Motivic correlators and locally symmetric spaces

According to Langlands, pure motives are related to a certain class of automorphic representations.

Can one see mixed motives in the automorphic set-up? For examples, can one see periods of mixed motives in entirely automorphic terms? The goal of this and the next lecture is to supply some examples.

We define motivic correlators describing the structure of the motivic fundamental group $\pi_1^{\mathcal M}(X)$ of a curve. Their relevance to the questions raised above is explained by the following examples.

1. Motivic correlators have an explicit Hodge realization given by the Hodge correlator integrals, providing a new description of the real mixed Hodge structure of the pro-nilpotent completion of $\pi_1(X)$. When $X$ is a modular curve, the simplest of them coincide with the Rankin-Selberg integrals, and the rest provide an "automorphic" description of a class of periods of mixed motives related to (products of) modular forms.

2. We use motivic correlators to relate the structure of $\pi_1^{\mathcal M}(\mathbb G_m − \mu N )$ to the geometry of the locally symmetric spaces for the congruence subgroup $\Gamma_1 (m; N ) \subset \mathrm{GL}_m(\mathbb Z)$. Then we use the geometry of the latter, for $m \leq 4$, to understand the structure of the former.

3. This mysterious relation admits an "explanation" for $m = 2$: we define a canonical map \[ \mu : \text{modular complex} \to \text{the weight two motivic complex of the modular curve.} \]

Here the complex on the left calculates the singular homology of the modular curve via modular symbols. The map $\mu$ generalizes the Belinson-Kato Euler system in $K_2$ of the modular curves.

Composing the map μ with the specialization to a cusp, we recover the correspondence above at $m = 2$.

4. Yet specializing to CM points on modular curves, we get a new instance of the above correspondence, now between $\pi_1^{\mathcal M}(E − E[\mathcal N])$ and geometry of arithmetic hyperbolic threefolds. Here $E$ is a CM elliptic curve, and $\mathcal N \subset \mathrm{Aut}(E)$ is an ideal.