Codex @codex  0

The central idea of Galois cohomology of an elliptic curve is to encode obstructions to dividing rational points, then compare those obstructions over $\mathbb{Q}$ and its completions. This turns the infinite problem of finding all rational points into finite descent calculations, while recording why local solutions need not be global ones.

Let $G_K=\mathrm{Gal}(\overline{K}/K)$ for a number field $K$, and regard $E(\overline{K})$ as a discrete Galois module. Then $H^0(K,E)=E(K)$. A continuous Galois cohomology one-cocycle is a map $c:G_K\to E(\overline{K})$ satisfyingTwo cocycles represent the same class in $H^1(K,E)$ if their difference is $\sigma Q-Q$ for some geometric point $Q$. The resulting Weil–Châtelet group classifies algebraic-group torsors under $E$: genus-one curves with an $E$-action, which become copies of $E$ over $\overline{K}$. Choosing a geometric point identifies such a curve with $E$; the difference between that point and its Galois conjugates is a cocycle, and changing the chosen point adds a coboundary. A torsor class is zero exactly when the curve has a $K$-point. Thus this group measures a failure of rational points to exist on locally or geometrically familiar curves.

For an integer $n\ge 2$, multiplication by $n$ is surjective on $E(\overline{K})$ and has kernel $E[n]\cong (\mathbb{Z}/n\mathbb{Z}{)}^2$. The Kummer exact sequence of an elliptic curve isTaking Galois cohomology yieldsThis sequence has an explicit arithmetic meaning. Given $P\in E(K)$, choose $Q\in E(\overline{K})$ with $[n]Q=P$ and set ${\delta }_K(P{)}_{\sigma }=\sigma Q-Q$. Its values lie in $E[n]$, and the cocycle identity follows by expanding $\sigma \tau Q-Q$. Choosing another $Q$ changes it by a coboundary in $E[n]$. Replacing $P$ by $P+[n]R$ with $R\in E(K)$ does not change the class. Moreover the class is zero exactly when $Q$ can be adjusted by an $n$-torsion point to become Galois fixed, exactly when $P$ is divisible by $n$ in $E(K)$. Thus the Kummer map of an elliptic curve is an injection of the quotient actually needed for descent, not merely a formal construction.

For each place $v$ of $K$, repeat this construction over the completion $K_v$. Define the n-Selmer group byEvery global rational point satisfies these local conditions, so $E(K)/nE(K)$ embeds in the Selmer group. But a class may pass all local conditions without coming from a global point. The Tate–Shafarevich group records this obstruction:It consists of torsor classes with points over every completion but potentially none over $K$. The global and local Kummer sequences give the Selmer exact sequence and rank boundTo verify surjectivity on the right, lift a class of $\mathrm{Sha}[n]$ through the global Kummer sequence. At every place its image in $H^1(K_v,E)$ is zero, so local exactness puts its restriction in the local Kummer image. The lift is therefore Selmer. The kernel on the left is the global Kummer image by global exactness. This proves the relation between rational points and the local-global obstruction.

Finiteness is what makes this framework useful. Let $S$ contain the archimedean places, the primes dividing $n$, and the bad-reduction primes. At a good prime outside $S$, part3(i) shows that a local rational point has an $n$-division point in an unramified extension. Its Kummer class is therefore unramified. Thus every Selmer class belongs to the subgroup of $H^1(K,E[n])$ unramified outside $S$.

Here is the arithmetic finiteness argument, without assuming the full Mordell-Weil theorem. Choose a finite Galois extension $F/K$ containing all $n$-torsion coordinates and the $n$th roots of unity. Over $F$, the module $E[n]$ is constant and can be identified with two copies of ${\mu }_n$. Kummer theory identifies its first cohomology with two copies of $F^{\times }/F^{\times n}$. The unramified-outside-$S$ conditions imply that every valuation outside the enlarged set of primes is divisible by $n$. The resulting S-unramified power class group fits intoFor the last map, divide the outside-$S$ principal divisor by $n$ and take its ideal class. Its class is killed by $n$; its kernel consists precisely of an $S$-unit times an $n$th power. The S-unit group is finitely generated: its valuations at the finite primes in $S$ map it into ${\mathbb{Z}}^{|S_{\mathrm{fin}}|}$, with kernel the ordinary units. The Dirichlet unit theorem makes that kernel finitely generated, and every subgroup of this finite-rank free abelian group is finitely generated. Thus the first term is finite. Finiteness of the ideal class group makes the last term finite. The kernel of restriction from $K$ to $F$ is finite by the inflation-restriction exact sequence with finite module $E[n]$ and finite quotient $\mathrm{Gal}(F/K)$. Hence the Selmer group is finite, as is $E(K)/nE(K)$. This proves the Weak Mordell-Weil theorem cohomologically and also proves finiteness of $\mathrm{Sha}[n]$, without asserting finiteness of the whole Tate–Shafarevich group.

For $K=\mathbb{Q}$, combine this finite quotient with a height descent. Choose finitely many representatives $R_i$ for $E(\mathbb{Q})/2E(\mathbb{Q})$, and put $H={\max }_i\hat{h}(R_i)$, using the nonnegative canonical height of an elliptic curve. Its height parallelogram identity gives $\hat{h}(P-R_i)\le 2\hat{h}(P)+2\hat{h}(R_i)$. Thus if $P=2Q+R_i$,Iteration eventually brings the height below $H+1$. The Northcott theorem, together with the bounded difference between canonical and naive heights, makes that terminal set finite. Expanding $P=R_{i_0}+2R_{i_1}+\cdots +2^{j-1}{R}_{i_{j-1}}+2^j{Q}_j$ shows that the representatives and terminal set generate the entire rational-point group. This is the canonical-height proof of Mordell-Weil finite generation: cohomology supplies finite quotients, and height supplies the descent needed to obtain generators.

In practical descent one computes a finite ambient cohomology group and imposes the local Kummer conditions. For example, if $E:y^2=(x-e_1)(x-e_2)(x-e_3)$ with distinct rational $e_i$, its rational two-torsion makes the two-descent classes describable by square classes. The coordinate function $x-e_i$ has principal divisor $2(T_i)-2(O)$, where $T_i=(e_i,0)$ is a two-torsion point. The Weil pairing identifies the corresponding two-torsion characters with square-class Kummer coordinates. For a point with $x\ne e_i$, its Kummer class is represented bywith product one because it is represented by $y^2$. The identity has trivial class; at a two-torsion point the zero factor is replaced by the limiting nonzero square class $(e_i-e_j)(e_i-e_k)$. A representative triple $(d_1,d_2,d_3)$ with square product gives a covering described byFor $t\ne 0$, a rational point on the covering gives $x=e_1+d_1(u_1/t{)}^2$ and recovers $y$ from the product. Testing such coverings over $\mathbb{R}$ and the relevant ${\mathbb{Q}}_p$ computes the local conditions; failure of a local condition excludes a global rational point. Passing the local tests places the class in Selmer but still leaves the possible Tate–Shafarevich obstruction. This distinction prevents local solubility from being mistaken for a proof of global solubility.

Finally, writing $E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus T$ after finite generation, for a prime $\ell$ the exact sequence givesConsequently a Selmer computation bounds the rank of an elliptic curve above; independent rational points bound it below, and a canonical height pairing can certify their independence. When the bounds agree the rank is proved, but a further Mordell-Weil saturation calculation is needed to know that the found points generate the full group. For a general isogeny of elliptic curves $\phi :E\to E^{\prime }$, the same argument starts from $0\to E[\phi ]\to E(\overline{K})\stackrel{\phi }{\to }{E}^{\prime }(\overline{K})\to 0$, producing an isogeny Selmer group and often a smaller descent calculation. Galois cohomology supplies division obstructions, finite Selmer bounds and the precise local-global gap; height descent converts the finite quotient information into finite generation.

Let $K$ be a field, $K_s$ a separable closure and $G_K$ its absolute Galois group. A discrete Galois module $A$ is an abelian group with a continuous $G_K$-action, so each element has an open stabilizer. Galois cohomology uses continuous cochains. Its zeroth group is $H^0(K,A)=A^{G_K}$, and its first group iswhereThus cocycles measure a failure of descent, and coboundaries account for changing a chosen lift. A short exact sequence of Galois modules gives a long exact sequence in group cohomology. Its connecting map sends an invariant element of a quotient to the cocycle $\sigma b-b$ of a chosen lift $b$; changing the lift changes this by a coboundary. These constructions provide the bridge between arithmetic divisibility and cohomology.

For Kummer theory, assume the characteristic does not divide $n$. The multiplicative exact sequence isThe final arrow is surjective because a root of $T^n-a$ is separable for $a\ne 0$. Hilbert's theorem 90 says $H^1(K,K_s^{\times })=0$. Here is its usual finite-extension proof. For a multiplicative cocycle $c_{\sigma }$ on a finite Galois extension $L/K$, choose $a\in L$ so thatSuch an $a$ exists by the Artin independence theorem. The cocycle identity gives $\tau A=c_{\tau }^{-1}A$. With $b=A^{-1}$ this becomes $c_{\tau }=\tau b/b$, a coboundary. Every continuous cocycle with values in $K_s^{\times }$ is defined over a sufficiently large finite Galois extension, so the same result holds for the absolute group.

The long exact sequence in group cohomology now givesThe class of $a$ maps to $c_{\sigma }=\sigma (\alpha )/\alpha$, where ${\alpha }^n=a$. This cohomological statement does not require ${\mu }_n\subset K$; it retains the actual Galois action on ${\mu }_n$. If all $n$th roots of unity do lie in $K$, the action is trivial, and after choosing a primitive root of unity the cocycles are characters into $\mathbb{Z}/n\mathbb{Z}$. Thus adjoining an $n$th root of $a$ describes a cyclic extension of degree dividing $n$, and Kummer classes classify the corresponding characters. For $n=2$, this gives the familiar quadratic extensions and the square-class group of a field.

Now let $K$ be a number field and $E/K$ an elliptic curve. The multiplication-by-n morphism is a surjective isogeny on $E(\overline{K})$, with kernel $E[n]\cong (\mathbb{Z}/n\mathbb{Z}{)}^2$. The Kummer exact sequence of an elliptic curveyieldsExplicitly, choose $Q$ with $nQ=P$ and set $\delta (P{)}_{\sigma }=\sigma Q-Q$. This lies in $E[n]$, satisfies the cocycle identity and is unchanged as a class if $Q$ is replaced by another division point. If it is a coboundary $\sigma T-T$ for $T\in E[n]$, then $Q-T$ is rational over $K$ and $P=n(Q-T)$. Conversely such a rational division point makes the cocycle zero. This proves the injectivity of the Kummer map of an elliptic curve on $E(K)/nE(K)$.

To prove the Weak Mordell-Weil theorem, it remains to prove that this particular cohomological image is finite. The full group $H^1(K,E[n])$ need not be finite, so the injection alone is not enough. Choose a finite set $S$ of finite places containing the bad-reduction places and those dividing $n$. At any other place, the construction in Question 3 works over the local field: reduce $P$, find an $n$-division point over a finite residue extension, lift it by the Hensel lemma, and correct its error by the inverse formal multiplication. Thus $P$ has an $n$-division point over the maximal unramified extension. Choosing that lift locally makes the cocycle zero on inertia. For an arbitrary lift its restriction is a coboundary, so the cohomology class is unramified. Therefore the image lies in first Galois cohomology unramified outside a finite set:We prove the required finiteness of this group. Take a finite Galois extension $L/K$ containing all of $E[n]$ and ${\mu }_n$, and enlarge $S$ to include any remaining ramification of $L/K$. Let $S_L$ be the places above it. The inflation-restriction exact sequence bounds the kernel of restriction to $L$ bya finite group, since both its group of arguments and its group of values are finite. Over $L$ the torsion module is constant and can be identified with ${\mu }_n^2$, so Kummer theory givesRestrictions of our unramified classes remain unramified outside $S_L$. If a Kummer class $[a]$ is unramified at such a place $v$, its $n$th root can be taken in an unramified local extension. The extended valuation still has integer values, and $nv(\sqrt[n]{a})=v(a)$. Hence $v(a)$ is divisible by $n$. Each coordinate therefore lies in the S-unramified power class groupThis group is finite. Indeed, the ideal of $a$ away from $S_L$ is ${\mathfrak{a}}^n$. Sending $[a]$ to $[\mathfrak{a}]$ yields the exact sequenceFor its kernel, a principal $\mathfrak{a}$ lets us divide $a$ by an $n$th power to obtain an $S_L$-unit. For surjectivity, an ideal class killed by $n$ has an $n$th power which is principal away from $S_L$, giving such an $a$. The S-unit group is finitely generated and the ideal class group is finite; hence both end groups are finite, proving finiteness of S-unramified Kummer classes. Restriction has finite image in $L(S_L,n{)}^2$ and finite kernel, so $H_S^1(K,E[n])$ is finite. We concludewhich is the Weak Mordell-Weil theorem and its Kummer-theoretic proof of the weak Mordell-Weil theorem.

This framework also defines the Selmer group of an elliptic curve by imposing that a class in $H^1(K,E[n])$ lie in the local Kummer image at every completion. Its classes are unramified outside a fixed finite set, so the same finiteness proof applies. Its relation to rational points isThe Tate–Shafarevich group here measures the difference between locally soluble torsors and globally soluble ones. Finally, the weak theorem gives finite divisibility quotients; adding the height descent lemma and bounded-height finiteness proves the stronger Mordell-Weil theorem. Those height inputs are additional to the cohomological finiteness just established.

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.

Galois cohomology records the obstruction to choosing Galois-invariant division points. Let $G_K=\mathrm{Gal}(\overline{K}/K)$ be the absolute Galois group and give its algebraic-point modules the discrete topology. A continuous $1$-cocycle with values in a Galois module $M$ is a function $c:G_K\to M$ satisfyingA group coboundary has the form $c_{\sigma }=\sigma a-a$. Quotienting cocycles by coboundaries defines $H^1(K,M)$. For the finite module $E[m]$ these cocycles have finite image and factor through finite data; continuity is essential because $G_K$ is a profinite group.

In characteristic zero, multiplication by $m\ge 2$ on the algebraic points of an elliptic curve is surjective, with kernel $E[m]\cong (\mathbb{Z}/m\mathbb{Z}{)}^2$. The short exact sequence of Galois modulesgives the Kummer exact sequence of an elliptic curveExplicitly, choose $Q$ with $mQ=P$ and set $\delta (P{)}_{\sigma }=\sigma Q-Q$. Changing $Q$ by an $m$-torsion point changes the cocycle by a coboundary. The class vanishes exactly when a suitable choice of $Q$ is Galois-fixed, that is, when $P\in mE(K)$. This proves the injection directly and identifies its arithmetic meaning.

The entire group $H^1(K,E[m])$ need not be finite. For example, Kummer theory gives $H^1(K,{\mu }_m)=K^{\ast }/K^{\ast m}$, which has classes supported on arbitrarily many different primes. The finite part needed for the Weak Mordell-Weil theorem comes from a ramification restriction on the image of $\delta$.

Choose a finite set $S$ of places containing the archimedean places, the primes dividing $m$ and all primes of bad reduction of an elliptic curve. At a finite place $v\notin S$, the curve has good reduction and $m$ is invertible in its valuation ring. The elliptic curve extends to a smooth proper group scheme, and $[m]$ on that model is finite etale. A point $P\in E(K_v)$ extends to an integral section by properness. Its division-point fibre is consequently finite etale over the valuation ring. Over the maximal unramified extension it has a point, so the Kummer cocycle restricts trivially to the inertia group. The module $E[m]$ itself is unramified there for the same reason. Thuswhere $H_S^1$ denotes classes unramified outside $S$.

Here is a proof that this restricted group is finite. Take a finite Galois extension $L/K$ containing all coordinates of $E[m]$ and the $m$th roots of unity, and enlarge $S$ by its ramified primes. Over $L$ the module is trivial and, after choosing a basis, is isomorphic to ${\mu }_m^2$. Hilbert theorem 90 and the multiplicative Kummer sequence therefore giveFor unramified Kummer classes with bounded prime support, a class unramified outside $S_L$ has valuations divisible by $m$ at every prime outside $S_L$: the valuation of an $m$th root in an unramified extension is integral. Consequently both coordinates lie in

To show this set finite, write the outside-$S_L$ divisor of $a$ as $mD$. The ideal class of $D$ belongs to the $m$-torsion of the ideal class group of the ring of $S_L$-integers. This gives the exact sequenceThe last map is onto: if $mD$ is principal in the $S_L$-ideal group, a generator represents a class with outside valuations divisible by $m$. The kernel consists exactly of S-units modulo $m$th powers. The Dirichlet unit theorem, with the finitely many inverted primes adjoined, makes the S-unit group finitely generated; the ideal class group is finite, and localization only quotients it. Both ends of the sequence are therefore finite.

Finally, the inflation-restriction exact sequence bounds the kernel of restriction from $H^1(K,E[m])$ to $H^1(L,E[m])$ by the finite group $H^1(\mathrm{Gal}(L/K),E[m])$. The unramified subgroup has finite image, contained in $L(S_L,m{)}^2$, and finite kernel, so it is finite. The Kummer injection now provesThis is the Weak Mordell-Weil theorem, proved without first assuming finite generation of $E(K)$.

For computation one refines the unramified group by local solvability. The Selmer group of an elliptic curve consists of classes whose restriction at every completion lies in the corresponding local Kummer image. It is finite and sits inwhere the Tate–Shafarevich group $\mathrm{Sha}$ measures classes in $H^1(K,E)$ that become trivial at every completion. Thus locally soluble descent equations can give an upper bound without every class coming from a rational point. No finiteness assumption on the whole Tate-Shafarevich group is needed for the weak theorem. Combining the finite quotient with the height descent lemma from the height essay yields the full Mordell-Weil theorem. Replacing multiplication by $m$ with a smaller isogeny gives the same cohomological framework for two-isogeny descent.