Discriminants and Arakelov Euler Characteristics

The study of discriminants has been a central part of algebraic number theory (c.f. [30]), and has recently led to striking results in arithmetic geometry (e.g. [6], [33], [2]). In this article we summarize two different generalizations ([16], [11]) of discriminants to arithmetic schemes having a tame action by a finite group. We also discuss the results proved in [11, 15, 16] concerning the connection of these discriminants to 0 and -factors in the functional equations of L-functions. These results relate invariants defined by coherent cohomology (discriminants) to ones defined by means of étale cohomology (conductors and -factors.) One consequence is a proof of a conjecture of Bloch concerning the conductor of an arithmetic scheme [15] when this scheme satisfies certain hypotheses (c.f. Theorem 2.6.4). In the last section of this paper we present an example involving integral models of elliptic curves. The discriminant dK of a number field K can be defined in (at least) three different ways. The definition closest to Arakelov theory arises from the fact that √ |dK | is the covolume of the ring of integers OK of K in R ⊗Q K with respect to a natural Haar measure on R ⊗Q K (c.f. [8, §4]). This Haar measure is the one which arises from the usual metrics at infinity one associates to Spec(K) as an arithmetic variety. A second definition of dK is that it is the discriminant of the bilinear form defined by the trace function TrK/Q : K → Q. The natural context in which to view this definition is in terms of the coherent duality theorem, since TrK/Q is the trace map which the duality theorem associates to the finite morphism Spec(K) → Spec(Q). A third definition of the ideal dKZ is that this is the norm of the annihilator ∗Supported in part by NSF grant DMS97-01411. †Supported in part by NSF grant DMS99-70378 and by a Sloan Research Fellowship. ‡EPSRC Senior Research Fellow.