Kummer generators and torsion points of elliptic curves with bad reduction at some primes (Q2866997)

Let \(E\) be an elliptic curve defined over a number field \(K\). For a prime \(p\), let us denote by \(K(E[p])\) the field generated by the \(p\)-torsion points of \(E\). Let \(\zeta_p\) be a fixed primitive \(p\)th root of unity. Let us assume that \(E\) has a \(p\)-torsion point rational over \(K\). Then the field \(K(E[p])\) is a Kummer extension over \(K(\zeta_p)\) (possibly trivial).NEWLINENEWLINEIn case \(p=5\) and \(p=7\), the author studies the ramification of the extension \(K(E[p])/K(\zeta_p)\) in terms of Kummer generators computed by Verdure in 2006.NEWLINENEWLINEMore precisely, it is known that there exists an element \(t\in K\) such that \(E\) is isomorphic over \(K\) to the elliptic curve given by the Weierstrass equation NEWLINE\[NEWLINEE_t^{(5)} : y^2+(1-t)xy-ty=x^3-tx^2\quad \text{if}\quad p=5,NEWLINE\]NEWLINE NEWLINE\[NEWLINEE_t^{(7)} : y^2+(1+t-t^2)xy+(t^2-t^3)y=x^3+(t^2-t^3)x^2\quad \text{if}\quad p=7.NEWLINE\]NEWLINE The discriminants of \(E_t^{(p)}\) are NEWLINE\[NEWLINE\Delta\bigl(E_t^{(5)}\bigr)=t^5Q_5(t)\quad \text{and}\quad \Delta\bigl(E_t^{(7)}\bigr)=t^7(1-t)^7Q_7(t),NEWLINE\]NEWLINE with NEWLINE\[NEWLINEQ_5(t)=t^2-11t-1\quad \text{and}\quad Q_7(t)=t^3-8t^2+5t+1.NEWLINE\]NEWLINE One has NEWLINE\[NEWLINE\begin{cases} Q_5(t)=(t-\alpha_5)(t-\beta_5) \cr \alpha_5=8+5\zeta_5+5\zeta_5^4 \cr \beta_5=3-5\zeta_5-5\zeta_5^4,\end{cases} \quad \begin{cases} Q_7(t)=(t-\alpha_7)(t-\beta_7)(t-\gamma_7) \cr \alpha_7=1-2\zeta_7-3\zeta_7^2-3\zeta_7^5-2\zeta_7^6 \cr \beta_7=1-2\zeta_7^2-3\zeta_7^3-3\zeta_7^4-2\zeta_7^5\cr \gamma_7=1-3\zeta_7-2\zeta_7^3-2\zeta_7^4-3\zeta_7^6.\end{cases} NEWLINE\]NEWLINE For any \(t\in K\) such that \(\Delta\bigl(E_t^{(p)}\bigr)\neq 0\), set NEWLINE\[NEWLINEa_5(t)={t-\alpha_5\over t-\beta_5}\quad \text{and}\quad a_7(t)={(t-\alpha_7)(t-\beta_7)^2\over (t-\gamma_7)^3}.NEWLINE\]NEWLINE Then Verdure has shown that NEWLINE\[NEWLINEK\bigl(E_t^{(p)}[p]\bigr)=K\Bigl(\zeta_p,\root p\of{a_p(t)}\Bigr).NEWLINE\]NEWLINE Let \(O_K\) be the ring of integers of \(K\) and \(t\) an element of \(O_K\) such that \(\Delta\bigl(E_t^{(p)}\bigr)\neq 0\). In this paper, the author has proved that the extension \(K\bigl(E_t^{(p)}[p]\bigr)/K(\zeta_p)\) is unramified outside the set of primes dividing \(Q_p(t)\). As a consequence of his result, for some quadratic fields \(K\), he gives examples of unramified Kummer extensions over \(K(\zeta_p)\) of degree \(p\).