
School of Mathematics and Statistics
David Kohel
University of Sydney
Quaternion algebras and invariants of Shimura curves.
Friday 1st October, 121pm, Carslaw 375.
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 ShimuraTaniyama
conjecture). In contrast to modular curves, it is notoriously
difficult to write down explicit models for these curves.
However the techniques of Eichler and MesterOesterle 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.
