Elliptic arrangements of complex multiplication type

Abstract We provide a natural definition of an elliptic arrangement, extending the classical framework to an elliptic curve $\mathcal {E}$ with complex multiplication. We analyze the intersections of elements of the arrangement and their connected components as $\operatorname {End}(\mathcal {E})$ -modules. Furthermore, we prove that the combinatorial data of elliptic arrangements define both an arithmetic matroid and a matroid over the ring $\operatorname {End}(\mathcal {E})$ . In this way, we obtain a class of arithmetic matroids that is different from the class of arithmetic matroids realizable via toric arrangements. Finally, we show that the Euler characteristic of the complement is an evaluation of the arithmetic Tutte polynomial.