Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T20:16:52.420Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Discontinuous groups and automorphic forms Arithmetic problems. Diophantine geometry 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Anderson, G. W., t-motives, Duke Math. J. 53 (1986), 457–502.CrossRefGoogle Scholar
Anderson, G. W., Log-algebraicity of twisted A-harmonic series and special values of L-series in characteristic p, J. Number Theory 60 (1996), 165–209.CrossRefGoogle Scholar
Anderson, G. and Thakur, D., Tensor powers of the Carlitz module and zeta values, Ann. of Math. (2) 132 (1990), 159–191.CrossRefGoogle Scholar
Anglès, B. and Pellarin, F., Functional identities for L-series values in positive characteristic, J. Number Theory 142 (2014), 223–251.CrossRefGoogle Scholar
Anglès, B. and Pellarin, F., Universal Gauss–Thakur sums and L-series, Invent. Math. 200 (2015), 653–669.CrossRefGoogle Scholar
Anglès, B. and Taelman, L., Arithmetic of characteristic p special L-values, Proc. Lond. Math. Soc. (3) 110 (2015), 1000–1032; with an appendix by V. Bosser.CrossRefGoogle Scholar
Böckle, G., Global L-functions over function fields, Math. Ann. 323 (2002), 737–795.Google Scholar
Böckle, G., Cohomological theory of crystals over function fields and applications (CRM, Bellaterra, 2010).Google Scholar
Böckle, G. and Pink, R., Cohomological theory of crystals over function fields, EMS Tracts in Mathematics, vol. 9 (European Mathematical Society, Zürich, 2010).Google Scholar
Carlitz, L., On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), 137–168.CrossRefGoogle Scholar
Fresnel, J. and van der Put, M., Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218 (Birkhäuser, 2004).CrossRefGoogle Scholar
Goss, D., Basic structures of function field arithmetic (Springer, Berlin, 1996).CrossRefGoogle Scholar
Goss, D., On the L-series of Pellarin, J. Number Theory 133 (2013), 955–962.CrossRefGoogle Scholar
Lang, S., Algebra, revised third edition, Graduate Texts in Mathematics (Springer, 2002).CrossRefGoogle Scholar
Pellarin, F., Values of certain L-series in positive characteristic, Ann. of Math. (2) 176 (2012), 2055–2093.CrossRefGoogle Scholar
Perkins, R., On Pellarin’s L-series, Proc. Amer. Math. Soc. 142 (2014), 3355–3368.CrossRefGoogle Scholar
Rosen, M., Number theory in function fields, Graduate Texts in Mathematics (Springer, 2002).CrossRefGoogle Scholar
Taelman, L., A Dirichlet unit theorem for Drinfeld modules, Math. Ann. 348 (2010), 899–907.CrossRefGoogle Scholar
Taelman, L., Special L-values of Drinfeld modules, Ann. of Math. (2) 75 (2012), 369–391.CrossRefGoogle Scholar
Taelman, L., A Herbrand–Ribet theorem for function fields, Invent. Math. 188 (2012), 253–275.CrossRefGoogle Scholar
Taelman, L., Sheaves and functions modulo $p$, lectures on the Woods-Hole trace formula, in London Mathematical Society Lecture Note Series (2015), to appear, available at:https://staff.fnwi.uva.nl/l.d.j.taelman/publications.html.Google Scholar
Taguchi, Y. and Wan, D., L-functions of 𝜙-sheaves and Drinfeld modules, J. Amer. Math. Soc. 9 (1996), 755–781.CrossRefGoogle Scholar
Thakur, D., Gauss sums for Fq[t], Invent. Math. 94 (1988), 105–112.CrossRefGoogle Scholar
van der Put, M. and Singer, M. F., Galois theory of linear differential equations (Springer, 2003).CrossRefGoogle Scholar
Yu, J., Transcendence and special zeta values in characteristic p, Ann. of Math. (2) 134 (1991), 1–23.CrossRefGoogle Scholar