Torsion points of elliptic curves with good reduction (Q2518781)

The torsion points of prime order of elliptic curves over certain number fields with good reduction everywhere are studied. The main result consists in proving that an elliptic curve \(E\) over a number field \(K\) having a real place has no \(K\)-rational points of order \(p\) assuming that \(p\) does not divide the class number of \(K(\zeta_p)\) and that the ramification index is inferior to \(p-1\) for all primes of \(K\) above \(p\). To prove this result, the extensions of a diagonalizable group scheme \(\mu_p\) by a constant group scheme \(\mathbb{Z}/p\mathbb{Z}\) over the ring of integers are examined.