We extend the Serre-Tate theory of canonical lifts of ordinary abelian varieties to arbitrary unpolarised families of ordinary abelian varieties parameterised by a $p$-adic formal scheme $S$. We show that the canonical lift is the unique lift to $W(S
)$ which admits a $delta$-structure in the sense of Joyal, Buium, and Bousfield. We prove analogous statements for families of ordinary $p$-groups and $p$-divisible groups.
We consider generalized $Lambda$-structures on algebras and schemes over the ring of integers $mathit{O}_K$ of a number field $K$. When $K=mathbb{Q}$, these agree with the $lambda$-ring structures of algebraic K-theory. We then study reduced finite f
lat $Lambda$-rings over $mathit{O}_K$ and show that the maximal ones are classified in a Galois theoretic manner by the ray class monoid of Deligne and Ribet. Second, we show that the periodic loci on any $Lambda$-scheme of finite type over $mathit{O}_K$ generate a canonical family of abelian extensions of $K$. This raises the possibility that $Lambda$-schemes could provide a framework for explicit class field theory, and we show that the classical explicit class field theories for the rational numbers and imaginary quadratic fields can be set naturally in this framework. This approach has the further merit of allowing for some precise questions in the spirit of Hilberts 12th Problem. In an interlude which might be of independent interest, we define rings of periodic big Witt vectors and relate them to the global class field theoretical mathematics of the rest of the paper.
We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for varia
nts of these functors which are in a certain sense their analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications.
A subtraction-free definition of the big Witt vector construction was recently given by the first author. This allows one to define the big Witt vectors of any semiring. Here we give an explicit combinatorial description of the big Witt vectors of th
e Boolean semiring. We do the same for two variants of the big Witt vector construction: the Schur Witt vectors and the $p$-typical Witt vectors. We use this to give an explicit description of the Schur Witt vectors of the natural numbers, which can be viewed as the classification of totally positive power series with integral coefficients, first obtained by Davydov. We also determine the cardinalities of some Witt vector algebras with entries in various concrete arithmetic semirings.
Let O be the ring of integers of a number field K. For an O-algebra R which is torsion free as an O-module we define what we mean by a Lambda_O-ring structure on R. We can determine whether a finite etale K-algebra E with Lambda_O-ring structure has
an integral model in terms of a Deligne-Ribet monoid of K. This a commutative monoid whose invertible elements form a ray class group.
This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, p-typical Witt vectors of p-adic schemes of finite type are already reasonably well understood. The main point here is to generalize this theory in two
ways. We allow not just p-typical Witt vectors but those taken with respect to any set of primes in any ring of integers in any global field, for example. This includes the big Witt vectors. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of Buiums formal arithmetic jet functor, which is dual to the Witt functor. We also give concrete geometric descriptions of Witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors.
We study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications we give a new interpretation of arithmetic Laplacians and we discuss the de Rham cohomology of some specific arithmetic jet spaces.
We show that any Lambda-ring, in the sense of Riemann-Roch theory, which is finite etale over the rational numbers and has an integral model as a Lambda-ring is contained in a product of cyclotomic fields. In fact, we show that the category of them i
s described in a Galois-theoretic way in terms of the monoid of pro-finite integers under multiplication and the cyclotomic character. We also study the maximality of these integral models and give a more precise, integral version of the result above. These results reveal an interesting relation between Lambda-rings and class field theory.
