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Quantum statistical mechanics over function fields

Consani, Caterina and Marcolli, Matilde (2007) Quantum statistical mechanics over function fields. Journal of Number Theory, 123 (2). pp. 487-528. ISSN 0022-314X. doi:10.1016/j.jnt.2006.12.002.

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In this paper we construct a noncommutative space of "pointed Drinfeld modules" that generalizes to the case of function fields the noncommutative spaces of commensurability classes of Q-lattices. It extends the usual moduli spaces of Drinfeld modules to possibly degenerate level structures. In the second part of the paper we develop some notions of quantum statistical mechanics in positive characteristic and we show that, in the case of Drinfeld modules of rank one, there is a natural time evolution on the associated noncommutative space, which is closely related to the positive characteristic L-functions introduced by Goss. The points of the usual moduli space of Drinfeld modules define KMS functionals for this time evolution. We also show that the scaling action on the dual system is induced by a Frobenius action, up to a Wick rotation to imaginary time.

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Additional Information:© 2006 Elsevier Inc. Received 26 July 2006; revised 7 December 2006. Available online 17 December 2006. Communicated by David Goss. Part of this work was completed during a visit of the first author to the Max-Planck Institute whose hospitality is gratefully acknowledged. We thank David Goss and Benoit Jacob for reading an early draft of the manuscript and providing useful feedback. We thank the referees for many useful comments.
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Record Number:CaltechAUTHORS:CONjnt07
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13479
Deposited By: Tony Diaz
Deposited On:10 Aug 2009 22:50
Last Modified:08 Nov 2021 22:38

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