This project will develop several aspects of a theory of real multiplication (RM), a long-sought counterpart of the theory of complex multiplication (CM) discovered in the 19th century. Classical CM theory is famed for its beauty and elegance, and is important in a variety of contexts. For instance:
1. in the classical era, it arose in the context of explicit class field theory. This feature is the subject of Kronecker’s Jugendtraum and Hilbert’s 12th problem,
2. in the modern era, it has been instrumental in proving known cases of the Birch–Swinnerton-Dyer conjecture, notably the results of Gross–Zagier which are a main theme of this proposal,
3. it has been used in elliptic and hyperelliptic curve cryptography, in cryptosystems based on supersingular isogeny graphs, one of the candidates for a secure post-quantum international standard.
The objectives are to develop analytic, computational, and geometric aspects of such an RM theory, and address the full scope of these features. The theory is based on the notion of arithmetic intersections of geodesics, and this project gives a new approach towards RM theory based on a notion of p-adic weak harmonic Maaß forms, and p-adic height pairings of geodesics attached to real quadratic fields.
Emphasis will lie on analytic aspects (p-adic Borcherds lifts and p-adic mock modular forms), computational aspects (development of user-friendly software for computations in RM theory), and geometric aspects (RM cycles on Shimura curves, and applications to the Birch–Swinnerton-Dyer conjecture).
