Number Theory Seminar

We describe the quaternionic analogue of classical holomorphic automorphic forms. They are defined as functions on quaternionic symmetric domains, with values in irreducible representations of SU(2), that satisfy an analogue of the Cauchy-Riemann equations. As an example, we construct the quaternionic Weierstrass $\wp$-function, and show it has a Taylor series expansion with coefficients given by values of quaternionic Eisenstein series. We speculate about the algebraic and differential equations satisfied by these functions.