Geodesics And Geometric-ARithmetic INtersections

Información del proyecto

GAGARIN

Identificador del acuerdo de subvención: 101076941

DOI

10.3030/101076941

Fecha de la firma de la CE 16 Febrero 2023

Fecha de inicio 1 Mayo 2023

Fecha de finalización 30 Abril 2028

Financiado con arreglo a

• European Research Council (ERC)

Coste total

€ 1 500 000,00

Aportación de la UE

€ 1 500 000,00

Coordinado por

UNIVERSITEIT LEIDEN
Netherlands

Periodic Reporting for period 1 - GAGARIN (Geodesics And Geometric-ARithmetic INtersections)

Período documentado: 2023-05-01 hasta 2025-10-31

Resumen del contexto y de los objetivos generales del proyecto

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).

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

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.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

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.