On shuffle of double Eisenstein series in positive characteristic

Chen, Huei-Jeng¹

¹ Taida Institute for Mathematical Sciences No.1, Sec.4, Roosevelt Road Taipei, Taiwan

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 815-825.

Abstract (VO)

The study of the present paper is inspired by Gangl, Kaneko and Zagier’s result of the connection with double zeta values and modular forms. We introduce double Eisenstein series $E_{r,s}$ in positive characteristic with double zeta values ${\zeta }_A(r,s)$ as their constant term and compute the t-expansions of the double Eisenstein series. Moreover, we derive the shuffle relations of double Eisenstein series which match the shuffle relations of double zeta values in [4].

Résumé

L’étude du présent article s’inspire du résultat de Gangl, Kaneko et Zagier sur connexion entre les valeurs zêta doubles et les formes modulaires. Nous introduisons la série d’Eisenstein double $E_{r,s}$ en caractéristique positive avec les valeurs zêta doubles ${\zeta }_A(r,s)$ comme terme constant et calculons les t-expansions de la série d’Eisenstein double. De plus, on en déduit les relations de shuffle de deux séries d’Eisenstein qui correspondent aux relations de shuffle des valeurs zêta doubles dans [4].

Détail

Reçu le : 2016-08-23
Révisé le : 2016-10-21
Accepté le : 2016-12-02
Publié le : 2017-12-13

DOI : 10.5802/jtnb.1002

Classification : 11J91, 11M36
Keywords: Double zeta values, Eisenstein series, $t$-expansions, shuffle relations.

Licence : CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

[1] Brown, Francis Mixed Tate motives over $\mathbb{Z}$, Ann. Math., Volume 175 (2012) no. 2, pp. 949-976 | DOI | Zbl

[2] Chang, Chieh-Yu Linear independence of monomials of multizeta values in positive characteristic, Compos. Math., Volume 150 (2014) no. 11, pp. 1789-1808 | DOI | Zbl

[3] Chang, Chieh-Yu Linear relations among double zeta values in positive characteristic, Camb. J. Math., Volume 4 (2016) no. 3, pp. 289-331 | DOI | Zbl

[4] Chen, Huei-Jeng On shuffle of double zeta values over ${𝔽}_q\left[t\right]$, J. Number Theory, Volume 148 (2015), pp. 153-163 | DOI | Zbl

[5] Gangl, Herbert; Kaneko, Masanobu; Zagier, Don Double zeta values and modular forms, Automorphic forms and zeta functions (Tokyo, 2004), World Scientific, 2006, pp. 71-106 | Zbl

[6] Gekeler, Ernst-Ulrich On the coefficients of Drinfeld modular forms, Invent. Math., Volume 93 (1988) no. 3, pp. 667-700 | DOI | Zbl

[7] Goss, David The algebraist’s upper half-plane, Bull. Am. Math. Soc. (1980) no. 2, pp. 391-415 | DOI | Zbl

[8] Goss, David $\pi$-adic Eisenstein series for function fields, Compos. Math., Volume 41 (1980), pp. 3-38 | Zbl

[9] Thakur, Dinesh S. Function field Arithmetic, World Scientific, 2004, xv+388 pages | Zbl

[10] Thakur, Dinesh S. Power sums with applications to multizeta and zeta zero distribution for ${𝔽}_q\left[t\right]$, Finite Fields Appl., Volume 15 (2009) no. 4, pp. 534-552 | DOI | Zbl

[11] Thakur, Dinesh S. Shuffle relations for function field multizeta values, Int. Math. Res. Not., Volume 2010 (2010) no. 11, pp. 1973-1980 | Zbl

[12] Zagier, Don Values of zeta functions and their applications, First European Congress of Mathematics Vol. II (Paris, 1992) (Progress in Mathematics), Volume 120, Birkhäuser, 1994, pp. 497-512 | Zbl

[13] Zhao, Jianqiang Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and Its Applications, 12, World Scientific, 2016, xxi+595 pages | Zbl