Cohomology and combinatorics of abelian arrangements

The study of hyperplane arrangements, originating in the 1960s, has seen recent advancements that renewed interest in generalizing classical results to broader contexts. This thesis aims to extend foundational results by investigating both cohomology and combinatorics in the wider framework of abelian arrangements. We begin by presenting the cohomology ring of the complement of abelian arrangements. Using a technique that pushes forward cohomological relations from the real hyperplane case, we develop an Orlik-Solomon type presentation for noncompact abelian arrangements. This approach provides both an original result for the general case and a new proof of the Orlik-Solomon presentation for complexified hyperplane arrangements, as well as the De Concini-Procesi presentation for unimodular toric arrangements. We then turn to combinatorial aspects, introducing definitions for inductively and divisionally free abelian arrangements based on poset structures. After proving the factorization of their characteristic polynomial, we show that inductively free arrangements include strictly supersolvable arrangements as a proper subclass, extending a well known result of Jambu and Terao. We further apply these findings to toric arrangements associated with ideals of root systems of types A, B and C, showing their inductiveness and providing a formula to compute all exponents. Finally, we move beyond the abelian context to study elliptic arrangements from a new perspective: we focus on elliptic curves with complex multiplication, where the endomorphism ring strictly contains Z, and this leads to significantly different behaviors. We compute the number of connected components in the intersections of any subset of the arrangement and use this result to associate an arithmetic matroid structure to an elliptic arrangement, opening new possibilities for further studies in generalized arrangement theory.