The arithmetic of motives and p-adic L functions (n. 20222B24AY)
Type
PRIN 2022, D.D. n. 104 del 2 febbraio 2022
CUP
C53D23002380006
Principal Investigator
Fabrizio Andreatta (Università degli Studi di Milano)
Other research units
UO1 - Fabrizio ANDREATTA - Università degli Studi di MILANO
UO2 - Matteo LONGO - Università degli Studi di PADOVA
UO3 - Stefano VIGNI - Università degli Studi di GENOVA
Duration
28/09/2023 - 31/10/2025
Description
L-functions (associated to varieties or more generally motives over a number field) are ubiquitous in modern number theory. Conjectures of Beilinson and Bloch-Kato predict a deep relation between their special values at integers and the underlying geometric objects. We propose to study instances of this relation in the classical setting and in the p-adic setting.
(A) p-adic L-functions and reciprocity laws. The main motivating problem is the Equivariant
Tamagawa Number conjecture (ETNC), formulated by Bloch-Kato for motives generalizing the Birch and Swinnerton-Dyer conjecture (BSD). As common in this research area, p-adic analogues of this conjecture are more accessible and are studied via p-adic tools. From the analytic side, the main actors are p-adic L-functions, expecially for families of modular forms; these analytic objects can be seen as p-adic avatars of more common complex L-functions. On the arithmetic side, p- regulators and étale Abel-Jacobi maps of specific arithmetic cycles are the main actors. The p-adic variation in families of these maps, including Faltings comparison isomorphism, is one of the main ingredients to fully understand the arithmetic side. The content of reciprocity laws is an explicit relation between these twe arithmetic and the analytic sides, which can be seen, again, as
a p-adic avatar of the Birch and Swinnerton-Dyer conjecture, or the ETNC.
(B) Motives and their realizations. If in (A) we consider explicit examples of motives, having the advantage that they can be studied from different perspective, a more foundational work on arithmetic properties of motives and their realisations will be addressed. This foundational work will be developed along different lines, the main of which is a systematic comparison between realisations via different cohomology theories (Betti, étale, de Rham, crystalline, syntomic). Such comparison isomorphisms are also crucial to establish some cases of the reciprocity laws in (A). We will make extensive use of p-adic Hodge theory, as well as the approach of Scholze and his collaborators via perfectoid techiniques.
(C) Complex L-functions. Complex L-functions can be understood, in large, as the analytic part in BSD, or in its generalisation by Bloch-Kato. In essence, the values at integer points of these L- functions are expected to have interesting arithmetic properties, one of which is the classical analytic class number formula. They also play a crucial role in the definition of p-adic L-functions, since their values at integer points, suitably normalized by complex periods, are algebraic and satisfy p-adic congruences that can then be packaged into p-adic L-functions. The study of complex L-functions and the relation of their special values with arithmetic properties of the underlying geometric objects is therefore one of the central themes of the project
Activities
Each node hired 1 post-doc. In Padova, A. Marrama
