Galois Symbols for a Jacobian and Multiplicative Groups

Let $C$ be a smooth projective geometrically connected curve over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian variety of $C$. For an integer $r\ge 1$ and a positive integer $n$ prime to the characteristic of $k$, we prove that the Galois symbol map $$K(k;J,{\mathbb{G}}_m,\dots ,{\mathbb{G}}_m)\mathrm{/}n\to H_{\acute{\mathrm{e}}\mathrm{t}}^{r+1}(k,J[n]\otimes {\mu }_n^{\otimes r})$$ is injective, where the multiplicative group ${\mathbb{G}}_m$ occurs $r$ times. The proof uses Akhtar’s description of higher Chow groups of zero-cycles and the Beilinson–Lichtenbaum theorem. The case $r=1$ recovers a theorem of Spiess.