Gross-stark conjectures, abelian L-functions and algebraic cycles

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Let K be a number field and let G_K be the absolute Galois group of K. In the case where one deals with one dimensional representations of G_K, Class field theory provides a correspondence between these L-functions and finite abelian extensions of K. The L-functions coming from one dimensional representations are called abelian L-functions. In some cases, these abelian L-functions vanish precisely with order one at s=0. The last assumption implies that K is either a totally real number field or an almost totally real field (ATR for short). In the early seventies, Harold Stark made deep conjectures which relate the first derivative of such L-functions at s=0 with units lying in abelian extensions of K. This conjecture is striking since it relates a special value of an L-function (analytic data) with a certain unit in an abelian extension of K (algebraic data). Harold Stark could prove this conjecture in the special case where K is the field of rational numbers or K is an imaginary quadratic number field. He could prove it by relating the first derivative of the L-function to a special value of a nice holomorphic function which satisfies many symmetries, a so-called modular function. The main goal of my research is to find the "right mathematical object" that should replace the role played by the modular functions in the general case where K is either a totally real field or an ATR field. So far, this mathematical object seems to be very elusive and it is expected that any progress made on this problem will bring together different areas like analysis, algebra and differential geometry. All these mentioned themes appear in various guises in number theory and we, number theorists, would like to clarify their relationships in order to have a more unified understanding.