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Coverings surviving all local solubility tests need not have rational points. The quotient of an isogeny Selmer group by the rational-point image is a kernel in the Tate–Shafarevich group. A nonzero class therefore passes every local test but contributes no rational point to a search. This explains why two-isogeny descent can leave a gap between a certified upper bound and found lower bound for the rank of an elliptic curve. Higher descents address finer obstructions; see Tom Fisher's paper on higher descents with rational two-torsion.

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.

Move the nonidentity rational 2-torsion point to $T=(0,0)$ and complete the square. After a rational change of coordinates, the elliptic curve has an integral modelThe two-isogeny formula gives a partnerand the isogeny of elliptic curves and its dual isogeny areBoth extend at the exceptional points; their kernels are the identities and the distinguished 2-torsion points, and $\hat{\varphi }\varphi =[2]$.

The two-torsion square-class homomorphism is $\alpha (O)=1$, $\alpha (T)=[b]$, and $\alpha (x,y)=[x]$ otherwise, with values in ${\mathbb{Q}}^{\ast }/{\mathbb{Q}}^{\ast 2}$. Its kernel is $\hat{\varphi }{E}^{\prime }(\mathbb{Q})$. Define ${\alpha }^{\prime }$ on the partner in the same way. By the prime-support bound in two-isogeny descent, representatives are signed squarefree divisors $d$ of $b$, or of $b^{\prime }$ for the partner. A class is realized precisely when its quartic covering in a two-isogeny descent has a rational solution:For $UV\ne 0$, this comes from $x=d(U/V{)}^2$ and $y=dUN/V^3$; the cases $U=0$ or $V=0$ account for the distinguished point and the identity. Test this finite list using signs and local congruences, and exhibit rational solutions for the surviving classes. The local-to-global gap in isogeny descent matters: local solubility gives an upper bound from the isogeny Selmer group, so a rank determination requires matching actual classes or further descent, not merely declaring every locally soluble cover to have a rational point.

Once the two actual images are determined, the two-isogeny rank formula isFor completeness, put $A=E(\mathbb{Q})$ and $B=E^{\prime }(\mathbb{Q})$. The Mordell-Weil theorem gives $[A:2A]=2^r|A[2]|$. The index of a subgroup formula through $\hat{\varphi }B$ givesHere $\ker \hat{\varphi }\cap \varphi A=\varphi (A[2])$, since $\hat{\varphi }\varphi =[2]$, and this image has size $|A[2]|/2$. The denominator is therefore $4/|A[2]|$, proving the displayed rank of an elliptic curve formula.

The obstruction is the local-to-global gap in isogeny descent. A two-isogeny descent quartic can have points over $\mathbb{R}$ and every p-adic field without a rational point. Local tests compute an isogeny Selmer group, which can strictly contain the rational-point quotient. For an isogeny of elliptic curves $\varphi :E\to E^{\prime }$, the comparison isThe last group is the kernel of $\varphi :\mathrm{Sha}(E/\mathbb{Q})\to \mathrm{Sha}(E^{\prime }/\mathbb{Q})$, so it records elements of the Tate–Shafarevich group invisible to all local tests. Searching a locally soluble covering which has no rational points never succeeds. Thus a Selmer group of an elliptic curve gives an upper bound and found independent points give a lower bound, and these need not coincide. Higher descent can improve the bounds, but this particular procedure has no unconditional guarantee of closing every gap.