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.

我们研究了多个zeta值，超过$ \ mathbb {Z} $的混合Tate动机的Galois动机组以及Grothendieck-Teichmüller组的深度过滤及其与模块形式的关系。使用$ \ mathrm {SL} _2（\ mathbb {Z}）$的尖峰形式的周期多项式，我们构造了线性双随机混排方程解的显式李代数，从而给出了多个zeta值之间所有恒等式的猜想描述对$ \ zeta（2）$取模，对较低的深度取模。我们对此李代数的同源性提出一个单一的猜想，这暗示了由于Broadhurst和Kreimer，Racinet，Zagier和Drinfeld对多个zeta值的结构以及Grothendieck-Teichmüller李代数的猜想。