Reduction and isogenies of elliptic curves (Q6606880)

Let \(R\) be a discrete valuation ring with perfect residue field \(k\) of characteristic \(p\) and \(K\) be the quotient field of \(R\). In this paper, the author studies the reduction property of elliptic curves over \(K\) admitting a \(K\)-rational isogeny of degree \(\ell\). The author obtains several results of the following type: if \(p\) and \(\ell\) are greater than \(3\), d \(p \neq \ell\), and \(\ell -1 \equiv 2 \text{ or } 10 \pmod{12}\), then such an elliptic curve over \(K\) has either semistable reduction or teduction of type \(I_n^* \) for some \(n \ge 0\).