Potentially good reduction of Barsotti–Tate groups

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck’s conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.

Torsors under some group schemes of order [formula omitted

A class of soluble groups

This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R . The focus is on lattices, namely, finitely generated G -modules that are projective as R -modules, and on the full subcategory of all G -modules projective over R generated by the lattices. The stable category of such G -modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G . Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

Michel Raynaud gave a criterion for a three-point G-cover f : Y \rightarrow X = P^1, defined over a p-adic field K, to have good reduction. In particular, if the order of a p-Sylow subgroup of G is p, and the number of conjugacy classes of elements of order p is greater than the absolute ramification index e of K, then f has potentially good reduction. We give a different proof of this criterion, which extends to the case where G has an arbitrarily large cyclic p-Sylow subgroup, answering a question of Raynaud. We then use the criterion to give a family of examples of three-point covers with good reduction to characteristic p and arbitrarily large p-Sylow subgroups.

Let E/ℚ be an elliptic curve with good ordinary reduction at a prime p > 2. It has a welldefined Iwasawa μ-invariant μ(E)p which encodes part of the information about the growth of the Selmer group ) as Kn ranges over the subfields of the cyclotomic Zp-extension K∞/ℚ. Ralph Greenberg has conjectured that any such E is isogenous to a curve E′ with μ(E′)p = 0. In this paper we prove Greenberg's conjecture for infinitely many curves E with a rational p-torsion point, p = 3 or 5, no two of our examples having isomorphic p-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

Let \({S}\) be a Dedekind scheme with perfect residue fields at closed points. Let \({f: X\rightarrow S}\) be a minimal regular arithmetic surface of fibre genus at least 2 and let \({f': X'\rightarrow S}\) be the canonical model of \({f}\). It is well known that \({\omega_{X'/S}}\) is relatively ample. In this paper we prove that \({\omega_{X'/S}^{\otimes n}}\) is relatively very ample for all \({n\geq 3}\).

Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p-adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.

Let k denote a field with nontrivial discrete valuation. We assume that k is complete with perfect residue field f. Let G denote the group of k-rational points of a reductive, linear algebraic group G defined over k. A subgroup in G is said to be unramified if it is a connected reductive subgroup of G whose reduced Bruhat-Tits building contains a hyperspecial vertex. Let C denote the set of G-conjugacy classes of pairs (H,x) where H is a maximal rank unramified subgroup of G and x is a hyperspecial point in the reduced Bruhat-Tits building of H . Let I denote the set of pairs (F,H) where F is a facet in the Bruhat-Tits building of G and H is a f-cuspidal maximal rank connected reductive subgroup in GF (the connected reductive f-group associated to F ). There is a natural equivalence relation, to be denoted ∼, on I. We show that there is a bijective correspondence between the set I/ ∼ and C. From this, we derive a classification of the conjugacy classes of maximal unramified tori in G and, when f is finite and G is unramified, we determine which of these conjugacy classes of tori are stably conjugate.

Let $F$ be a discretely valued complete field with valuation ring $\mathcal{O}_F$ and perfect residue field $k$ of cohomological dimension $\leq 1$. In this paper, we generalize the Bruhat decomposition in Bruhat and Tits from the case of simply connected $F$-groups to the case of arbitrary connected reductive $F$-groups. If $k$ is algebraically closed, Haines and Rapoport define the Iwahori-Weyl group, and use it to solve this problem. Here we define the Iwahori-Weyl group in general, and relate our definition of the Iwahori-Weyl group to that of Haines and Rapoport. Furthermore, we study the length function on the Iwahori-Weyl group, and use it to determine the number of points in a Bruhat cell, when $k$ is a finite field.

The authors establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. They consider fields K that are complete discrete valuation fields of characteristic zero with perfect residue fields k of characteristic p > 2. They evaluate the K-theory with Z/p^v-coefficients of K, and verify the Lichtenbaum-Quillen conjecture for K.

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

We prove that a valued field of positive characteristic [Formula: see text] that has only finitely many distinct Artin–Schreier extensions (which is a property of infinite NTP2 fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has [Formula: see text]-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin–Schreier defect extensions.

Let $R$ be a commutative Noetherian Cohen-Macaulay local ring that has positive dimension and prime characteristic. Li proved that the tensor product of a finitely generated non-free $R$-module $M$ with the Frobenius endomorphism ${}^{\varphi^n}\!R$ is not maximal Cohen-Macaulay provided that $M$ has rank and $n\gg 0$. We replace the rank hypothesis with the weaker assumption that $M$ is locally free on the minimal prime ideals of $R$. As a consequence, we obtain, if $R$ is a one-dimensional non-regular complete reduced local ring that has a perfect residue field and prime characteristic, then ${}^{\varphi^n}\!R \otimes_{R}{}^{\varphi^n}\!R$ has torsion for all $n\gg0$. This property of the Frobenius endomorphism came as a surprise to us since, over such rings $R$, there exist non-free modules $M$ such that $M\otimes_{R}M$ is torsion-free.