Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 4

For $[x]\in \Delta$, choose ${\alpha }_x$ with ${\alpha }_x^n=x$ and define the Kummer pairingChanging $x$ by an $n$th power or changing its root leaves the quotient unchanged because ${\mu }_n\subset K$. Multiplication of radicals proves bilinearity. If an automorphism pairs trivially with every class, it fixes all generating radicals and hence all of $L$. If a class pairs trivially with every automorphism, its radical lies in $K$, so the class is trivial. Thus the pairing is well-defined, bilinear, and nondegenerate on both sides.

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For $[P]\in \Gamma$, choose $Q$ with $nQ=P$ and define the elliptic Kummer pairingReplacing $Q$ by $Q+T$ with $T\in E[n]$ changes nothing because all $n$-torsion is $K$-rational; replacing $P$ by $P+nR$ permits replacing $Q$ by $Q+R$, again without changing the value. The group laws prove bilinearity. An automorphism pairing trivially with every class fixes every division point and hence $L$. Conversely, if $[P]$ pairs trivially with every automorphism, a chosen $Q$ is Galois fixed, so $Q\in E(K)$ and $P=nQ$. The pairing is therefore nondegenerate.

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Define the S-unramified power class group byThere is an exact sequence from the $S$-unit group modulo $n$th powers into $K(S,n)$ and then into the $n$-torsion of the ideal class group. By the Dirichlet unit theorem,after absorbing the fixed roots of unity into the exponent. The ideal class group has fixed finite order $h_K$. Since $n\ge 2$, choose a constant $c$ depending only on $K$ with $h_K$ and the unit contribution bounded by $n^c$. Then

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Take $S$ to contain every finite prime dividing $n$ and every prime of bad reduction of $E$. The local theory of reduction of an elliptic curve shows that Kummer classes of rational points are unramified outside $S$. Choose a basis of the constant group $E[n]\cong (\mathbb{Z}/n\mathbb{Z}{)}^2$. Kummer theory and the Weil pairing identify the resulting two scalar coordinates ofwith power classes in $K^{\ast }/(K^{\ast }{)}^n$. The ramification statement places both coordinates in $K(S,n)$. Restriction to $\Gamma$ is injective by the nondegeneracy proved in part b, and therefore

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