Algebra Seminar

Quaternion algebras are the simplest of noncommutative rings, yet hold a rich arithmetic structure. Defined in terms of the ideal theory of quaternions, Eichler developed the theory of Brandt matrices for producing explicit bases of modular forms. Mestre and Oesterle described an essentially equivalent theory of graphs of isogenies of supersingular elliptic curves -- whose endomorphism rings are orders in quaternion algebras -- for similar computations of invariants of modular curves. Shimura curves generalize modular curves (and play a role in Ribet's reduction of Fermat's Last Theorem to the Shimura-Taniyama conjecture). In contrast to modular curves, it is notoriously difficult to write down explicit models for these curves. However the techniques of Eichler and Mester-Oesterle have similar application to the analysis of Shimura curves. We describe the connection between Shimura curves and the arithmetic of quaternions, then describe how we may use the ideal theory of quaternions to compute invariants of these generalized modular curves. Explicit examples provided.