Reduction types of CM curves (Q6568718)

Let \(R\) be a complete discrete valuation ring with fraction field \(L\) of characteristic \(0\) and finite residue field. If \(C/L\) is a curve of genus \(g\geq 1\), let Jac\((C)\) be the Jacobian variety of \(C\) defined over \(L\). The curve \(C/L\) is said to have complex multiplication by a CM field \(K\) of degree \(2g\) over \(\mathbb{Q}\) if Jac\((C)/L\) has complex multiplication over \(L\) in the sense that there is an embedding of \(K\) in the endomorphism algebra \(\mathrm{End}_L^0(J(C)):=\mathrm{End}_L(J(C))\otimes\mathbb{Q}\). (If \(K\) injects to \(\mathrm{End}^0(\mathrm{Jac}(C))):=\mathrm{End}_{\bar{\mathbb{Q}}}(\mathrm{Jac}(C)))\), \(\mathrm{Jac}(C)/L\) is said to have potential complex multiplication).\N\NThe paper classifies all the possible Kodaira types of reduction that can occur for elliptic curves with complex (resp.potential complex) multiplication. For genus \(2\) curves defined over \(L\), the possible Namikawa-Ueno reduction types are investigated under the assumptions that the Jacobian varieties are simple and have complex multiplication by quadratic CM fields over \(L\). There are two cases: \(C/L\) has potentially good reduction, and does not have potentially good reduction. In each of these cases, possible special fibers of the maximal proper regular model for \(C_{L^{\mathrm{unr}}}/L^{\mathrm{unr}}\) are presented. The classification results depend on the order of the group of roots of unity in a quadratic CM field \(K\) over \(L\).