finiteness property); Silverman 2009 *The Arithmetic of Elliptic Curves* GTM 106 Ch. VIII; Bombieri-Gubler 2006 *Heights in Diophantine Geometry* Ch. 1-9; Lang 1983 *Fundamentals of Diophantine Geometry* (Springer) Ch. 4-5; Manin-Panchishkin 2005 *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed.) Ch. 4 §3, Ch. 6 §3

References

Néron, A. — Quasi-fonctions et hauteurs sur les variétés abéliennes · *Annals of Mathematics* 82 (1965), 249-331. The originator paper for the canonical (Néron-Tate) height on an abelian variety, constructed through a decomposition $\hat h(P) = \sum_v \lambda_v(P)$ into Néron local height functions $\lambda_v$, one for each place $v$ of the number field, each a quasi-function (a function well-defined up to a bounded error governed by an intersection-theoretic local term). The local decomposition is the archimedean / non-archimedean refinement that connects the canonical height to Arakelov intersection numbers.
Tate, J. — The canonical height as a limit (unpublished, c. 1962) · Tate's limit construction $\hat h(P) = \lim_{n \to \infty} h(2^n P)/4^n$ (equivalently $\lim_n h(nP)/n^2$), exposed in Lang, *Elliptic Curves: Diophantine Analysis* (Springer 1978) Ch. 5 and in Silverman GTM 106 Ch. VIII §9. The defect $h(2P) - 4h(P)$ is bounded, so the sequence $h(2^n P)/4^n$ is Cauchy and its limit is the quadratic-form refinement of the naive height. The cleanest route to the parallelogram law and positive-definiteness.
Weil, A. — L'arithmétique sur les courbes algébriques / Arithmetic on algebraic varieties · *Acta Mathematica* 52 (1929), 281-315 (the Mordell-Weil theorem over number fields, the weak Mordell-Weil group $E(K)/mE(K)$ and the descent in which the height function first appears as the bookkeeping device controlling the infinite descent); and the later height-machine formalism (*Journal de Mathématiques Pures et Appliquées* 29 (1950), 281-314) attaching to each divisor class a height function well-defined up to $O(1)$, functorial under morphisms.
Northcott, D. G. — An inequality in the theory of arithmetic on algebraic varieties · *Proceedings of the Cambridge Philosophical Society* 45 (1949), 502-509. The Northcott finiteness property: for any bounds $B, d$, the set of algebraic points $P \in \mathbb{P}^n(\overline{\mathbb{Q}})$ with logarithmic height $h(P) \leq B$ and degree $[\mathbb{Q}(P) : \mathbb{Q}] \leq d$ is finite. The pigeonhole principle behind the finiteness half of Mordell-Weil and of the descent argument.
Silverman, J. H. — The Arithmetic of Elliptic Curves · Springer Graduate Texts in Mathematics 106, 2nd edition (2009). Ch. VIII: the Weil height on projective space, the height machine, the canonical height on an elliptic curve via Tate's limit, the parallelogram law, positive-definiteness, the height pairing and the regulator, and the descent theorem proving Mordell-Weil. The anchor textbook for the present unit.
Bombieri, E. & Gubler, W. — Heights in Diophantine Geometry · Cambridge New Mathematical Monographs 4 (Cambridge University Press 2006). Ch. 1-2 develop the absolute Weil height, the product formula, the height machine, Northcott and Kronecker; Ch. 9 develops the Néron-Tate height on abelian varieties as a quadratic form and the height regulator. The systematic modern reference for the height-theoretic foundations of Diophantine geometry.
Manin, Yu. I. & Panchishkin, A. — Introduction to Modern Number Theory · Springer Encyclopaedia of Mathematical Sciences 49, 2nd edition (2005). Ch. 4 §3 and Ch. 6 §3: heights on projective space and on abelian varieties, the Néron-Tate canonical height as the quadratic form entering the Birch-Swinnerton-Dyer regulator, and the role of heights in the proofs of Mordell-Weil and the Mordell conjecture. The source-book context for the present unit.

Estimated time

beginner: 20m
intermediate: 45m
master: 85m