1 Introduction. This paper is concerned with developing the theory of Hilbert modular forms along the lines of the theory of elliptic modular forms. Our main interests in this paper are: (i) to determine the ideal of congruences between Hilbert modular forms in char- acteristic p and to find conditions on the existence of congruences over artinian local rings. This allows us to derive explicit congruences between special values of zeta functions of totally real fields, to establish the existence of filtration for Hilbert modular forms, to establish the existence of p-adic weight for p-adic modular forms (defined as p-adic uniform limit of classical modular forms) and more (ii) to construct operators U, V, ©^ (one for each suitable weight ip) on modular forms in characteristic p and to study the variation of the filtration under these operators. This allows us to prove that every ordinary form has filtration bounded from above (iii) to show that there are well defined notions of a Serre p-adic modular form and of a Katz p-adic modular form and to show that the two notions agree for a suitable class of weights containing all the classical ones. Our argument involves showing that every g-expansion of a mod p modular form lifts to a ^-expansion of a characteristic zero modular form. We extend the theta operators 8^ to the p-adic setting - their Galois theoretic interpretation is that of twisting a representation by a Hecke character. Our approach to modular forms is emphatically geometric. Our goal is to develop systematically the geometric and arithmetic aspects of Hilbert modular varieties and to apply them to modular forms. As to be expected in such a project, we use extensively the ideas of N. Katz [Kal], [Ka2], [Ka3], [Ka4] and J.-P. Serre [Se], of the founders of the theory in the case of elliptic modular forms, and we have benefited much from B. Gross' paper [Gr]. In regard to previous work on the subject, we mention that some of the constructions and methods in this paper were introduced by the second named author, in the unramified case, in [Gol], [Go2], and the congruences we list for zeta functions may be derived from the work of P. Deligne and K. Ribet [DeRi]. For this reason we restrict our discussion to zeta functions, though the same reasoning applies to a wide class of L-functions. We now describe in more detail the main results of this paper. Let L be a totally real field of degree g over Q. Let K be a normal closure of L. Let T V 4 be an integer prime to p. Let 9JT(5, //jv) be the fine moduli scheme parameterizing polarized abelian schemes over S with RM by OL and JIN~ level structure see 3.2. A Hilbert modular form defined over an Ox-scheme S has a weight ip £ Xs, where X5 is the group of characters of the algebraic group Ss ResoL/zGm?oL x Spec(Z) S. We shall mostly be concerned with weights obtained from the characters X of SoK ~ ^esoL/z^m,oL x Spec(z) Spec(Ox) we shall use the notation X^ ("[/" for universal) to the denote the group of characters of 5s induced from X by base change. See 4.1. The group X is a free abelian group of rank g and has a positive cone X^ gener- ated by the characters coming from the embeddings G\,..., ag: L K. Indeed, the Received by the editor August 1, 2001. 1
Previous Page Next Page