Euler Systems of Garrett-Rankin-Selberg type and Stark-Heegner points

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Around twelve years ago I discovered a completely explicit, but for now entirely conjectural, construction of global points on elliptic curves, which I called "Stark-Heegner points". In the simplest non-trivial setting, these points are expected to be defined over abelian extensions of real quadratic fields and would therefore contribute to our understanding of "explicit class field theory" for such fields. The main objective of this Discovery Grant proposal is to give an unconditional construction of the global cohomology classes in Selmer groups of elliptic curves that ought to arise from Stark-Heegner points via the connecting homomorphism of Kummer theory. The key tools in this construction are two new types of Euler systems arising from p-adic deformations of Gross-Kudla-Schoen diagonal cycles and Beilinson-Flach elements which I have been exploring with my collaborators (most importantly, Victor Rotger and Massimo Bertolini) since 2010. These two Euler systems are a natural generalisation of Kato's Euler system of Beilinson elements, and I refer to all three as "Euler systems of Garrett-Rankin-Selberg type" because of the key role played by the formulae of Garrett and Rankin-Selberg in relating them to special values of L-functions. The study of these Euler systems has already led my collaborators and me to new cases of the Birch and Swinnerton-Dyer conjecture in the spirit of the fundamental early results of Coates and Wiles. Most relevant to the project at hand is the finiteness of components of Mordell-Weil groups of modular elliptic curves over Q attached to characters of real quadratic fields when the associated L-function is non-zero at the central point. This new inroad into the Birch and Swinnerton-Dyer for abelian characters of real quadratic fields in "analytic rank zero" raises the hope that extensions of the method will lead to the desired information about the mysterious Stark-Heegner points, corresponding to cases of the Birch and Swinnerton-Dyer conjecture "in analytic rank one".