Functor of points and height functions for noncommutative Arakelov geometry
- Creators
- Lima, Alicia
- Marcolli, Matilde
Abstract
We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function.
Additional Information
© 2021 Elsevier. Received 1 March 2021, Revised 15 July 2021, Accepted 17 July 2021, Available online 22 July 2021. The second author was partially supported by NSF grant DMS-1707882 and DMS-2104330, and by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593.Attached Files
Accepted Version - 2012.15276.pdf
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Additional details
- Eprint ID
- 110761
- Resolver ID
- CaltechAUTHORS:20210908-171121986
- DMS-1707882
- NSF
- DMS-2104330
- NSF
- RGPIN-2018-04937
- Natural Sciences and Engineering Research Council of Canada (NSERC)
- RGPAS-2018-522593
- Natural Sciences and Engineering Research Council of Canada (NSERC)
- Created
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2021-09-08Created from EPrint's datestamp field
- Updated
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2021-09-13Created from EPrint's last_modified field