000343029 001__ 343029
000343029 005__ 20230827173411.0
000343029 0247_ $$2CORDIS$$aG:(EU-Grant)101076941$$d101076941
000343029 0247_ $$2CORDIS$$aG:(EU-Call)ERC-2022-STG$$dERC-2022-STG
000343029 0247_ $$2originalID$$acorda_____he::101076941
000343029 035__ $$aG:(EU-Grant)101076941
000343029 150__ $$aGeodesics And Geometric-ARithmetic INtersections$$y2023-05-01 - 2028-04-30
000343029 372__ $$aERC-2022-STG$$s2023-05-01$$t2028-04-30
000343029 450__ $$aGAGARIN$$wd$$y2023-05-01 - 2028-04-30
000343029 5101_ $$0I:(DE-588b)5098525-5$$2CORDIS$$aEuropean Union
000343029 680__ $$aThis project will develop several aspects of a theory of real multiplication (RM), seeking to be a 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 front runners 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).
000343029 909CO $$ooai:juser.fz-juelich.de:1012389$$pauthority:GRANT$$pauthority
000343029 909CO $$ooai:juser.fz-juelich.de:1012389
000343029 980__ $$aG
000343029 980__ $$aCORDIS
000343029 980__ $$aAUTHORITY