Galois Symbols for a Jacobian and Multiplicative Groups

Abstract

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 $r\geq 1$ and a positive integer $n$ prime to the characteristic of $k$, we prove that the Galois symbol map \[ K(k;J,\mathbb{G}_{m},\ldots,\mathbb{G}_{m})/n \to H_{\mathrm{\acute et}}^{r+1}\bigl(k,J[n]\otimes \mu_n^{\otimes r}\bigr) \] 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.