Explicit form of Hilbert symbol for polynomial formal groups over multidimensional local field.~I

S. V. Vostokov ; V. V. Volkov ; M. V. Bondarko

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 53-60

Résumé

Let $K$ be a multidimensional local field with characteristic different from characteristic of its residue field, $c$ be a unit of $K$ and $F_c(X,Y)=X+Y+cXY$ be a polynomial formal group, which defines formal module $F_c(\mathfrak M)$ over maximal ideal of ring of integers in $K$. Assume that $K$ contains group of the roots of isogeny $[p^m]_c(X)$, which we denote by $\mu_{F_c,m}$. Let $\mathcal H$ be the multiplicative group of Cartier curves and $\mathcal H_c$ be a formal analogue of the module $F_c(\mathfrak M)$. In the current work we construct formal symbol $\{\cdot,\cdot\}_c\colon K_n(\mathcal H)\times\mathcal H_c\to\mu_{F_c,m}$ and check its basic properties. This is the first step in construction of the explicit formula for the Hilbert symbol.