General Relativity, the Cosmological Constant and Modular forms

Strong field (exact) solutions of the gravitational field equations of General Relativity in the presence of a Cosmological Constant are investigated. In particular, a full exact solution is derived within the inhomogeneous Szekeres-Szafron family of space-time line element with a nonzero Cosmological Constant. The resulting solution connects, in an intrinsic way, General Relativity with the theory of modular forms and elliptic curves. The homogeneous FLRW limit of the above space-time elements is recovered and we solve exactly the resulting Friedmann Robertson field equation with the appropriate matter density for generic values of the Cosmological Constant $% \Lambda $ and curvature constant $K$. A formal expression for the Hubble constant is derived. The cosmological implications of the resulting non-linear solutions are systematically investigated. Two particularly interesting solutions i) the case of a flat universe $K=0,\Lambda \not=0$ and ii) a case with all three cosmological parameters non-zero, are described by elliptic curves with the property of complex multiplication and absolute modular invariant $j=0$ and 1728, respectively. The possibility that all non-linear solutions of General Relativity are expressed in terms of theta functions associated with Riemann-surfaces is discussed.