On bilinear structures on divisor class groups
Let \(K_s\) be the separable closure of a field \(K\) with units \(K_s^*\) and let \(G_K\) be the absolute Galois group \(\Aut_K(K_s)\) of \(K\). Denote by \(\mathcal{O}\), resp. \(\bar{\mathcal{O}}\), the ring of holomorphic functions of an affine curve \(C_O\) defined over \(K\), resp. the ring of holomorphic functions of the curve \(\bar{C_O}\) obtained from \(C_O\) by constant field extension to \(K_s\). Denote by \(\text{Pic}(\mathcal{O})\) and \(\text{Pic}(\bar{\mathcal{O}})\) the ideal class group of \(\mathcal{O}\) and \(\bar{\mathcal{O}}\). For all \(n\) prime to \(\text{char}(K)\), the author first defines the Lichtenbaum-Tate pairing \[ T_{L,n}: \text{Pic}(\mathcal{O})/n\text{Pic}(\mathcal{O})\times H^1(G_K, \text{Pic}(\bar{\mathcal{O}}))[n]\rightarrow H^2(G_K,K_s^*)[n], \] induced from the Lichtenbaum pairing [\textit{S. Lichtenbaum}, Invent. Math. 7, 120--136 (1969; Zbl 0186.26402)]. When \(K=\mathbb{F}_q\) is a finite field with \(q\) elements, the author shows that the Lichtenbaum-Tate pairing induces a bilinear structure on \(\text{Pic}(\mathcal{O})\) of complexity \(\mathcal{O}(k\log(q))\) with value group \(\text{Br}(K)=H^2(G_K,K_s^*)\), where \(k\) is the smallest number such that \(n|q^k-1\). Thus the discrete logarithm in \(\text{Pic}(\mathcal{O})\) is reduced to the computation of invariants in \(\text{Br}(K)\). Lastly the author gives an index-calculus approach for the computation of invariants using the Hasse-Brauer-Noether sequence and the duality theorem of Tate-Poitou [\textit{B. Mazur}, Ann. Sci. Éc. Norm. Supér. (4) 6, 521--552 (1973; Zbl 0282.14004)].
scientific article; zbMATH DE number 5554656
scientific article