Kyushu Journal of Mathematics

We develop the theory of Frobenioids, which may be regarded as a category-theoretic abstraction of the theory of divisors and line bundles on models of finite separable extensions of a given function field or number field. This sort of abstraction is analogous to the role of Galois categories in Galo is theory or monoids in the geometry of log schemes. This abstract category-theoretic framework preserves many o f the important features of the classical theory of divisors and line bundles on models of finite separable extensions of a function field or number field such as the global degree of an arithmetic line bundle over a number field, but also exhibits interesting new phenomena, such as a ‘Frobenius endomorphism’ of the Frobenioid associated to a number field.

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We develop the theory of Frobenioids associated to non-archimedean (mixed-characteristic) and archimedean local fields. Inparticular, 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-archim edean 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.

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