Periodic Reporting for period 1 - GAGARIN (Geodesics And Geometric-ARithmetic INtersections)
Okres sprawozdawczy: 2023-05-01 do 2025-10-31
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).
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).
The activities take place within the three themes discussed above, in the context and overall objectives.
The analytic goals are approached through the theory of p-adic deformations of modular forms, exhibiting modular generating series for the objects of primary interest in this theory, particularly the RM singular moduli constructed in earlier work. The p-adic nature of the theory has the advantage of being able to leverage p-adic deformations of Artin representations to relate them to algebraic numbers in abelian extensions of real quadratic fields through class field theory. Further investigations into the nature of p-adic overconvergent modular forms (notably, equidistribution of their zero locus) are investigated through the theory of canonical subgroups.
The computational activities serve two distinct purposes. The first is to keep theoretical investigations grounded in reality, and to build a reliable and extensive collection of examples by which theoretical progress can be guided, earlier observations and conjectures verified/falsified, and new observations can be made. The second purpose is to develop algorithms that can be used by a wider community of researchers that are not part of the team, to ensure the possibility of wider adoption and experimentation.
The geometric activities have, at this intermediate stage of the project, only just come into view, as was expected in the planning. Nonetheless, first results have started to emerge.
The analytic goals are approached through the theory of p-adic deformations of modular forms, exhibiting modular generating series for the objects of primary interest in this theory, particularly the RM singular moduli constructed in earlier work. The p-adic nature of the theory has the advantage of being able to leverage p-adic deformations of Artin representations to relate them to algebraic numbers in abelian extensions of real quadratic fields through class field theory. Further investigations into the nature of p-adic overconvergent modular forms (notably, equidistribution of their zero locus) are investigated through the theory of canonical subgroups.
The computational activities serve two distinct purposes. The first is to keep theoretical investigations grounded in reality, and to build a reliable and extensive collection of examples by which theoretical progress can be guided, earlier observations and conjectures verified/falsified, and new observations can be made. The second purpose is to develop algorithms that can be used by a wider community of researchers that are not part of the team, to ensure the possibility of wider adoption and experimentation.
The geometric activities have, at this intermediate stage of the project, only just come into view, as was expected in the planning. Nonetheless, first results have started to emerge.
The main results of the investigations have so far been in the analytic and computational objectives, attacking the main objective of showing algebraicity and arithmetic factorisations of RM singular moduli through the introduction of a p-adic height symbol of two geodesics, which bears a striking resemblance to the theory of p-adic heights associated to a pair of global points on a modular curve as it appears, for instance, in the work of Gross-Zagier. Through the deformation theory of certain Hilbert modular cusp forms, modular generating series are constructed, whose modularity implies the conjectures set forth in earlier work with Darmon, up to a local norm from Qp^2 to Qp.