Jonathan Lubin

Formal Complex Multiplication in Local Fields

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The norm map for ordinary abelian varieties

In this paper we present an appreciable simplification of the proof of an important result of Barry Mazur. Let K be a finite extension of Q, with residue class field k. Let A be a d-dimensional Abelian variety over K with good ordinary reduction and let u be a twist matrix of A (the precise definition of u will be given in Section 1). Suppose L is a totally ramified &-extension of K and let M&A(L) be the intersection of the groups NKnlKA(Kn) h w ere KC K, CL and [K, : K] = p”. Then we have an exact sequence

Raynaud’s group-scheme and reduction of coverings

Let Y K → X K be a Galois covering of smooth curves over a field of characteristic 0, with Galois group G. We assume K is the fraction field of a discrete valuation ring R with residue characteristic p. Assuming p 2 ∤ G and the p-Sylow subgroup of G is normal, we consider the possible reductions of the covering modulo p. In our main theorem we show the existence, after base change, of a twisted curve \(\mathcal{X} \rightarrow Spec (R)\), a group scheme \(\mathcal{G}\rightarrow \mathcal{X}\) and a covering \(Y \rightarrow \mathcal{X}\) extending Y K → X K , with Y a stable curve, such that Y is a \(\mathcal{G}\)-torsor.In case p 2 | G counterexamples to the analogous statement are given; in the appendix a strong counterexample is given, where a non-free effective action of α p 2 on a smooth 1-dimensional formal group is shown to lift to characteristic 0.

Formal Flows on the Non-Archimedean Open Unit Disk

This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series Φ(t,x) satisfying the conditions Φ(0,x)=x and Φ(F(s,t),x)=Φ(s,Φ(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows Φ with any specified divisor of fixed points, where a point ξ of the open unit disk is a fixed point of order ≤n if (x−ξ)n|(Φ(t,x)−x). Furthermore, if γ is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow Φ, an element α of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of γ(x) is equal to Φ(α,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.

Entireness of the endomorphism rings of one-dimensional formal groups

If, for a one-dimensional formal group of height h which is defined over the integers in a local field of characteristic zero, all the coefficients in degree less than phlie in an unramified extension of the p-adic numbers, then the endomorphism ring of the formal group is inte-