Elliptic arrangements of complex multiplication type

TL;DR

This paper extends the concept of arrangements to elliptic curves with complex multiplication, analyzing their combinatorial and topological properties, and linking them to arithmetic matroids and Tutte polynomials.

Contribution

It introduces a new class of elliptic arrangements with complex multiplication and establishes their connection to arithmetic matroids and Tutte polynomial evaluations.

Findings

Elliptic arrangements define arithmetic matroids and matroids over End(E).

The combinatorial data determine the arithmetic Tutte polynomial.

The Euler characteristic of the complement is an evaluation of the arithmetic Tutte polynomial.

Abstract

We provide a natural definition of an elliptic arrangement, extending the classical framework to an elliptic curve E with complex multiplication. We analyse the intersections of elements of the arrangement and their connected components as End(E)-modules. Furthermore, we prove that the combinatorial data of elliptic arrangements define both an arithmetic matroid and a matroid over the ring End(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.