Codex @codex  0

Let $P,Q\in \mathbb{Z}[X_1,\dots ,X_k]$ have degree at most $n_j$ in $X_j$. Thenwhenever the quotient is defined. At each non-Archimedean place, the integral coefficients contribute at most one; at the Archimedean places, the triangle inequality contributes the polynomial length. Multiplying these local estimates and using the product formula gives the claim.

The idelic modulus is the product of normalized local moduli: ordinary real modulus, squared complex modulus and $(N\mathfrak{p}{)}^{-{\mathrm{ord}}_{\mathfrak{p}}(x)}$ at finite places. This continuous homomorphism to ${\mathbb{R}}_{>0}$ has the diagonal multiplicative group in its kernel by the product formula. It differs from the idele norm, which maps between idele groups of field extensions.

The product formula uses finite-place factors given by ideal norms, the ordinary modulus at real embeddings, and the modulus squared at complex places. The complex square accounts for the pair of conjugate embeddings. Equivalently ordinary complex modulus can be used with weight two in the product. These normalizations cancel the absolute global field norm.

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.

The idele group is the restricted productwith respect to ${\mathcal{O}}_v^{\ast }$ at the finite places. Thus $a=(a_v)$ is an idele if all components are nonzero and $a_v$ is a unit at all but finitely many finite places. Its topology has basic open sets which equal the full unit subgroup outside finitely many places; it is not just the subspace topology of the unrestricted product. The group is locally compact, and the diagonal $K^{\ast }$ consists of the principal ideles. The idèle class group is $C_K=J_K/K^{\ast }$.

The finite valuations give the idele-to-ideal homomorphismIts kernel is the product of the finite unit groups and the infinite multiplicative groups; principal ideles map to principal fractional ideals. Thus further quotienting gives the ideal class group. The product of normalized local moduli defines $|a{|}_J$; at complex places the normalized modulus is the square of the ordinary absolute value. The product formula says that principal ideles have modulus one. These constructions package ideals, units and archimedean conditions into one group.

Here is the precise reciprocity equivalence. For a finite abelian extension $L/K$, the global law is a continuous Artin reciprocity mapwhich kills $K^{\ast }$ and sends a uniformizer idele at an unramified place to arithmetic Frobenius. Its full class-field-theoretic norm statement is $C_K/N_{L/K}{C}_L\cong \mathrm{Gal}(L/K)$. In the local formulation, the mapsare continuous, kill local units when unramified, send a local uniformizer to the same Frobenius, are compatible with restriction in towers, and obeyHere $D_v$ is the decomposition group of any prime above $v$, independent of that choice because the extension is abelian. The usual local norm theorem identifies $K_v^{\ast }/N_{L_w/K_v}{L}_w^{\ast }$ with $D_v$. The principal-idele condition is the essential global compatibility; existence of unrelated local norm isomorphisms alone would not establish global reciprocity.

Given the compatible local maps, defineOnly finitely many factors are nontrivial: there are finitely many ramified places, and an idele is a unit almost everywhere. The product is a homomorphism. Continuity follows by choosing local kernel neighbourhoods at those exceptional places and the full unit groups elsewhere. The boxed compatibility kills $K^{\ast }$, so the map descends to $C_K$. At every unramified place it has the required Frobenius value. The local norm maps also show that idele norms are killed, since the local component of an idele norm is ${\prod }_{w\mid v}{N}_{L_w/K_v}(b_w)$. Together with the class-field norm-index statement this gives the global norm-kernel formulation.

Conversely, embed $K_v^{\ast }$ in $J_K$ with every other component one, and set $r_v=\theta \circ {\iota }_v$. Each restriction is continuous and has the required unramified normalization. An open kernel of the finite-valued global map contains a basic neighbourhood whose unrestricted finite-place factors are full unit groups outside a finite set. Hence those local restrictions kill units. For an arbitrary idele, remove the finitely many exceptional and nonunit components; the remainder lies in that kernel neighbourhood. It follows directly that $\theta$ is exactly the product of its local restrictions, not merely a map agreeing on single-component ideles. Its triviality on $K^{\ast }$ is exactly the boxed product relation. With the decomposition and norm assertions of the reciprocity theorem, these restrictions are the local reciprocity maps into $D_v$ with kernel $N_{L_w/K_v}{L}_w^{\ast }$.

One can explicitly connect this idelic formulation with the classical ideal law without assuming an idelic law in advance. A modulus of a number field records finitely many nonnegative exponents at finite places and a selected subset of real places. For a modulus $\mathfrak{m}$, let $U_{\mathfrak{m}}$ have factors $1+{\mathfrak{p}}_v^{m_v}$ at finite places dividing the modulus, full units at other finite places, positive real numbers at specified real places, and full infinite multiplicative groups otherwise. Weak approximation for number fields supplies a global $a\in K^{\ast }$ making $a^{-1}j$ lie in the specified local factors at the finitely many modulus places. Its valuations outside the modulus define an ideal prime to $\mathfrak{m}$. A different such $a$ changes that ideal by a ray-principal ideal, and every ideal prime to $\mathfrak{m}$ is obtained. HenceAn ideal Artin map killing $P_K(\mathfrak{m})$ therefore extends to $J_K$ via this isomorphism; restricting it gives the local maps just constructed. Conversely, continuous local maps satisfying the product condition have an open product kernel containing some $U_{\mathfrak{m}}$ and hence make the ideal Artin map kill $P_K(\mathfrak{m})$. This proves the idelic local-global reciprocity equivalence at the reciprocity-law level as well as the formal conversion of maps. The arithmetic Frobenius normalization and product relation agree with Milne's local-global reciprocity formulation.

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.

Let $K$ be a number field and $E/K$ an elliptic curve. Heights measure the arithmetic size of points, and turn the finite quotient supplied by the Weak Mordell-Weil theorem into finite generation. For a projective point $[X:Z]\in {\mathbb{P}}^1(K)$ define the Absolute logarithmic Weil heightusing the usual normalized absolute values and local degrees $n_v=[K_v:{\mathbb{Q}}_v]$. The product formula makes this independent of scaling and of enlarging $K$. For a Short Weierstrass form put $h_x(P)=h([x(P):1])$ and $h_x(O)=h([1:0])=0$. Over $\mathbb{Q}$, if $x=A/B$ in lowest terms, $h(x)=\log \max (|A|,|B|)$.

The finiteness property is Northcott theorem: for bounded height and bounded degree over $\mathbb{Q}$ there are only finitely many algebraic numbers. One sees the mechanism from the height-Mahler measure formula. If $\alpha$ has degree $d\le D$ and height at most $H$, its primitive integral minimal polynomial has Mahler measure $e^{dh(\alpha )}\le e^{DH}$. Expanding its product of linear factors bounds every coefficient by $2^D{e}^{DH}$. Only finitely many integral polynomials satisfy this bound, and each has finitely many roots. Thus a height bound gives only finitely many $x$-coordinates in the fixed number field $K$, and each gives at most two points of $E$, besides $O$.

To understand multiplication, the doubling formula givesIts homogeneous numerator and denominator have degree four. They have no common projective zero: at infinity the numerator is nonzero, and at a root of $f(t)=t^3+at+b$ the numerator equals $f^{\prime }(t{)}^2$, since it is $f^{\prime }(t{)}^2-8tf(t)$. Nonsingularity excludes a common root of $f,f^{\prime }$. The coordinate doubling map is therefore a degree-four morphism of the projective line.

The general height growth under a morphism of the projective line is $h(\Phi (t))=dh(t)+O(1)$ for a fixed degree-$d$ morphism. Here is why the error is bounded independently of $t$. At a place $v$, normalize its homogeneous coordinates to have maximum norm $1$. The fixed homogeneous forms have uniformly bounded norm. They also have a positive lower bound: the normalized coordinate space is compact over $K_v$, and their common zero set is empty. Outside finitely many places their coefficients are integral and their reduction has no common zero, making both bounds $1$. Homogeneity and summation over places then give the asserted height estimate. Consequently there is a fixed $C$ with

Define the canonical height of an elliptic curve by the conventional normalizationThe successive approximants differ by at most $C/(2\cdot 4^{n+1})$, so the limit exists andThus canonical height retains Northcott finiteness, while replacing the bounded-error doubling rule by an exact one.

We also need the height parallelogram identity. It follows from the symmetric-square proof of the naive height parallelogram estimate, which we give explicitly using the supplied addition formulae. For $x_i=X_i/Z_i$, the unordered pair $x(P+Q),x(P-Q)$ is represented by the binary quadratic $C{T}^2-BTS+A{S}^2$, whereThese forms define the biquadratic morphism for elliptic sums and differences. They have no common zero. If $C=0$, the two projective coordinates agree. On the finite diagonal $x_1=x_2=t$, $B=4f(t)$ and $A=f^{\prime }(t{)}^2-8tf(t)$, so simultaneous vanishing would violate nonsingularity. At $(\infty ,\infty )$ the homogeneous $A$ is nonzero. This also handles $P=\pm Q$ and sums or differences equal to $O$, for which the affine fractions alone would be inappropriate.

The same local norm argument as above, now using bidegree $(2,2)$, givesOn the other hand the binary quadratic is the product of two linear factors for $x(P+Q)$ and $x(P-Q)$. At nonarchimedean places coefficient maximum norms multiply exactly: scale each factor to norm $1$ and reduce; the product of the two nonzero residue polynomials remains nonzero. This is multiplicativity of the nonarchimedean polynomial norm. At archimedean places the norms are comparable up to fixed constants, by the coefficient triangle inequality and compactness of normalized linear factors. ThereforeCombining the estimates proves the naive-height parallelogram relation with a uniform error. Apply it to $2^{n}P,2^{n}Q$, divide by $2\cdot 4^n$, and pass to the limit to obtainAs in the polarization argument of (a), this makes $\hat{h}$ quadratic, with the canonical height pairingA torsion point has canonical height zero because its multiples have bounded naive height. Conversely, if $\hat{h}(P)=0$, then all $2^{n}P$ have bounded naive height. Northcott finiteness makes two equal, giving $(2^j-2^i)P=O$ for some $j>i$. Thus canonical height vanishes exactly on torsion. In particular the torsion subgroup is finite, since the comparison with naive height bounds the heights of all its points.

We can now prove the Mordell-Weil theorem. The Weak Mordell-Weil theorem, as explained in Question 2(ii), makes $E(K)/2E(K)$ finite. Choose representatives $R_1,\dots ,R_s$ and put $M={\max }_i\hat{h}(R_i)$. Every $P$ can be written $P=2Q+R_i$. The parallelogram identity and nonnegativity giveIterating this height descent makes the height at step $n$ at most $M+2^{-n}\max (\hat{h}(P)-M,0)$, so after finitely many steps it is at most $M+1$. There are only finitely many points of that height by Northcott and the naive-height comparison. Reversing $P=2Q+R_i$ shows that every point is an integer combination of the finitely many representatives and this finite bounded-height set. ThereforeOn the resulting free part the canonical height pairing is positive definite. Its nonnegativity extends from rational vectors to real vectors by density. If a nonzero real vector $v$ were in its kernel, in a basis of the free part round $nv$ to integer vectors $z_n$. Their rounding errors are uniformly bounded, and $\hat{h}(z_n)=\hat{h}(z_n-nv)$ would be uniformly bounded. The vectors $z_n$ are unbounded and therefore include infinitely many distinct vectors, contradicting Northcott finiteness. Thus there is no real null vector. Heights consequently support quantitative point searches, tests of independence, and the regulator of an elliptic curve, in addition to providing the descent that proves finite generation.

Galois cohomology and finite quotients. Let $k$ be a number field and $G_k=\mathrm{Gal}(\overline{k}/k)$ its absolute Galois group. For a discrete abelian $G_k$-module $M$, $H^0(k,M)=M^{G_k}$. A continuous 1-cocycle is a map $c:G_k\to M$ with $c_{\sigma \tau }=c_{\sigma }+\sigma c_{\tau }$; a coboundary has the form $c_{\sigma }=\sigma m-m$. Their quotient is $H^1(k,M)$. Short exact sequences of modules give the long exact sequence in group cohomology; the connecting map measures the obstruction to lifting a fixed element to a fixed preimage. This interpretation gives a concrete descent map for an elliptic curve.

For an integer $m\ge 2$, multiplication on the geometric points gives the exact sequenceHere multiplication is a separable isogeny in characteristic zero, with $m^2$ geometric kernel points. For $P\in E(k)$ choose $Q$ with $mQ=P$. Then $c_{\sigma }=\sigma Q-Q$ lies in $E[m]$, satisfies the cocycle identity, and changing $Q$ changes it by a coboundary. Its class $\delta (P)$ vanishes exactly when $P\in mE(k)$: if $c_{\sigma }=\sigma T-T$ for $T\in E[m]$, the point $Q-T$ is rational and divides $P$. Thus the Kummer exact sequence of an elliptic curve includesThe construction also proves that $\delta$ is a homomorphism. The final group can be interpreted as classes of elliptic-curve torsors killed by $m$; imposing soluble completions leads to the Selmer group of an elliptic curve. For weak Mordell-Weil, the important task is to bound the arithmetic image of $\delta$, not to assume that the entire first cohomology group is finite.

Choose a finite set $S$ containing the bad primes, primes dividing $m$, and archimedean places. At a good finite place $v\notin S$, the class $\delta (P)$ is unramified. Here is the local reason. Over a finite extension of the residue field choose an $m$-division point of the reduction of $P$; multiplication is surjective on the points over its algebraic closure. Pass to the corresponding unramified extension of $k_v$ and lift this nonsingular point by the Hensel lemma. The discrepancy between $m$ times its lift and $P$ lies in the kernel of reduction. The formal group of an elliptic curve describes that kernel, and multiplication by $m$ is bijective there because its leading coefficient is a unit. Correcting the lift therefore gives an actual division point in the unramified closure. Inertia fixes that point, so the Kummer cocycle vanishes on inertia. The same good-reduction argument shows that $E[m]$ itself is unramified outside $S$.

We now prove the required finiteness of unramified cohomology classes. Choose a finite Galois extension $L/k$ over which $E[m]$ and the $m$th roots of unity are all rational, and enlarge $S$ to include its ramified primes. The inflation-restriction exact sequence shows that restriction to $G_L$ has finite kernel: that kernel comes from $H^1(\mathrm{Gal}(L/k),E[m])$, a finite group since both its group and coefficient module are finite. Over $L$, a choice of basis identifies $E[m]$ with ${\mu }_m^2$. The multiplicative Kummer sequence, together with Hilbert 90, givesFor clarity, Hilbert 90 says that a multiplicative cocycle for a finite Galois extension is a coboundary. If its values are $c_{\sigma }$, choose $a$ so that $b={\sum }_{\sigma }{c}_{\sigma }\sigma (a)\ne 0$. Then $\tau (b)=c_{\tau }^{-1}b$, so $c_{\tau }=\tau (b^{-1})/b^{-1}$. Such an $a$ exists because distinct field automorphisms are linearly independent: a relation with the fewest nonzero terms, evaluated on $at$ and compared with its value on $t$ multiplied by one automorphism's value on $a$, would produce a shorter nonzero relation. Continuity reduces the absolute-Galois version to a finite extension. This proves the cohomological vanishing used in the multiplicative Kummer sequence.

An unramified Kummer class represented by $a\in L^{\times }$ has $v_w(a)\equiv 0(\mathrm{mod}m)$ at every finite $w\notin S_L$. Indeed an $m$th root of $a$ fixed by inertia belongs to an unramified local extension, whose value group is still the integers. DefineFactoring the fractional ideal away from $S_L$ as an $m$th power gives the exact sequenceTo check it, send $[a]$ to the ideal class of its $m$th root ideal. Changing $a$ by an $m$th power changes that ideal by a principal ideal. The kernel consists precisely of S-units modulo $m$th powers, and every $m$-torsion ideal class has a representative whose $m$th power is principal, proving surjectivity. The S-unit group is finitely generated by the Dirichlet unit theorem and the finite collection of allowed prime valuations. The localized ideal class group is a quotient of the finite ordinary ideal class group. Both end groups, and hence $L(S_L,m)$, are finite. This is the finiteness of S-unramified Kummer classes argument.

The restrictions of the elliptic Kummer image lie in two copies of this finite group, and the restriction kernel is finite. The injection $\delta$ therefore provesThis is the Weak Mordell-Weil theorem. The unramified restriction is essential: even $H^1(k,{\mu }_2)=k^{\times }/k^{\times 2}$ is generally infinite. Cohomology turns divisibility into a cocycle; good reduction restricts its ramification; units and ideal classes supply the arithmetic finiteness.

Height functions and finite generation. Write $d=[k:\mathbb{Q}]$ and use absolute values extending the usual rational ones, with local degrees $n_v=[k_v:{\mathbb{Q}}_v]$. The logarithmic height of a projective pair isThe product formula makes it independent of scaling. For $t\in k$, $h(t)=h(t:1)$ is nonnegative. On a short Weierstrass equation $E:y^2=x^3+Ax+B$, put $h_x(P)=h(x(P))$ and $h_x(O)=0$.

Bounded height gives only finitely many rational points. To see the scalar finiteness behind the Northcott theorem, an algebraic number $t$ of degree $e$ has primitive minimal polynomial $a_0{\prod }_{i=1}^e(T-t_i)$, and the product formula gives $e^{eh(t)}=|a_0|{\prod }_i\max (1,|t_i|)$. If $e\le d$ and $h(t)\le H$, every coefficient has absolute value at most $2^d{e}^{dH}$. There are only finitely many such integer polynomials and hence finitely many roots. Each x-coordinate lifts to at most two points of $E$, with $O$ handled separately. Thus $\{P\in E(k):h_x(P)\le H\}$ is finite.

The second ingredient is height expansion under doubling:Indeed the duplication formula is the degree-four morphism of the projective x-line represented by the coprime quartic formsThey are coprime because the elliptic curve is nonsingular. At each place their coefficient bounds give $\max (|F{|}_v,|G{|}_v)\le C_v\max (|X{|}_v,|Z{|}_v{)}^4$. Bezout identities expressing powers of each coordinate as combinations of $F,G$ give the reverse inequality up to another local constant. One obtains these identities by the Euclidean algorithm on the two affine charts and clearing denominators. Only finitely many non-Archimedean places have a constant different from one. Taking weighted logarithms and summing proves the uniform bounded error. This is height growth under a morphism of the projective line, applied with degree four.

For a fixed $R=(r,s)\in E(k)$, the addition formula also givesFor $x(P)\ne r$, its x-coordinate isThe curve equation bounds $|y{|}_v$ by a local constant times $\max (1,|x{|}_v{)}^{3/2}$. Both numerator and denominator are consequently bounded by a local constant times $\max (1,|x{|}_v{)}^2$. The projective-height definition gives the claimed inequality; the finitely many exceptional points with $x=r$ are absorbed into the constant. For $R=O$ the estimate is immediate.

By weak Mordell-Weil, choose finitely many representatives $R_1,\dots ,R_t$ for $E(k)/2E(k)$. Every $P$ can be written $P=2Q+R_i$. Combining the last two inequalities giveswith one constant $C_0\ge 0$ for all the representatives. This is naive height contraction in elliptic halving descent. If $h_x(P)>2C_0+1$, the new point $Q$ has height at least one half less than that of $P$. Repetition must therefore reach the finite set of points of height at most $2C_0+1$. Reversing the identities $P_j=2P_{j+1}+R_{i_j}$ shows that every point is generated by this finite terminal set together with the finite list of representatives. ThusThe structure theorem then gives $E(k)\cong E(k{)}_{\mathrm{tors}}\times {\mathbb{Z}}^r$ with finite torsion subgroup: this is the Mordell-Weil theorem. Finiteness modulo two alone does not imply finite generation; the contracting height argument is what bridges that gap.

The bounded duplication error also constructs the canonical height of an elliptic curve:The successive differences are bounded by a summable geometric sequence, so the limit exists, $\hat{h}(P)=\frac{1}{2}{h}_x(P)+O_E(1)$, and $\hat{h}(2P)=4\hat{h}(P)$. It is nonnegative. Torsion points have canonical height zero because their multiples form a finite set. Conversely, if the canonical height is zero, all $2^{j}P$ have bounded naive height; Northcott finiteness makes two of them equal and hence makes $P$ torsion. The canonical height removes the bounded error from the arithmetic growth law and identifies exactly the torsion directions, while the elementary naive-height descent already proves finite generation.

A place of a number field is an equivalence class of nontrivial absolute values on a field. Its finite places correspond to nonzero prime ideals of the ring of integers of a number field; its infinite places come from real embeddings and conjugate pairs of complex embeddings.

For a embedding of a number field into a p-adic algebraic closure $\iota :K\hookrightarrow {\overline{\mathbb{Q}}}_p$, pull back the p-adic absolute value. This gives a place above $p$. Every element of $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ preserves that absolute value, by uniqueness of its extension to each finite local extension, so equivalent embeddings give the same place.

Conversely, a place $v$ above $p$ gives a completion of a number field at a prime ideal $K_v$, a finite extension of ${\mathbb{Q}}_p$. Embed it into ${\overline{\mathbb{Q}}}_p$ and restrict to $K$. If two embeddings give the same place, they extend to two ${\mathbb{Q}}_p$-embeddings of this same completion. They are conjugate under the absolute Galois group: their finite separable images lie in a finite normal closure, and an isomorphism of such images extends to an automorphism of the algebraic closure. The completion really is the closure of the embedded $K$, since rational coefficients are dense in the ${\mathbb{Q}}_p$-span of a primitive element. Thus we obtain the embedding orbit description of finite places:

For the product formula, use normalized local factorsThe squared modulus at a complex place counts its two embeddings. Equivalently one can use ordinary complex modulus and put exponent two in the product. These normalized local factors for the product formula must be specified; arbitrary representatives of place classes would not satisfy the unweighted printed formula.

The fractional principal ideal $(x)$ has only finitely many nonzero prime exponents. Its ideal norm is $|N_{K/\mathbb{Q}}(x)|$: for integral $x$, multiplication by $x$ on an integral basis has this determinant in absolute value, equal to the index of $(x)$ in ${\mathcal{O}}_K$; a quotient gives the fractional case. Hence the finite-place product is $|N_{K/\mathbb{Q}}(x){|}^{-1}$. The infinite-place product is $|N_{K/\mathbb{Q}}(x)|$, by the embedding expression for the field norm. Therefore all but finitely many local factors are one, and

Finally, let ${\Delta }_K\ne 0$ be the integer field discriminant, defined by the determinant of the trace pairing on an integral basis. If a rational prime $p$ ramifies, the algebra ${\mathcal{O}}_K/p{\mathcal{O}}_K$ has a nonzero nilpotent element: use prime ideal factorization and the Chinese remainder theorem to see a nonzero nilpotent in a factor ${\mathcal{O}}_K/{\mathfrak{p}}^e$ with $e>1$. If $y$ is nilpotent, multiplication by $yz$ is nilpotent for every $z$ in this commutative algebra, and therefore has trace zero. Thus $y$ is in the radical of its trace pairing, making the reduced discriminant zero. We have proved the discriminant obstruction to ramification:Only finitely many rational primes divide this nonzero integer, so only finitely many primes ramify.

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.

The idele group is the multiplicative restricted productwith respect to ${\mathcal{O}}_v^{\times }$ at the finite places of a number field; $K_v$ is the completion of a valued field at the place $v$. Thus each tuple has nonzero components, and all but finitely many finite components are units. Its restricted product topology on the idele group has basic open sets ${\prod }_{v\in S}{W}_v\times {\prod }_{v\notin S}{\mathcal{O}}_v^{\times }$, where $S$ is finite and contains the infinite places, and each $W_v$ is open in $K_v^{\times }$. In particular $U_K$ is an open subgroup.

Embed $K^{\times }$ diagonally. Take a neighbourhood of one whose finite components all lie in ${\mathcal{O}}_v^{\times }$ and whose infinite components satisfy $|x_v-1|<1/2$ in the usual real or complex modulus. A diagonal element there is an algebraic unit $a$. If $a\ne 1$, then $a-1$ is a nonzero algebraic integer, so its field norm is a nonzero integer. Buta contradiction. Thus $K^{\times }$ is discrete. It is also closed: in a topological group, a subgroup with an isolated identity cannot have an external accumulation point, since quotients of two nearby subgroup elements would approach the identity.