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For the primitive minimal polynomial $a_d{\prod }_j(T-{\alpha }_j)$, the height-Mahler measure formula gives $H(\alpha {)}^d=a_d{\prod }_j\max (1,|{\alpha }_j|)$. Expanding the product bounds every integer coefficient by $2^D{B}^D$. There are finitely many bounded coefficient lists and each has finitely many roots, proving finiteness of the algebraic numbers. On a projective coordinate chart, every coordinate ratio has height at most the point's projective height; applying the scalar bound proves the projective Northcott theorem too. A degree bound is essential, as the infinitely many roots of unity all have height one.

The naive height of a polynomial is the largest absolute value of its coefficients. For a primitive polynomial in $\mathbb{Z}[X_1,\dots ,X_n]$, this equals the projective height of its coefficient vector.

Let $K$ be a number field of degree $d$, and normalize its absolute values so that the product formula is ${\prod }_v|a{|}_v^{n_v}=1$ for $a\ne 0$, where $n_v=[K_v:{\mathbb{Q}}_v]$; at a complex place use the ordinary modulus with weight two. For $P=[a_0:\cdots :a_N]$, defineOnly finitely many factors differ from one. Rescaling all coordinates multiplies the local maxima by $|a{|}_v$, whose weighted product is one, so the projective height is well-defined. It is at least one by comparison with any nonzero coordinate. On enlarging $K$ to $L$, the identity ${\sum }_{w\mid v}[L_w:{\mathbb{Q}}_v]=[L:K][K_v:{\mathbb{Q}}_v]$ makes the normalized product unchanged. Thus this is an absolute height on algebraic projective points. For coprime rational integer coordinates it is simply their maximum absolute value.

For a projective variety $V/K$ and an embedding $i:V\hookrightarrow {\mathbb{P}}^N$, take $h_i=h\circ i$. Changing the basis of the same projective linear system changes the height by a bounded amount: the coordinate transformation and its inverse have fixed local coefficient bounds, equal to one outside finitely many places. If $L$ is ample and $L^m$ is very ample, set $h_L=m^{-1}{h}_i$ for an embedding from $L^m$, up to a bounded function. A general line bundle can be written $(L\otimes A)\otimes A^{-1}$ with both $A$ and $L\otimes A$ very ample, defining its height as the difference of their embedding heights.

This is the Weil height machine, withThe first relation follows from the Segre embedding, whose local maximum is the product of the two maxima; tensor powers similarly use the Veronese embedding. The second follows by pulling back the defining sections. These constructions make the choices equivalent modulo bounded functions. For a fixed homogeneous morphism with no base point, upper local height bounds come from its coefficients, and lower bounds from the Projective Nullstellensatz expressing coordinate powers in the ideal of its defining forms. The constants are one outside a finite set of places. This gives the uniform bounded errors and shows why heights belong to line-bundle classes rather than being independent of every projective embedding.

The Northcott lemma states that for fixed $B$ and $D$, the algebraic points of ${\mathbb{P}}^N$ with $H(P)\le B$ and $[\mathbb{Q}(P):\mathbb{Q}]\le D$ form a finite set. In particular the $K$-rational points of bounded projective height are finite. Here is a proof, including the degree bound.

First let $\alpha$ have degree $e\le D$ and $H(\alpha )=H([1:\alpha ])\le B$. Write its primitive minimal polynomial as $a_e{\prod }_{j=1}^e(T-{\alpha }_j)$, with positive integer $a_e$. The height-Mahler measure formula isIts finite-place part follows from multiplicativity of the nonarchimedean polynomial coefficient norm: a primitive integer polynomial has norm one at each prime, so factoring it bounds the product of the roots' local maxima by the inverse leading-coefficient norm. The archimedean part is the displayed product over conjugates. Combining them with the product formula gives the identity.

Expanding the polynomial now bounds each integer coefficient byThere are finitely many degrees and finitely many such integer coefficient lists, each with finitely many roots. Hence there are only finitely many possible $\alpha$. This proves the minimal-polynomial coefficient proof of Northcott finiteness for algebraic numbers.

On the projective chart $a_j\ne 0$, each ratio ${\alpha }_i=a_i/a_j$ has degree at most the point's field degree andThe equality uses the product formula. Each coordinate ratio therefore belongs to a finite set, and there are finitely many charts. This proves the projective Northcott theorem. Restricting to $V$ gives the result for a chosen embedding. For an ample line-bundle height, the comparison $m{h}_L=h_i+O(1)$ reduces its bounded-height sets to this case.

Both qualifications matter: a degree bound cannot be dropped, since infinitely many roots of unity have height one; and the finiteness assertion need not hold for a nonample line bundle, such as the trivial bundle's bounded height on a variety with infinitely many rational points.

Arithmetic heights measure arithmetic complexity. For a reduced rational number $x=a/b$, its multiplicative height is $H(x)=\max (|a|,|b|)$ and its logarithmic height is $h(x)=\log H(x)$. Over a number field $K$, use the absolute projective heightNormalize the absolute values on a field by the product formula. That formula makes arithmetic height independent of the chosen homogeneous representative, and the normalization makes it independent of enlarging $K$. The Northcott theorem says that algebraic points of bounded degree and bounded arithmetic height form a finite set. For rational numbers this follows immediately by bounding the numerator and denominator. In a fixed number field, the same finiteness follows by bounding the coefficients of the minimal polynomial using its conjugates and the arithmetic height.

For an elliptic curve put $h_x(P)=h([x(P):1])$ and $h_x(O)=0$. The two points above a given $x$ show that bounded $h_x$ gives finitely many $K$-rational points. More importantly, multiplication expands this arithmetic height. To see the mechanism, choose a short Weierstrass equation of an elliptic curve $y^2=x^3+Ax+B$ over $K$. Doubling givesAfter homogenization the numerator and denominator are quartics with no common zero. At infinity the numerator is nonzero; at a root $t$ of the cubic the numerator becomes $(3t^2+A{)}^2$, nonzero by nonsingularity. For a degree-$d$ morphism of the projective line, fixed homogeneous coefficients give the upper arithmetic height bound $h(f(x))\le dh(x)+C$. A nonzero resultant supplies the reverse bound: at each place it bounds the common cancellation of the two homogeneous values, with a nontrivial constant needed at only finitely many places. Applying these bounds to duplication gives a constant $C$ independent of $P$ withThis includes points mapping to $O$, by the projective formulation.

The canonical height removes the bounded errors. With the conventional factor one half, defineThe successive terms before multiplying by one half differ by at most $C{4}^{-n-1}$. The limit exists, and summing this geometric bound givesHence bounded canonical height still gives finitely many points over the fixed number field.

To obtain the pairing, the quadratic nature comes from a uniform arithmetic height parallelogram estimate, not just from doubling. If $x=x(P)$ and $z=x(Q)$, the unordered coordinates $x(P+Q),x(P-Q)$ are the roots of the binary quadratic with coefficientsThese formulas follow by adding and multiplying the two chord expressions with slopes $(y(P)-y(Q))/(x-z)$ and $(y(P)+y(Q))/(x-z)$. Their homogenizations have bidegree $(2,2)$ and no simultaneous zero. Off the diagonal $C_2\ne 0$; on the finite diagonal $x=z=t$, $C_1=4f(t)$ and $C_0=f^{\prime }(t{)}^2-8tf(t)$ for $f(t)=t^3+At+B$, so simultaneous vanishing would contradict nonsingularity. At the double point at infinity the homogeneous $C_0$ is nonzero. Local coefficient norms, or the corresponding resultant bounds, therefore giveThe arithmetic height of a binary quadratic's coefficient vector differs by $O(1)$ from the sum of the heights of its two roots. At finite places this follows from multiplicativity of the maximum coefficient norm of products; at infinite places the fixed-degree norms are comparable. ThusApply this to $2^{n}P,2^{n}Q$, divide by $2\cdot 4^n$ and take limits. The errors vanish and give the exact height parallelogram identity. Polarization consequently makesa symmetric bilinear pairing. Nonnegativity of $\hat{h}(aP+bQ)$ for integers $a,b$, and density of rational ratios, give the Cauchy-Schwarz inequality. In particular $\hat{h}(P-Q)\le 2\hat{h}(P)+2\hat{h}(Q)$. Also $\hat{h}(mP)=m^2\hat{h}(P)$ for integers $m$. Torsion points of an elliptic curve have canonical height zero because their doubling orbit is finite; conversely, if a point has canonical height zero, all its integer multiples have bounded naive height, and finiteness forces two multiples to coincide. Thus canonical height vanishes exactly on torsion.

There is an arithmetic finiteness input before height descent: $E(K)/2E(K)$ is finite, the Weak Mordell-Weil theorem. Here is its mechanism. The multiplication exact sequence gives an injective Kummer map $E(K)/2E(K)\to H^1(K,E[2])$: choose a half-point $Q$ and use the cocycle $\sigma \mapsto \sigma Q-Q$; changing $Q$ changes a coboundary, and a trivial class means a half-point can be chosen over $K$. Let $L=K(E[2])$, a Finite Galois extension. Over $L$ the torsion module is $(\mathbb{Z}/2\mathbb{Z}{)}^2$, so Kummer theory identifies its first Galois cohomology with two copies of $L^{\times }/L^{\times 2}$.

Only classes unramified outside a fixed finite set $S$ occur. Include the primes over $2$, bad-reduction primes and the ramified primes of $L/K$. At a good prime away from $2$, multiplication by two on the proper smooth elliptic model is finite étale morphism, so the torsor of halves of a point is unramified. In square-class language, valuations outside $S$ are even. Such classes form a finite group: write the principal ideal of a representative as a square times an ideal supported on $S$. The ideal class gives an element of the finite two-torsion of the localized ideal class group; after fixing that class, ambiguity is an $S$-unit modulo squares. The $S$-unit theorem makes the latter finite. The group kernel of restriction from $K$ to $L$ is contained in $H^1(\mathrm{Gal}(L/K),E[2])$, which is finite because both the group and module are finite. This proves the weak theorem, without assuming finite generation of $E(K)$.

Finally choose representatives $R_1,\dots ,R_s$ of $E(K)/2E(K)$ and put $M={\max }_i\hat{h}(R_i)$. Every $P$ has the form $2Q+R_i$, andIf $\hat{h}(P)>M+1$, this reduces canonical height by more than one half. Iterate until a point of canonical height at most $M+1$ is reached. There are only finitely many such points, by Northcott and the bounded difference of heights. Reconstructing $P=2Q+R_i$ at every stage proves that these finitely many small points and the finitely many representatives generate $E(K)$. ThereforeThe canonical pairing is positive definite on the free part. It turns rank computations into an arithmetic lattice problem, supplies independence tests for proposed generators, and bounds searches once a arithmetic height bound is available. Weak finite-quotient finiteness alone would not prove this result: the arithmetic height contraction is the step that converts it into finite generation.

A height function measures arithmetic size, which is what makes an otherwise infinite descent terminate. For a number field $K$, normalize absolute values to extend the standard real and $p$-adic ones, and write $n_v=[K_v:{\mathbb{Q}}_v]$. The logarithmic projective height isThe product formula makes it independent of the chosen homogeneous coordinates, and the local-degree normalization makes it independent of the number field containing them. Over $\mathbb{Q}$, $h([a:b])=\log \max (|a|,|b|)$ for coprime integer coordinates. Set $h_x(P)=h(x(P))$ on an elliptic curve, with $h_x(O)=0$.

Two features are essential. First, the Northcott theorem says that points of bounded projective height and bounded field degree form a finite set. For points on a fixed elliptic curve over $K$, each $x$-coordinate has at most two preimages, so bounded $h_x$ gives finitely many points. Second, a degree-$d$ morphism of the projective line satisfies $h(f(z))=dh(z)+O(1)$, uniformly in $z$. The upper bound comes from evaluating its homogeneous polynomials; for the lower bound, their lack of a common zero gives a resultant identity bounding the input coordinates by the output coordinates at each place. Summing the local bounds gives the asserted uniform constant.

The duplication map on the $x$-line has degree four. Nonsingularity ensures that its numerator and denominator have no common projective zero. ThereforeTelescoping defines the canonical height of an elliptic curveThe error in successive terms is bounded by a geometric series, proving convergence and the uniform bounded difference. It also gives $\hat{h}(2P)=4\hat{h}(P)$ and nonnegativity. The usual addition formula gives the approximate height parallelogram identity for $h_x$; equivalently, the unordered pair of sum and difference on the $x$-line has bidegree $(2,2)$. Applying that identity to $2^{j}P,2^{j}Q$ and passing to the limit givesIn particular $\hat{h}(nP)=n^2\hat{h}(P)$. Its polarization is the canonical height pairing, a positive semidefinite bilinear form even before finite generation has been proved. The Cauchy-Schwarz inequality for this pairing givesAlso $\hat{h}(P)=0$ precisely for torsion points of an elliptic curve: one direction follows from periodic multiples, and in the other direction all multiples have bounded $h_x$, so the Northcott theorem makes two multiples equal. Bounded canonical height of an elliptic curve likewise gives a finite set of $K$-rational points.

Now the Weak Mordell-Weil theorem gives finitely many representatives $R_1,\dots ,R_s$ for $E(K)/2E(K)$. Put $H={\max }_i\hat{h}(R_i)$. Write any point as $P=2Q+R_i$. ThenRepeatedly applying this height descent lemma eventually reaches height at most $H+1$: after $j$ steps the height is at most $H+2^{-j}(\hat{h}(P)-H)$. The set of points with height at most $H+1$ is finite. Reading the relations $P=2Q+R_i$ backwards shows that this finite set together with the $R_i$ generates $E(K)$. ConsequentlyHeights turn weak Mordell-Weil finiteness into the Mordell-Weil theorem. The canonical height pairing subsequently equips the free part with a positive definite quadratic form, useful for bounding searches and measuring independent generators; this interpretation is a consequence of the proof, not an assumption used in the descent.

Heights measure arithmetic size, and their finiteness and growth properties turn finite divisibility information into finite generation. For an elliptic curve $E$ over a number field $K$, the resulting Mordell-Weil theorem isThe distinction between this statement and the Weak Mordell-Weil theorem, which only asserts that $E(K)/mE(K)$ is finite, is exactly where heights enter.

For a rational point of the projective line, choose coprime integers $a,b$ and define the logarithmic projective heightFor a fixed number field, its absolute logarithmic version iswith the usual normalized absolute values. The product formula makes this independent of scaling $(a,b)$, and the local degree formula makes it independent of enlarging the number field. Heights are nonnegative and are useful precisely because they control numerator and denominator together, unlike an ordinary real absolute value.

The Northcott theorem states that algebraic points of bounded degree and bounded height form a finite set. Over $\mathbb{Q}$ this is immediate from the coprime numerator-denominator description. In general, bounded degree and height bound the coefficients of the primitive minimal polynomial: their elementary symmetric functions are bounded by its Mahler measure, which is the appropriate power of the multiplicative height. There are only finitely many such integer polynomials.

On a Weierstrass equation of an elliptic curve, define the naive logarithmic height $h_x(P)=h(x(P))$ and $h_x(O)=0$. Since an $x$-coordinate has at most two points above it, Northcott theorem implies that only finitely many points of $E(K)$ have bounded $h_x$.

Duplication descends to a rational map of the projective $x$-line of degree $4$. More generally, if a morphism $\varphi :{\mathbb{P}}^1\to {\mathbb{P}}^1$ has degree $d$, thenuniformly in $z$. Write $\varphi$ as two homogeneous degree-$d$ polynomials without common zeros. Bounding their coefficients gives the upper estimate at each place. A nonzero resultant, or homogeneous Bezout identities, prevents simultaneous cancellation and gives the lower estimate; only finitely many places contribute a nonzero constant. Applied to duplication, this gives the height growth under a morphism of the projective line estimate

Define the canonical height of an elliptic curve byThe factor $1/2$ is a convention making it the height for the degree-one symmetric divisor $O$. If $a_n=4^{-n}{h}_x(2^{n}P)$, then $|a_{n+1}-a_n|\le C_E{4}^{-(n+1)}$, so the limit exists by a geometric-series estimate. Summing that estimate also provesThus bounded canonical height still gives only finitely many $K$-points, and $\hat{h}(2P)=4\hat{h}(P)$ exactly.

The addition formula on the quotient by negation gives the uniform approximate parallelogram identityThis height parallelogram identity follows as follows. One way to obtain it is to express the unordered pair of their $x$-coordinates as a morphism ${\mathbb{P}}^1\times {\mathbb{P}}^1\to {\mathrm{Sym}}^2{\mathbb{P}}^1$ of bidegree $(2,2)$; the height of that unordered pair is the sum of the two heights, up to a bounded constant. It is also the height identity for the symmetric divisor $O$. Apply this identity to $2^{n}P,2^{n}Q$, divide by $2\cdot 4^n$, and take limits. The errors disappear, givingTogether with evenness and $\hat{h}(O)=0$, the recurrence implies $\hat{h}(mP)=m^2\hat{h}(P)$ for every integer $m$. The associated canonical height pairingis bilinear. Nonnegativity of the height implies its Cauchy-Schwarz inequality, by applying it to integer combinations and then rational approximations. In particular,The height vanishes precisely at torsion points: torsion gives a finite set of multiples, whereas if $\hat{h}(P)=0$, every multiple of $P$ has bounded naive height and therefore the set of multiples is finite by Northcott theorem.

Now apply the Weak Mordell-Weil theorem with $m=2$, proved cohomologically in the other essay. Choose a finite set $R$ of representatives for $E(K)/2E(K)$, and put $H={\max }_{R_i\in R}\hat{h}(R_i)$. Write an arbitrary point as $P=2Q+R_i$. ThenIf $\hat{h}(P)>H+1$, this decreases the height by more than $1/2$. Repeat until the current point has height at most $H+1$. There are only finitely many such points. Unwinding the equations $P=2Q+R_i$ shows that $E(K)$ is generated by this finite bounded-height set together with $R$. This is the height descent lemma, and proves the Mordell-Weil theorem.

Once finite generation is established, the structure theorem for finitely generated abelian groups gives the displayed decomposition. The canonical height pairing is positive definite on the real vector space of the free part: a null direction would, by approximation with integer combinations of generators, give infinitely many lattice points of bounded height, contradicting Northcott theorem. Its determinant is the regulator of an elliptic curve, and its geometry gives practical tools for searching for generators and comparing independence. Thus heights supply both the finiteness mechanism in the proof and the quadratic size function used in arithmetic computations.

For $P=[x_0:\cdots :x_N]\in {\mathbb{P}}^N(\mathbb{Q})$, choose coprime integer coordinates and define the projective heightWrite the morphism $F:{\mathbb{P}}^1\to {\mathbb{P}}^1$ as $F=[F_0:F_1]$, where $F_0,F_1\in \mathbb{Z}[X,Y]$ are homogeneous of degree $d$ with no common zero. Bounding each polynomial by the sum of the absolute values of its coefficients gives

Because $F_0$ and $F_1$ have no common projective zero, the Projective Nullstellensatz gives an integer $m\ge d$ and homogeneous polynomials $A_{ij}$ such that suitable nonzero integer multiples of $X^m$ and $Y^m$ lie in the ideal $(F_0,F_1)$. Evaluating at primitive coordinates and using the same coefficient bound givesafter absorbing the bounded common divisor of $F_0(x,y)$ and $F_1(x,y)$ into $C$. ThereforeThis is height growth under a morphism of the projective line.

Since $\hat{h}-h$ is bounded, a bound on $\hat{h}(P)$ bounds the projective height of $x(P)$. The Northcott theorem gives only finitely many possible rational $x$-coordinates, and each has at most two points above it. Hence

By the Mordell-Weil theorem, $E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus E(\mathbb{Q}{)}_{\mathrm{tors}}$. The canonical height is a positive-definite quadratic form on the lattice ${\mathbb{Z}}^r$, so canonical-height lattice-point growth givesIf $r=0$, this count is bounded; if $r=1$, it is $O(\sqrt{B})$. Consequently

For a linear form $L=a_1{X}_1+\cdots +a_N{X}_N$ over a number field $K$, define its height to be the projective height of its coefficient vector:The product formula makes this independent of multiplying $L$ by a nonzero scalar.

The Siegel lemma says the following. Let $D=[K:\mathbb{Q}]$, let $L_1,\dots ,L_M$ be linear forms in $N$ variables with $DM<N$, and suppose $H_p(L_i)\le \mathcal{H}$ for every $i$, where $\mathcal{H}\ge 1$. Then there is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ annihilated by all the forms and satisfying

Writeso there are $N=2(n+1)$ unknown integer coefficients. Let $t$ be the number of integers $r$ withFor each such $r$, impose the linear equation over $K=\mathbb{Q}(\alpha )$where $D_r$ is the normalized derivative of a polynomial. The coefficient of $p_j$ is $\left(\genfrac{}{}{0ex}{}{j}{r}\right){\alpha }^{j-r}$ and that of $q_j$ is $\left(\genfrac{}{}{0ex}{}{j}{r}\right){\alpha }^{j-r+1}$. The local definition of projective height, together with $\left(\genfrac{}{}{0ex}{}{j}{r}\right)\le 2^n$, givesfor a constant $C_0$ depending only on $\alpha$.

There are $M=t$ forms over the degree-$d$ field $K$, andMoreover $N-dM\ge \kappa n+2$. Applying Siegel lemma gives a nonzero integral coefficient vector withbecause the exponent is bounded in terms of $\kappa$ and every fixed power of $n$ is at most exponential in $n$. These $P,Q$ have all the required vanishing normalized derivatives.

Let $K$ be a number field of degree $D$, and let $L_1,\dots ,L_M$ be linear forms in $N>DM$ variables with coefficients in $K$ and projective height at most $\mathcal{H}\ge 1$. There is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ satisfying every $L_j(\mathbf{x})=0$ andFor $K=\mathbb{Q}$, this follows by applying the pigeonhole principle to the images of the integer box $\{0,\dots ,Y{\}}^N$ under $(L_1,\dots ,L_M)$. Expanding coefficients in a rational basis of $K$ gives the factor $D$ in the general count.

The biquadratic morphism for elliptic sums and differences has no common zero, so the local coefficient norms giveuniformly in $P,Q$. Upper bounds use the fixed coefficients. Lower bounds follow from the absence of common zeros over every local field; the constants are $1$ outside finitely many places, where the integral morphism still has no base point after reduction.

For the binary quadratic with roots $u,v\in {\mathbb{P}}^1(K)$, the projective height of its coefficient vector is $h(u)+h(v)+O(1)$. At nonarchimedean places the coefficient maximum norm of a product of linear polynomials is the product of their maximum norms by the multiplicativity of the nonarchimedean polynomial norm. At archimedean places the same norms are comparable up to a fixed factor for degree two. This also includes roots at infinity. Combining the comparisons yields $h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$. After dividing the identity for $2^{n}P,2^{n}Q$ by $4^n$ and taking limits, it gives the exact height parallelogram identity for the canonical height of an elliptic curve.

On a projective variety over a number field, a very ample line bundle gives a logarithmic projective height by an embedding. Tensor products correspond to sums by the Segre embedding, and a sufficiently ample twist expresses any line bundle as a difference of very ample ones. This defines its height modulo bounded functions. It is additive and functorial under morphism pullback. An ample bundle has the Northcott theorem finiteness property for bounded height and bounded point degree; a trivial bundle need not. The product formula makes these absolute heights independent of scalar coordinates and of the number field used to compute them.