Explicit class field theory and the Birch and Swinnerton-Dyer conjecture

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My research program will revolve around two central open questions in number theory: firstly, the construction of global points on elliptic curves with the goal of better understanding the Birch and Swinnerton-Dyer conjecture, and secondly, Hilbert's twelfth problem and explicit class field theory, of which the analytic construction of class fields of real quadratic fields is the simplest prototypical special case. I intend to build on the significant progress I have achieved towards these questions in the last five years, most notably, (1) my work with Victor Rotger on p-adic deformations of diagonal cycles in the Chow groups of triple products of modular curves and Kuga-Sato varieties, which has led in particular to the proof of new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank zero, for elliptic curves over Q twisted by ring class characters of real quadratic fields, and (2) my more recent work with Jan Vonk in the past year, which has revealed a previously unexpected theory of singular moduli for real quadratic fields enjoying striking parallels with the classical theory of complex multiplication. Both works offer complementary and promising avenues for better understanding the construction of Stark-Heegner points that I introduced around 2000, whose shoring up has been my primary research focus since that time. My work with Victor revolves around objects which we call "generalised Kato classes": global classes in the Selmer groups of elliptic curves (over appropriate number fields, class fields of real quadratic fields being a particularly tantalising special case) arising from p-adic deformations of special geometric objects in Chow groups or higher Chow groups of Shimura varieties. Our ongoing efforts aim to compare these classes with the images of Stark-Heegner points under the connecting homomorphism of Kummer theory. While not sufficient to establish the global nature of Stark-Heegner points, which are defined analytically as purely local objects, relating them to global Selmer classes is a decisive step in that direction. From an ostensibly quite different angle, my discovery with Jan that Stark-Heegner points can be recast in the broader framework of a (still conjectural) theory of complex multiplication for real quadratic fields in which the role of meromorphic modular functions is played by what we call "rigid meromorphic cocycles", seems to be full of promise for future progress. Indeed, we now dispose of convincing strategies for making some parts of this picture unconditional, potentially leading to a satisfying solution to Hilbert's twelfth problem for real quadratic fields based on extending fundamental work of Gross-Zagier and of Kudla-Rapoport-Yang to a p-adic setting. That an eventual extension of Kudla's program to the p-adics could offer a key to Hilbert's twelfth problem is perhaps the most significant insight to emerge from my recent work with Jan Vonk.