Anderson-Stark units for 𝔽_{𝕢}[𝜃]

Anglès, Bruno; Pellarin, Federico; Tavares Ribeiro, Floric

doi:10.1090/tran/6994

Anderson-Stark units for ${\mathbb F}_{q}[\theta ]$
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by Bruno Anglès, Federico Pellarin and Floric Tavares Ribeiro PDF

Trans. Amer. Math. Soc. 370 (2018), 1603-1627 Request permission

Abstract:

We investigate the arithmetic of special values of a new class of $L$-functions recently introduced by the second author. We prove that these special values are encoded in some particular polynomials which we call Anderson-Stark units. We then use these Anderson-Stark units to prove that $L$-functions can be expressed as sums of polylogarithms.

References

Greg W. Anderson, Log-algebraicity of twisted $A$-harmonic series and special values of $L$-series in characteristic $p$, J. Number Theory 60 (1996), no. 1, 165–209. MR 1405732, DOI 10.1006/jnth.1996.0119
Bruno Anglès and Federico Pellarin, Functional identities for $L$-series values in positive characteristic, J. Number Theory 142 (2014), 223–251. MR 3208400, DOI 10.1016/j.jnt.2014.03.004
Bruno Anglès and Federico Pellarin, Universal Gauss-Thakur sums and $L$-series, Invent. Math. 200 (2015), no. 2, 653–669. MR 3338012, DOI 10.1007/s00222-014-0546-8
B. Anglès, F. Pellarin, and F. Tavares Ribeiro, Arithmetic of positive characteristic $L$-series values in Tate algebras, Compos. Math. 152 (2016), no. 1, 1–61. With an appendix by F. Demeslay. MR 3453387, DOI 10.1112/S0010437X15007563
Greg W. Anderson and Dinesh S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. of Math. (2) 132 (1990), no. 1, 159–191. MR 1059938, DOI 10.2307/1971503
N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1308, Hermann, Paris, 1964 (French). MR 0194450
W. Dale Brownawell and Matthew A. Papanikolas, A rapid introduction to Drinfeld modules, $t$-modules, and $t$-motives, Preprint.
Demeslay, Formules de classes en caractéristique positive, Ph.D. Thesis, 2015.
Jean Fresnel and Marius van der Put, Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2014891, DOI 10.1007/978-1-4612-0041-3
David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR 1423131, DOI 10.1007/978-3-642-61480-4
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
Matthew A. Papanikolas, Log-algebraicity on tensor powers of the Carlitz module and special values of Goss $L$-functions, In preparation.
Federico Pellarin, Values of certain $L$-series in positive characteristic, Ann. of Math. (2) 176 (2012), no. 3, 2055–2093. MR 2979866, DOI 10.4007/annals.2012.176.3.13
Rudolph Bronson Perkins, Explicit formulae for $L$-values in positive characteristic, Math. Z. 278 (2014), no. 1-2, 279–299. MR 3267579, DOI 10.1007/s00209-014-1315-5
Rudolph Bronson Perkins, On Pellarin’s $L$-series, Proc. Amer. Math. Soc. 142 (2014), no. 10, 3355–3368. MR 3238413, DOI 10.1090/S0002-9939-2014-12080-6
Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR 1876657, DOI 10.1007/978-1-4757-6046-0
Dinesh S. Thakur, Function field arithmetic, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR 2091265, DOI 10.1142/9789812562388