Special functions and twisted L-series
Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 931-961.

Nous donnons une généralisation de la fonction spéciale d’Anderson–Thakur et nous prouvons un théorème de rationalité pour les séries L à plusieurs variables associées aux fonctions chtoucas.

We present a generalization of the Anderson–Thakur special function, and we prove a rationality result for several variable twisted L-series associated to shtuka functions.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1007
Classification : 11M38,  11F52,  11G09
Mots clés : Goss L-functions, several variable L-series, Drinfeld modules
@article{JTNB_2017__29_3_931_0,
     author = {Bruno Angl\`es and Tuan Ngo Dac and Floric Tavares Ribeiro},
     title = {Special functions and twisted $L$-series},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {931--961},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {3},
     year = {2017},
     doi = {10.5802/jtnb.1007},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1007/}
}
Bruno Anglès; Tuan Ngo Dac; Floric Tavares Ribeiro. Special functions and twisted $L$-series. Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 931-961. doi : 10.5802/jtnb.1007. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1007/

[1] Greg W. Anderson Rank one elliptic A-modules and A-harmonic series, Duke Math. J., Volume 73 (1994) no. 3, pp. 491-542 | Article | Zbl 0807.11032

[2] Greg W. Anderson Log-Algebraicity of Twisted A-Harmonic Series and Special Values of L-series in Characteristic p, J. Number Theory, Volume 60 (1996) no. 1, pp. 165-209 | Article | Zbl 0868.11031

[3] Greg W. Anderson; Dinesh S. Thakur Tensor Powers of the Carlitz Module and Zeta Values, Ann. Math., Volume 132 (1990) no. 1, pp. 159-191 | Article | Zbl 0713.11082

[4] Bruno Anglès; Tuan Ngo Dac; Floric Tavares Ribeiro Exceptional Zeros of L-series and Bernoulli–Carlitz Numbers (2015) (https://arxiv.org/abs/1511.06209)

[5] Bruno Anglès; Tuan Ngo Dac; Floric Tavares Ribeiro Twisted Characteristic p Zeta Functions, J. Number Theory, Volume 168 (2016), pp. 180-214 | Article | Zbl 06599778

[6] Bruno Anglès; Tuan Ngo Dac; Floric Tavares Ribeiro Stark units in positive characteristic, Proc. Lond. Math. Soc., Volume 115 (2017), pp. 763-812 | Article | Zbl 06797296

[7] Bruno Anglès; Federico Pellarin Universal Gauss-Thakur sums and L-series, Invent. Math., Volume 200 (2015) no. 2, pp. 653-669 | Article | Zbl 1321.11053

[8] Bruno Anglès; Federico Pellarin; Floric Tavares Ribeiro Anderson-Stark units for 𝔽 q [θ] (2016) (https://arxiv.org/abs/1501.06804, to appear in Trans. Am. Math. Soc.) | Article

[9] Bruno Anglès; Federico Pellarin; Floric Tavares Ribeiro Arithmetic of positive characteristic L-series values in Tate algebras, Compos. Math., Volume 152 (2016) no. 1, pp. 1-61 (with an appendix by F. Demeslay) | Article | Zbl 1336.11042

[10] Bruno Anglès; Lenny Taelman Arithmetic of characteristic p special L-values, Proc. Lond. Math. Soc., Volume 110 (2015) no. 4, pp. 1000-1032 (with an appendix by V. Bosser) | Article | Zbl 1328.11065

[11] Bruno Anglès; Floric Tavares Ribeiro Arithmetic of function fields units, Math. Ann., Volume 367 (2017), pp. 501-579 | Article

[12] Florent Demeslay A class formula for L-series in positive characteristic (2014) (https://arxiv.org/abs/1412.3704)

[13] Jiangxue Fang Equivariant Special L-values of abelian t-modules (2015) (https://arxiv.org/abs/1503.07243)

[14] Jiangxue Fang Special L-values of abelian t-modules, J. Number Theory, Volume 147 (2015), pp. 300-325 | Article | Zbl 1360.11074

[15] Jiangxue Fang Equivariant trace formula mod p, C. R., Math., Acad. Sci. Paris, Volume 354 (2016) no. 4, pp. 335-338 | Article

[16] David Goss Basic Structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, Volume 35, Springer, 1996, xiii+422 pages | Zbl 0874.11004

[17] Nathan Green; Matthew A. Papanikolas Special L-values and shtuka functions for Drinfeld modules on elliptic curves (2016) (https://arxiv.org/abs/1607.04211, to appear in Research in the Mathematical Sciences)

[18] Matthew A. Papanikolas Log-Algebraicity on Tensor Powers of the Carlitz Module and Special Values of Goss L-Functions (work in progress)

[19] Federico Pellarin Values of certain L-series in positive characteristic, Ann. Math., Volume 176 (2012) no. 3, pp. 2055-2093 | Article | Zbl 1336.11064

[20] Lenny Taelman A Dirichlet unit theorem for Drinfeld modules, Math. Ann., Volume 348 (2010) no. 4, pp. 899-907 | Article | Zbl 1217.11062

[21] Lenny Taelman Special L-values of Drinfeld modules, Ann. Math., Volume 175 (2012) no. 1, pp. 369-391 | Article | Zbl 1323.11039

[22] Dinesh S. Thakur Gauss sums for 𝔽 q [T], Invent. Math., Volume 94 (1988) no. 1, pp. 105-112 | Article | Zbl 0629.12014

[23] Dinesh S. Thakur Shtukas and Jacobi sums, Invent. Math., Volume 111 (1993) no. 3, pp. 557-570 | Article | Zbl 0770.11032