S-arithmetic groups

S-arithmetic groups are mathematical objects that generalize the concept of integer lattice structures within algebraic groups, considering number systems beyond ordinary integers. They are defined over rings of S-integers, which include integers and some of their fractional counterparts, associated with a set of primes S. These groups play a crucial role in number theory, algebra, and geometry, notably in understanding symmetries and automorphisms of algebraic varieties and the structure of solutions to polynomial equations. Essentially, S-arithmetic groups extend the idea of integer-based symmetries to include broader number systems, providing a rich framework for studying arithmetic and geometric properties.