Special functions and twisted $L$-series

Anglès, Bruno; Ngo Dac, Tuan; Tavares Ribeiro, Floric

Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 931-961.

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.

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.

Classification: 11M38, 11F52, 11G09
Mots-clés : Goss $L$-functions, several variable $L$-series, Drinfeld modules

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

[2] Anderson, Greg W. 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 | DOI | Zbl

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

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

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

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

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

[8] Anglès, Bruno; Pellarin, Federico; Tavares Ribeiro, Floric Anderson-Stark units for ${𝔽}_q\left[\theta \right]$ (2016) (https://arxiv.org/abs/1501.06804, to appear in Trans. Am. Math. Soc.) | DOI

[9] Anglès, Bruno; Pellarin, Federico; Tavares Ribeiro, Floric 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) | DOI | Zbl

[10] Anglès, Bruno; Taelman, Lenny 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) | DOI | Zbl

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

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

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

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

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

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

[17] Green, Nathan; Papanikolas, Matthew A. 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] Papanikolas, Matthew A. Log-Algebraicity on Tensor Powers of the Carlitz Module and Special Values of Goss $L$-Functions (work in progress)

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

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

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

[22] Thakur, Dinesh S. Gauss sums for ${𝔽}_q\left[T\right]$, Invent. Math., Volume 94 (1988) no. 1, pp. 105-112 | DOI | Zbl

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