Arithmetic of values of L-functions and generalized multiple zeta values over number fields

Abstract : The principal objective of this thesis is to generalize multiple zeta values to the case when the ground field Q is replaced by an arbitrary number field. The motivation behind the construction comes from the work of A. Goncharov on Hodge correlators and the plectic philosophy of J. Nekovar and A. Scholl. We start by constructing the higher plectic Green functions. Hecke once proved that the integral of the restriction of a suitable Eisenstein series over $\mathbb{Q}$ to the idele class group of a given number field multipled an idele class character of finite order is equal to the L-function of this charator. By replacing Eisenstein seris with our higher plectic Green functions, a similar integration gives new results, which give the generalization of classical multiple zeta values and multiple polyloarithms. According to the plectic principle, a non-trivial subgroup of the ring of integers of a given number field plays an essential role in this work.
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Xiaohua Ai. Arithmetic of values of L-functions and generalized multiple zeta values over number fields. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066394⟩. ⟨tel-02303034⟩

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