Jan Stienstra

FORMAL GROUPS AND CONGRUENCES FOR L-FUNCTIONS

Introduction. In this note we show congruences, similar to those of Atkin and Swinnerton-Dyer [2, 6], for a large class of schemes, including branched double coverings of pN of arbitrary dimension and genus, defined over any ring which is flat and of finite type over Z. The results of sections 1-4 together yield the following theorem. THEOREM 0.1. Let K be a ring which is flat and offinite type over Z. Let R E K[T0, . . *, TN] be a homogeneous polynomial of degree 2d. Assume 2d > 2N > 0. Let 9C be the double covering of PZ given by the equation U2 = R (where U is a new variable of weight d). Let (P be a maximal ideal of K with residue field K/1@ of characteristic p and of order q = p f. Let e be an integer such that I < e ? p - 1 and p e (We.

Motives from Diffraction

We look at geometrical and arithmetical patterns created from a finite subset of Zn by diffracting waves and bipartite graphs. We hope that this can make a link between Motives and the Melting Crystals/Dimer models in String Theory. - http://www.arxiv.org/abs/math.NT/0511485

The Generalized De Rham-Witt Complex and Congruence Differential Equations

The De Rham-Witt complex is a powerful instrument for studying the crystalline cohomology of a smooth projective variety over a perfect field of positive characteristic. In [9] the De Rham-Witt complex is constructed for schemes on which some prime number p is zero. Here in section 2 we construct on every scheme X on which 2 is invertible the generalized De Rham-Witt complex W Ω X this is a Zariski sheaf of anti-commutative differential graded algebras with the additional structures and properties described in (2.1)–(2.6). Section 3 gives the (obvious) definition of the relative generalized De Rham-Witt complex W Ω X/S for f: X → S a morphism of schemes over Z[1/2].