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Arithmetic of positive characteristic $L$-series values in Tate algebras

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms Algebraic groups

Published online by Cambridge University Press:  07 September 2015

B. Anglès ,
F. Pellarin ,
F. Tavares Ribeiro  and
F. Demeslay
B. Anglès
Affiliation:
Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme, UMR 6139, Campus II, Boulevard Maréchal Juin, B.P. 5186, 14032 Caen Cedex, France email bruno.angles@unicaen.fr
F. Pellarin
Affiliation:
Institut Camille Jordan, UMR 5208, Site de Saint-Etienne, 23 rue du Dr. P. Michelon, 42023 Saint-Etienne, France email federico.pellarin@univ-st-etienne.fr
F. Tavares Ribeiro
Affiliation:
Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme, UMR 6139, Campus II, Boulevard Maréchal Juin, B.P. 5186, 14032 Caen Cedex, France email floric.tavares-ribeiro@unicaen.fr
F. Demeslay
Affiliation:
Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme, UMR 6139, Campus II, Boulevard Maréchal Juin, B.P. 5186, 14032 Caen Cedex, France email florent.demeslay@unicaen.fr
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Abstract

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The second author has recently introduced a new class of $L$-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these $L$-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.

Keywords

L-values in positive characteristiclog-algebraic theoremclass modulesBernoulli–Carlitz fractions

MSC classification

Primary: 11F52: Modular forms associated to Drinfel'd modules 14G25: Global ground fields 14L05: Formal groups, $p$-divisible groups
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 1 , January 2016 , pp. 1 - 61
Copyright
© The Authors 2015 

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