# Chloe Xiaohua Ai, From Feynman Amplitudes to multiple L-values (Thursday, July 21)

:::{.question}
Recall that the multiple zeta function is defined as 
\[
\zeta(\vector s) = \sum_{n_1> n_2 > \cdots > 0} n_1^{-s_1} n_2^{-s_2}\cdots n_k^{-s_k}
.\]
The classical zeta generalizes to Dedekind zeta functions $\zeta_K(S) = \sum_{0\neq n\subset \OO_K} N_{K/\QQ}(n)^{-s}$, is there an analog for the multiple zeta function?
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:::{.remark}
The Hecke formula: takes automorphic periods to $L\dash$functions by integrating a well-chosen Eisenstein series with respect to a normalized Haar measure.
Can this idea be extended to multiple variables?
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:::{.remark}
Goncharov's theory on Hodge correlators: complex numbers given by integrals assigned to complex curves or varieties.
Can construct integrals all arising from a single object: Green functions/currents.
Many $L\dash$functions can be interpreted as Hodge correlators: Rankin-Selberg integral, polylogarithms, multiple Eisenstein-Kronecker series.
Hodge correlators are periods of motivic correlators.
The Green functions in Goncharov's construction work over $\CC$, what about number fields?
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:::{.remark}
Plectic conjecture: in the presence of real multiplication by a totally real number field, motives have canonical functorial additional structure.
See Hilbert modular varieties, Hilbert modular forms.
Goal: define higher plectic Green functions, and define multivariate zeta functions by integrating them.
These new multivariate functions will depend on some combinatorial data: a graph $\Gamma$.
If $\Gamma$ is a special tree, then the generalized multiple zeta values $Z_{I, v}(\Gamma, \bd \Gamma)$ can be expressed as finite $\ZZ\dash$linear combinations of classical multiple zeta values.
In the totally real case, $Z$ be expanded as sum in values of $\mcl_m$, a generalized $m\dash$logarithm which is a nontrivial iterated integral.
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