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For nontrivial even $\chi$ in the full cyclotomic tower of $\mathbb{Q}$, Kummer theory relates the p-ramified Iwasawa module to the reflected odd unramified Iwasawa module, up to pseudo-isomorphism. Here $\iota (\gamma )={\gamma }^{-1}$ and $(1)$ is the Tate twist. The induced substitution on characteristic series is $T\mapsto u(1+T{)}^{-1}-1$, where $u$ is the cyclotomic value of $\gamma$. Omitting either operation changes the interpolation convention.

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.

The hypotheses for Kummer theory are that $n\ge 1$ is prime to the characteristic of $K$ and that all $n$th roots of unity lie in $K$. Under these hypotheses, adjoining radicals produces abelian extensions of exponent dividing $n$, and conversely every such finite extension is obtained this way. The roots-of-unity hypothesis matters: without it, adjoining one radical need not give a Galois extension.

We first prove Hilbert's theorem 90, including its cocycle form. Distinct field automorphisms are linearly independent as functions into the field. Indeed, in a shortest nontrivial relation ${\sum }_i{c}_i{\tau }_i(y)=0$, choose $x$ with ${\tau }_1(x)\ne {\tau }_r(x)$. Subtract ${\tau }_r(x)$ times the relation at $y$ from the relation at $xy$. This eliminates the last term while leaving the first nonzero, contradicting minimality. This is linear independence of distinct field embeddings.

Let $L/K$ be finite Galois with group $G$, and suppose $a_{\sigma }\in L^{\ast }$ satisfy $a_{\sigma \tau }=a_{\sigma }\sigma (a_{\tau })$. By independence of the automorphisms, choose $c\in L$ for whichThenThus every multiplicative cocycle is a coboundary:For a cyclic group generated by $\sigma$ of order $m$, an element $a$ of norm one defines the cocycle $a_{{\sigma }^i}={\prod }_{j=0}^{i-1}{\sigma }^j(a)$; the condition at $i=m$ is exactly $N_{L/K}(a)=1$. HenceThe reverse implication follows by telescoping the product of conjugates.

Now let $\Delta$ be a finite subgroup of $K^{\ast }/K^{\ast n}$ and put $L=K(\sqrt[n]{\Delta })$. Finitely many representatives generate it; since ${\mu }_n\subset K$, the polynomials $X^n-a$ split in $L$ and are separable. Thus $L/K$ is finite Galois. For $\sigma \in G$ and $[a]\in \Delta$, choose ${\alpha }^n=a$ and define the Kummer pairingChanging the root or multiplying $a$ by an $n$th power does not change this value. It is multiplicative in both arguments. Since the radicals generate $L$, it embeds $G$ in $\mathrm{Hom}(\Delta ,{\mu }_n)$. Consequently $G$ is abelian of exponent dividing $n$.

For completeness, establish perfection rather than just this injection. Let ${\Delta }_L$ consist of all classes in $K^{\ast }/K^{\ast n}$ having an $n$th root in $L$. The same construction gives an injective map ${\Delta }_L\to \mathrm{Hom}(G,{\mu }_n)$: the character is trivial precisely when the root lies in $K$. It is surjective by Hilbert 90. Given $\chi :G\to {\mu }_n$, apply the cocycle theorem to $a_{\sigma }=\chi (\sigma {)}^{-1}$; it gives $\sigma (b)/b=\chi (\sigma )$, so $b^n\in K$ and its class has the required character. A finite abelian group of exponent dividing $n$ has as many ${\mu }_n$-characters as elements, by decomposition into cyclic factors. ThereforeIt follows that $\Delta ={\Delta }_L$ andConversely, for any finite abelian $L/K$ of exponent dividing $n$, Hilbert 90 supplies eigenvectors for every character. Their $n$th powers lie in $K$, and the intersection of their character kernels is trivial, so they generate $L$. This proves the Kummer classification and the inclusion-reversing duality between subgroups and intermediate extensions.

The simplest case is a cyclic extension of degree $m\mid n$: it has a generator $\alpha$ with ${\alpha }^m\in K$, and its automorphisms multiply $\alpha$ by $m$th roots of unity. Quadratic extensions in characteristic different from two are the case $n=2$. The cohomological summary is the exact sequence $1\to {\mu }_n\to K_s^{\ast }\stackrel{z\mapsto z^n}{\to }{K}_s^{\ast }\to 1$ together with Hilbert 90, giving $H^1(G_K,{\mu }_n)\cong K^{\ast }/K^{\ast n}$.

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 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.

A one-variable main conjecture for CM elliptic curves compares a torsion arithmetic Iwasawa module with a measure interpolating special Hecke L-function values. The classical bounded-measure formulation uses a prime of good reduction which splits in the CM field, hence gives ordinary reduction. We first describe the one-prime division-tower formulation, including the analytic construction, and then explain its cyclotomic Selmer counterpart and the change at supersingular primes.

For a precise basic setting, take $E/K$ with complex multiplication by ${\mathcal{O}}_K$, $K$ of class number one, and a split prime $p>3$ of good reduction, prime to the Hecke conductor $\mathfrak{f}$. Write $p{\mathcal{O}}_K=\mathfrak{p}\bar{\mathfrak{p}}$, choose the $p$-adic embedding corresponding to $\mathfrak{p}$, and let $\pi ={\psi }_E(\mathfrak{p})$. Its formal group of an elliptic curve has height one, with $[\pi ](t)\equiv t^p(\mathrm{mod}\mathfrak{p})$. The one-prime division towerhas $\Delta$ of order prime to $p$ and $\Gamma \simeq {\mathbb{Z}}_p$. Its CM character $\kappa :G\to {\mathbb{Z}}_p^{\times }$ describes the action on the prime-primary Tate module. Let $F=F_{\infty }^{\Gamma }$ and $\Lambda =I[[\Gamma ]]\simeq I[[T]]$. We extend coefficients to the complete unramified ring $I$ to include character values and a CM period.

Let $X$ be the Galois group over $F_{\infty }$ of the maximal abelian pro-$p$ extension unramified outside primes above $\mathfrak{p}$. Let $A$ be the analogous everywhere-unramified group, equivalently the norm inverse limit of the $p$-parts of the ideal class groups. Write $U$ for the norm limit of principal local units at $\mathfrak{p}$, $\mathcal{E}$ for the closures of global units in $U$, and $\mathcal{C}$ for the closures of elliptic units. Roots of unity and finite initial-layer errors are retained when formulating integral statements; after passage to a finite-character component they do not change a characteristic ideal.

The class-field unit sequence, obtained by taking norm limits in global Artin reciprocity, givesIn this ordinary one-prime tower, the unit and global-unit terms have the same Iwasawa rank, and the displayed quotients and $X,A$ are torsion on their character components. The equalityis the global-units/class-groups form of the CM main theorem. Multiplicativity of characteristic ideals in the sequence then givesThe analytic construction must identify the latter ideal with a genuinely interpolating p-adic L-function; it is not enough to rename an arbitrary characteristic power series as an L-function.

Here is that construction. For an auxiliary ideal $\mathfrak{a}$ prime to $6p\mathfrak{f}$, a suitably normalized rational elliptic function has divisor a multiple ofTranslate by a generator of the conductor torsion and take the product over its ray-class orbit. This produces a rational function $f_{\mathfrak{a}}(t)$ defined over $K$ which is a unit series at the identity. Evaluating it on compatible ${\mathfrak{p}}^n$-division points gives elliptic units. The reason for the norm relations is geometric: multiplication by $\pi$ groups the translated zeros and poles into fibers, so the product over $E[\mathfrak{p}]$ has the divisor of $f_{\mathfrak{a}}([\pi ]t)$. The normalization fixes the constant, giving the local identityThis is also the norm relation of the Coleman power series for a CM formal group interpolating the resulting local units.

An integral logarithmic correction makes the passage from those units to a bounded measure explicit. In the chosen split completion $K_{\mathfrak{p}}={\mathbb{Q}}_p$, putSince $[\pi ]t\equiv t^p(\mathrm{mod}p)$ and the coefficients of $f$ reduce to ${\mathbb{F}}_p$, the ratio is in $1+p{\mathbb{Z}}_p[[t]]$. For $p>2$, its logarithm divided by $p$ is integral. Constants are treated using the usual $p$-adic logarithm on units. This proves the integrality needed for the Frobenius-corrected logarithm; applying an uncorrected logarithm alone would not prove boundedness.

Choose an isomorphism $\eta :{\hat{\mathbb{G}}}_m\to \hat{E}$ over $I$, and normalize the CM period byThen $A_f(T)={\mathcal{L}}_f(\eta (T))\in I[[T]]$ is the Amice transform of a unique $I$-valued p-adic measure on ${\mathbb{Z}}_p$. The norm relation above givesthe first logarithmic sum is $\log f([\pi ]t)$ and the second is the same sum, because $[\pi ]Q=0$ and there are $p$ points. On the measure side the sum selects $p{\mathbb{Z}}_p$ and multiplies by $p$, so this identity says that the measure is supported on ${\mathbb{Z}}_p^{\times }$. Transport it to $G$ by the CM character and project to the desired component.

The moments can now be calculated rather than postulated. With $D=(1+T)d/dT$,The factor ${\pi }^k$ appears because $[\pi ]$ is multiplication by $\pi$ in the additive formal-logarithm coordinate. The remaining derivative is a Coates–Wiles homomorphism. The complex product for the elliptic function makes it a sum of shifted inverse powers of lattice elements. Grouping these lattice elements by ray-class ideals identifies that sum with $(k-1)!L_{\mathfrak{f}}({\bar{\psi }}_E^k,k)/{\Omega }_{\infty }^k$, multiplied by the auxiliary factor $N\mathfrak{a}-{\psi }_E(\mathfrak{a}{)}^k$. Here $L_{\mathfrak{f}}$ omits primes dividing the original conductor, including when the conductor of the power character becomes smaller.

Normalize the rational functions and matching complex and $p$-adic periods consistently, and remove the auxiliary factors by the distribution relations. They are evaluations of $N\mathfrak{a}-[\mathfrak{a}]$ in the Iwasawa algebra. On each finite-character component an auxiliary ideal can be chosen making this element a unit: its residues at $\mathfrak{p}$ and $\bar{\mathfrak{p}}$ can be prescribed independently by the Chinese remainder theorem. The resulting integral measure ${\mu }_E$ satisfies, on its branch and with $k$ in the corresponding residue class,The exact normalization of periods matters for this equality; multiplication by an integral unit does not affect the ideal in the main conjecture. Finite-order twists are obtained by inserting a character in the ray-class sums, with its local Gauss sum. The lattice calculation proves algebraicity of the normalized complex values, while the integral series construction proves their $p$-adic interpolation. These are different steps.

The Coleman measure map identifies $U_{\chi }$ with a rank-one Iwasawa algebra lattice, up to finite errors, and takes ${\mathcal{C}}_{\chi }$ to the ideal generated by the corresponding power series ${\mathcal{L}}_{\chi }$. The conjecture therefore has the concrete formIts proof uses the Euler system of elliptic units. Adjoining auxiliary ray moduli supplies norm relations with factors $1-{\mathrm{Frob}}_{\mathfrak{q}}^{-1}$. Derivative operators in the auxiliary cyclic groups produce Kolyvagin classes; their controlled valuations relate a new auxiliary prime to its ideal class. Chebotarev density theorem permits the required Frobenius choices. Successively choosing these primes bounds the class-group elementary divisors by the elliptic-unit index, giving the difficult characteristic-ideal divisibility. The opposite comparison comes from the class-number/unit-index formula and passage through the tower. Their equality, combined with the exact sequence above and the logarithmic computation, is the proof mechanism. The ordinary split-prime setting of this formulation is specified in Rubin's one-variable main-conjecture paper.

For completeness, the familiar cyclotomic formulation concerns a different tower. If $E/\mathbb{Q}$ has complex multiplication and good ordinary reduction at $p$, put ${\Gamma }_{\mathrm{cyc}}=\mathrm{Gal}({\mathbb{Q}}_{\infty }/\mathbb{Q})$ andFor a finite layer $L$, its Selmer group of an elliptic curve iswhere $S$ contains the bad primes and primes above $p$; at infinite level take the direct limit. Thus it imposes the local Kummer conditions and sits between rational points tensored with ${\mathbb{Q}}_p/{\mathbb{Z}}_p$ and the $p$-primary Tate–Shafarevich group. Its dual ${\mathcal{X}}_E$ is a torsion Iwasawa module, and the cyclotomic main conjecture isWith compatible reciprocity conventions, the ordinary function has the finite-character interpolationwhere $\alpha$ is the $p$-adic unit root of $X^2-a_{p}X+p$ and $\tau (\chi )={\sum }_{a\mathrm{mod}{p}^n}\chi (a)e^{2\pi ia/p^n}$; characters of this cyclotomic ${\mathbb{Z}}_p$-extension are even. The two-variable CM measure specializes to this line after the CM-character twist. Kummer theory, class-field descriptions of the prime-primary local conditions and control of finite-layer errors identify its algebraic counterpart with the displayed Selmer dual. The one-prime division tower is not itself the cyclotomic tower, so this requires the twist and specialization argument, not just changing its name.

Finally, at a good inert prime the curve has supersingular reduction. The ordinary unit root is absent, the unmodified cyclotomic Selmer dual is not torsion, and the ordinary bounded-function argument does not apply. For CM curves over $\mathbb{Q}$ and odd supersingular $p$, signed local trace conditions and signed functions give instead $\mathrm{char}{\mathcal{X}}_E^{\pm }=(L_p^{\pm }(E))$, as proved in Pollack and Rubin's supersingular main-conjecture theorem. Thus the ordinary hypotheses and the analytic local correction are essential parts of the essay's main-conjecture statement.

Kummer theory studies extracting roots through Galois actions, and on an elliptic curve the corresponding operation is division of points. The bridge is a cocycle: different Galois conjugates of a chosen division point differ by torsion. Controlling the ramification of these cocycles is what proves the Weak Mordell-Weil theorem.

Classically, for a characteristic-zero field $K$, the sequenceis exact. Hilbert's theorem 90 gives $H^1(K,{\overline{K}}^{\times })=0$, so Galois cohomology identifiesThe class of $a$ is represented by $\sigma \mapsto \sigma (\sqrt[m]{a})/\sqrt[m]{a}$. If ${\mu }_m\subset K$, the module is constant and these are continuous characters into ${\mu }_m$; the associated fields are cyclic Kummer extensions of degree dividing $m$. Several such classes give abelian extensions of exponent dividing $m$. Without the roots-of-unity hypothesis the cohomological description still holds, but the action on ${\mu }_m$ must not be discarded.

For an elliptic curve over a number field $K$, multiplication by $m\ge 2$ gives the exact sequence of Galois modulesThe map is onto because it is a nonconstant isogeny over an algebraically closed field. For $P\in E(K)$ choose $Q\in E(\overline{K})$ with $mQ=P$, and setIt satisfies $\delta (P{)}_{\sigma \tau }=\delta (P{)}_{\sigma }+\sigma \delta (P{)}_{\tau }$, the Galois 1-cocycle identity. Changing $Q$ by a torsion point of an elliptic curve changes the cocycle by a coboundary. If the class is zero, a corresponding change of $Q$ makes it Galois-fixed, so $P\in mE(K)$. Conversely a rational division point gives the zero class. Thus the Kummer map of an elliptic curve is injective on the quotient, and the cohomology sequence givesThis is the Kummer exact sequence of an elliptic curve. The last group parametrizes the appropriate torsor classes. It is important that the entire middle group need not be finite; an unrestricted Kummer class can ramify at arbitrarily many primes.

The Weak Mordell-Weil theorem states $E(K)/mE(K)$ is finite for every number field $K$ and every $m\ge 1$. The case $m=1$ is trivial. To prove the remaining cases, choose a finite set $S$ containing the primes of bad reduction and the primes dividing $m$. At a prime outside $S$, the elliptic curve extends to a smooth proper group scheme over the valuation ring, and $[m]$ is finite étale. Properness extends a local rational point to a section; its division fibre is a finite étale torsor over that ring. Therefore its Kummer class is unramified. This is unramified division torsors at good primes, and shows that ${\delta }_m(E(K))$ lies in first Galois cohomology unramified outside a finite set.

Let $L=K(E[m])$, enlarged if necessary to contain ${\mu }_m$, and include in $S$ the finitely many primes ramified in $L/K$. This is a finite Galois extension. Over $L$ the torsion module is constant, isomorphic to $(\mathbb{Z}/m\mathbb{Z}{)}^2$, and so to two copies of ${\mu }_m$. Restricting a cocycle to $G_L$ therefore reduces it to two classical Kummer classes unramified outside the primes above $S$. Such a class is represented by $a\in L^{\times }$ satisfyingWrite this power-class group as $L(S_L,m)$. The implication follows locally because, away from residue characteristic dividing $m$, the valuation of an unramified $m$th-root extension must be divisible by $m$.

Its finiteness comes from two standard arithmetic finiteness results, rather than from an assertion that all of $L^{\times }/(L^{\times }{)}^m$ is finite. The valuations outside $S_L$ associate to $a$ an ideal whose $m$th power is principal. There is an exact sequenceThe left group is finite by the finite generation of the S-unit group, and the right group is finite by finiteness of the ideal class group. This proves finiteness of S-unramified Kummer classes. Hence there are only finitely many restricted cocycles over $L$. The kernel of restriction from $K$ is contained in $H^1(\mathrm{Gal}(L/K),E[m])$, which is finite because both group and module are finite. The restricted cohomology group over $K$ is therefore finite, and its injected subgroup $E(K)/mE(K)$ is finite too. This completes a Kummer-theoretic proof of the weak Mordell-Weil theorem.

For arithmetic computation one adds local conditions. The Selmer group of an elliptic curve consists of classes in $H^1(K,E[m])$ whose restriction at every completion $K_v$ lies in the image of the corresponding local Kummer map. They are unramified outside the same finite set, so this group is finite. The failure of a everywhere locally soluble torsor to have a global point is measured by the Tate–Shafarevich group, defined as the kernel of the local restriction map on $H^1(K,E)$. The resulting exact sequence isThus the Selmer group of an elliptic curve gives a computable upper bound for the quotient; actual rational points supply lower bounds. Equality need not follow just from local solubility. Finiteness of this fixed torsion subgroup does not prove finiteness of the whole Tate–Shafarevich group.

Finally, weak finiteness alone is not finite generation: the additive group $\mathbb{Q}$ has trivial quotients by multiplication and is not finitely generated. The additional ingredient for the Mordell-Weil theorem is height descent. Choose representatives $P_i$ of $E(K)/mE(K)$ and write $P=P_i+mQ$. The quadratic growth of the canonical height of an elliptic curve, together with bounds for translation by the finitely many $P_i$, makes $Q$ have smaller height whenever $P$ is sufficiently large. Northcott theorem gives only finitely many points of bounded height, and repeated descent yields a finite generating set. This explains the roles of Kummer theory, local information and heights without conflating the weak and full theorems.

Classical Kummer theory relates extraction of $n$th roots to Galois cohomology. In characteristic zero the exact sequenceand Hilbert theorem 90 identify $H^1(K,{\mu }_n)$ with $K^{\times }/K^{\times n}$. For an elliptic curve, the corresponding Kummer exact sequence of an elliptic curve isMultiplication by $n$ is surjective over the algebraic closure. If $nQ=P\in E(K)$, the cocycle $\sigma \mapsto \sigma Q-Q$ takes values in $E[n]$. Changing the choice of $Q$ changes it by a coboundary, and changing $P$ by an element of $nE(K)$ does not change its class. Conversely, a trivial cocycle class allows $Q$ to be adjusted by an $n$-torsion point to become $K$-rational. Thus the Kummer map of an elliptic curve is an injectionThis is the elliptic form of Kummer theory. The full cohomology group need not be finite; the arithmetic restriction on these classes is essential.