Let be a number field, with ring of integers . Let be an Arakelov divisor on ; i.e., the formal sum of a divisor (in the usual sense) on the affine scheme , and multiples , with , for all infinite places of . This paper discusses a new definition of in the context of Arakelov theory, with the goal of proving results analogous to those that are true in the function field case.
Let be the fractional ideal of corresponding to (in the sense that an element lies in if and only if the divisor is effective, ignoring the infinite places). Instead of the naïve definition , this paper defines the effectivity of an Arakelov divisor to be a real number in the interval given by . (Functions other than may be used here; this choice was based on a letter of K. Iwasawa [in: N. Kurokawa et al. (ed.), Zeta functions in geometry. Tokyo, Kinokuniya, Adv. Stud. Pure Math. 21, 445-450 (1992; Zbl 0835.11002)].) The authors then define and (where the summand is presumably 1 when ). The latter is called the size of and corresponds to the dimension of in the case of a function field over a finite field. It depends only on the linear equivalence class of .
Also define a canonical divisor on to be the Arakelov divisor whose finite part is the different of and whose infinite components are all zero. Then a Riemann-Roch theorem is proved, where is the discriminant of . It is noted that this is a special case of a Riemann-Roch theorem due to J. Tate [Thesis, printed in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic Number Theory. Academic Press (1967; Zbl 0153.07403)].
Additional results are given, again in the spirit of furthering the analogy with the function field case. These results include expressing the Riemann zeta function as an integral of the effectivity function, an analogue of the inequality , and an analogue of the genus of .
The authors express a hope that this paper will stimulate others to continue investigating this definition of .