The geometry of Frobenioids II: Poly-Frobenioids

Abstract

We develop the theory of Frobenioids associated to non-archimedean (mixed-characteristic) and archimedean local fields. In particular, we show that the resulting Frobenioids satisfy the properties necessary to apply the main results of the general theory of Frobenioids. Moreover, we show that the reciprocity map in the non-archimedean case, as well as a certain archimedean analogue of this reciprocity map, admit natural Frobenioid-theoretic translations, which are, moreover, purely category-theoretic, to a substantial extent (i.e., except for the extent to which this category-theoreticity is obstructed by certain 'Frobenius endomorphisms' of the relevant Frobenioids). Finally, we show that certain Frobenioids which naturally encode the global arithmetic of a number field may be 'grafted' (i.e., glued) onto the Frobenioids associated to non-archimedean and archimedean primes of the number field to obtain 'poly-Frobenioids'. These poly-Frobenioids encode, in a purely category-theoretic fashion, most of the important aspects of the classical framework of the arithmetic geometry of number fields. © 2008 Faculty of Mathematics, Kyushu University.