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arXiv · 1909.06705

On $\ell$-adic Galois polylogarithms and triple $\ell$-th power residue symbols

Densuke Shiraishi
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Abstract

The $\ell$-adic Galois polylogarithm is an arithmetic function on an absolute Galois group with values in $\ell$-adic numbers, which arises from Galois actions on $\ell$-adic \'etale paths on ${\mathbb P}^1 \backslash \{0,1,\infty\}$. In the present paper, we discuss a relationship between $\ell$-adic Galois polylogarithms and triple $\ell$-th power residue symbols in some special cases studied by a work of Hirano-Morishita. We show that a functional equation of $\ell$-adic Galois polylogarithms by Nakamura-Wojtkowiak implies a reciprocity law of triple $\ell$-th power residue symbols.

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Densuke Shiraishi. 2019-09-15. On $\ell$-adic Galois polylogarithms and triple $\ell$-th power residue symbols. https://doi.org/10.2206/kyushujm.75.95

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