staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Huei-Jeng Chen
On shuffle of double Eisenstein series in positive characteristic
Journal de théorie des nombres de Bordeaux, 29 no. 3 (2017), p. 815-825
Article PDF
Class. Math.: 11J91, 11M36
Keywords: Double zeta values, Eisenstein series, $t$-expansions, shuffle relations.

Résumé - Abstract

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].

Bibliography

[1] Francis Brown, Mixed Tate motives over $\mathbb{Z}$, Ann. Math. 175 (2012), p. 949-976
[2] Chieh-Yu Chang, Linear independence of monomials of multizeta values in positive characteristic, Compos. Math. 150 (2014), p. 1789-1808
[3] Chieh-Yu Chang, Linear relations among double zeta values in positive characteristic, Camb. J. Math. 4 (2016), p. 289-331
[4] Huei-Jeng Chen, On shuffle of double zeta values over $\mathbb{F}_q[t]$, J. Number Theory 148 (2015), p. 153-163
[5] Herbert Gangl, Masanobu Kaneko & Don Zagier, Double zeta values and modular forms, Automorphic forms and zeta functions (Tokyo, 2004), World Scientific, 2006, p. 71–106
[6] Ernst-Ulrich Gekeler, On the coefficients of Drinfeld modular forms, Invent. Math. 93 (1988), p. 667-700
[7] David Goss, The algebraist’s upper half-plane, Bull. Am. Math. Soc. (1980), p. 391-415
[8] David Goss, $\pi $-adic Eisenstein series for function fields, Compos. Math. 41 (1980), p. 3-38
[9] Dinesh S. Thakur, Function field Arithmetic, World Scientific, 2004
[10] Dinesh S. Thakur, Power sums with applications to multizeta and zeta zero distribution for $\mathbb{F}_q[t]$, Finite Fields Appl. 15 (2009), p. 534-552
[11] Dinesh S. Thakur, Shuffle relations for function field multizeta values, Int. Math. Res. Not. 2010 (2010), p. 1973-1980
[12] Don Zagier, Values of zeta functions and their applications, First European Congress of Mathematics Vol. II (Paris, 1992), Progress in Mathematics 120, Birkhäuser, 1994, p. 497–512
[13] Jianqiang Zhao, Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and Its Applications 12, World Scientific, 2016