scholarly journals ZETA ELEMENTS IN DEPTH 3 AND THE FUNDAMENTAL LIE ALGEBRA OF THE INFINITESIMAL TATE CURVE

2017 ◽  
Vol 5 ◽  
Author(s):  
FRANCIS BROWN
Keyword(s):  
Lie Algebra ◽  
Modular Forms ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Explicit Formulae ◽  
Tate Curve ◽  
Tate Motives

This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for $\text{SL}_{2}(\mathbb{Z})$. We write down explicit formulae for zeta elements $\unicode[STIX]{x1D70E}_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst–Kreimer conjecture, and solve the double shuffle equations for multiple zeta values in depths two and three.

scholarly journals Depth-graded motivic multiple zeta values

2021 ◽  
Vol 157 (3) ◽  
pp. 529-572
Author(s):  
Francis Brown
Keyword(s):  
Galois Group ◽  
Lie Algebra ◽  
Modular Forms ◽  
Cusp Forms ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Depth Filtration ◽  
Zeta Values ◽  
Lower Depth ◽  
Tate Motives

We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.

scholarly journals Elliptic Multizetas and the Elliptic Double Shuffle Relations

2020 ◽  
Author(s):  
Pierre Lochak ◽  
Nils Matthes ◽  
Leila Schneps
Keyword(s):  
Lie Algebra ◽  
Direct Product ◽  
Half Plane ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Generating Series ◽  
Zeta Values ◽  
Upper Half Plane ◽  
Precise Relationship ◽  
Ring Of Functions

Abstract We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in $\mathcal{O}({{\mathfrak{H}}})$, the ring of functions on the Poincaré upper half-plane ${{\mathfrak{H}}}$. The elliptic multizetas generate a ${{\mathbb{Q}}}$-algebra ${{\mathcal{E}}}$, which is an elliptic analogue of the algebra of multiple zeta values. Working modulo $2\pi i$, we show that the algebra ${{\mathcal{E}}}$ decomposes into a geometric and an arithmetic part and study the precise relationship between the elliptic generating series and the elliptic associator defined by Enriquez. We show that the elliptic multizetas satisfy a double shuffle type family of algebraic relations similar to the double shuffle relations satisfied by multiple zeta values. We prove that these elliptic double shuffle relations give all algebraic relations among elliptic multizetas if (1) the classical double shuffle relations give all algebraic relations among multiple zeta values and (2) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure analogous to that established by Enriquez for the elliptic Grothendieck–Teichmüller Lie algebra.

The quantum ${\mathfrak g}_2$ invariant and relations of multiple zeta values

2001 ◽  
Vol 10 (07) ◽  
pp. 983-997
Author(s):  
Kentaro Ihara ◽  
Takashi Takamuki
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Multiple Zeta Values ◽  
Oriented Link ◽  
Simple Lie Algebra ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Dimensional Irreducible Representation

In this paper we obtain a family of relations among the multiple zeta values by calculating the quantum [Formula: see text]-invariant of a framed oriented link, where Γ1,0 is the 7-dimensional irreducible representation of the exceptional simple Lie algebra [Formula: see text] over [Formula: see text].

scholarly journals A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS

2020 ◽  
Vol 8 ◽  
Author(s):  
FRANCIS BROWN
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Multiple Zeta Values ◽  
New Family ◽  
Algebra Of Functions ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Upper Half Plane ◽  
Real Analytic ◽  
Holomorphic Modular Forms

We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.

scholarly journals Mixed Tate motives and multiple zeta values

2002 ◽  
Vol 149 (2) ◽  
pp. 339-369 ◽  
Author(s):  
Tomohide Terasoma
Keyword(s):  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Tate Motives

Motivic multiple zeta values relative to μ2

2020 ◽  
Vol 14 (10) ◽  
pp. 2685-2712
Author(s):  
Zhongyu Jin ◽  
Jiangtao Li
Keyword(s):  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values

scholarly journals ON MULTIPLE ZETA VALUES OF EXTREMAL HEIGHT

2015 ◽  
Vol 93 (2) ◽  
pp. 186-193 ◽  
Author(s):  
MASANOBU KANEKO ◽  
MIKA SAKATA
Keyword(s):  
Explicit Formula ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Maximal Height ◽  
Zeta Values ◽  
Sum Formula

We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.

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