RAMANUJAN MODULAR FORMS AND THE KLEIN QUARTIC

In one of his notebooks, Ramanujan gave some algebraic relations between three theta functions of order 7. We describe the automorphic character of a vector-valued mapping constructed from these theta series. This provides a systematic way to establish old and new identities on modular forms for the congruence subgroup of level 7, above all, a parametrization of the Klein quartic. From a historical point of view, this shows that Ramanujan discovered the main properties of this curve with his own means. As an application, we introduce an L-series in four different ways, generating the number of points of the Klein quartic over finite fields. From this, we derive the structure of the Jacobian of a suitable form of the Klein quartic over finite fields and some congruence properties on the number of its points.