Quaternion algebras, Shimura curves, and modular forms: Algorithms and arithmetic

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"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The principal investigator (PI) will advance the algorithmic theory of Shimura curves and quaternion algebras, with applications for the computation of Hilbert modular forms and automorphic forms on arithmetic Fuchsian groups. In continuation of joint work with Matthew Greenberg, the PI will generalize the scope of an algorithm to compute the Hecke module of Hilbert modular forms via the (degree 1) cohomology of a Shimura curve and will build exhaustive tables of modular elliptic curves over totally real fields. The PI will at the same time investigate the underlying algorithmic problems for quaternion algebras from both the perspective of computational complexity as well as practical implementation.Classical unsolved problems often serve as the genesis for the formulation of a rich and uni fied mathematical fabric. Diophantus of Alexandria first sought solutions to algebraic equations in integers almost two thousand years ago. Today, mathematicians recognize that geometric properties often govern the behavior of arithmetic objects.Furthermore, computational tools provide a means to test conjectures and can sometimes furnish partial solutions; at the same time, theoretical advances fuel dramatic improvements in computation. The theory, design, and implementation of algorithms in arithmetic geometry is a burgeoning area, and there are many exciting applications of these methods to diverse fields.