Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus
Journal für die Reine und Angewandte Mathematik
We develop a localized intersection theory for arithmetic schemes on the model of Fulton's intersection theory. We prove a Lefschetz ¢xed point formula for arithmetic surfaces, and give an application to a conjecture of Serre on the existence of Artin's representations for regular local rings of dimension 2 and unequal characteristic. Let S peR be the spectrum of a characteristic 0 complete discrete valuation ring R with algebraically closed residue ¢eld k of characteristic p b 0 and fraction
... b 0 and fraction ¢eld K. s, Z and Z denote, respectively, its closed point, its generic point and a geometric generic point, corresponding to an algebraic closure K of K. Let G be the Galois group of K over K. We ¢x a prime number l T p and let L be one of the rings Zal n Z, Z l or Q l . To any ¢nitely generated L-module M with a continuous action of G, we associate its Swan conductor swM which is a ¢nitely generated L-module (see Section 2 for the de¢nition). Any G-equivariant endomorphism of M induces an endomorphism of swM. We work in the category of separated schemes of ¢nite type over S. The subscripts s and Z, associated with an object in this category, denote, respectively, its closed and its generic ¢bers. Associated with a morphism, they denote the induced morphisms over the closed and the generic ¢bers. Let X be a separated scheme of ¢nite type over S. The group of cycles ZX È n Z n X and the Chow group AX È n A n X are graded by the absolute dimension over S. The latter is the sum of the relative dimension over S and the dimension of S. In this paper, dimension stands for the absolute dimension over S. Notice that if X is proper over S, the absolute dimension coincides with the Krull dimension. The group of n-bivariant classes associated with a map X 3 Y is denoted A n X 3 Y ( chapter 17).