Discrete Valuation Research Articles

Herr complex of (φ,τ)-modules

Let p be an odd prime number and K a mixed characteristic complete discrete valuation field with perfect residue field of characteristic p. We construct a three-term complex, defined in terms of the (φ,τ)-module of a p-adic representation and prove its homology is isomorphic to the Galois cohomology of the representation. We further show that our complex is quasi-isomorphic to the four-term complex constructed by Tavares Ribeiro, providing an alternative proof of our result. As an application, we describe Galois cohomology of the Tate module associated to a p-divisible group in terms of corresponding Breuil-Kisin modules.

On 𝑝-adic Simpson and Riemann–Hilbert correspondences in the imperfect residue field case

Let K K be a mixed characteristic complete discrete valuation field with residue field admitting a finite p p -basis, and let G K G_K be the Galois group. Inspired by Liu and Zhu’s construction of p p -adic Simpson and Riemann–Hilbert correspondences over rigid analytic varieties, we construct such correspondences for representations of G K G_K . As an application, we prove a Hodge–Tate (resp. de Rham) “rigidity” theorem for p p -adic representations of G K G_K , generalizing a result of Morita.

EXTENDING TORSORS UNDER QUASI-FINITE FLAT GROUP SCHEMES

Abstract Let R be a discrete valuation ring of field of fractions K and of residue field k of characteristic $p> 0$ . In an earlier work, we studied the question of extending torsors over K-curves into torsors over R-regular models of the curves in the case when the structural K-group scheme of the torsor admits a finite flat model over R. In this paper, we first give a simpler description of the problem in the case where the curve is semistable using recent work in Holmes, Molcho, Orecchia, and Poiret (2023, Journal für die Reine und Angewandte Mathematik [Crelle’s Journal] 230, 115–159) and Molcho and Wise (2022, Compositio Mathematica 158, 1477–1562). Second, if R is assumed to be Henselian and Japanese, we solve the problem of extending torsors by combining our previous work together with results in Antei and Emsalem (2018, Nagoya Mathematical Journal 230, 18–34) and Phung and Dos Santos (2023, Algebraic Geometry 230, 1–40), including the case where the structural group does not admit a finite flat R-model.

Singular loci of algebras over ramified discrete valuation rings

AbstractWhen is a field, the classical Jacobian criterion computes the singular locus of an equidimensional, finitely generated ‐algebra as the closed subset of an ideal generated by appropriate minors of the so‐called Jacobian matrix. Recently, Hochster‐Jeffries and Saito have extended this result for algebras over any unramified discrete valuation ring of mixed characteristic via the use of ‐derivations. Motivated by these results, in this paper, we state and prove an analogous Jacobian criterion for algebras over ramified discrete valuation rings of mixed characteristic.

Blocks whose defect groups are Suzuki 2-groups

We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.

Linking invariants for valuations and orderings on fields

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document} with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology.

A local-global principle for similarities over function fields of p-adic curves

Let p∈N be a prime with p≠2, and let K be a p-adic field. Let F be the function field of a curve over K. Let ΩF be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups.

Boater’s Preference for Coral Reef Restoration and Sustainable Resource Management in Florida

The Florida Reef is under intense pressure from excessive human disturbance and climate change. A group of Florida residents including boaters and fishers was surveyed about their concerns and willingness to pay (WTP) for restorative services. A discrete choice valuation experiment was conducted, involving nearly 1,300 respondents, to estimate WTP for coral reef restoration, wide beaches, and clean water. The boaters demonstrated a somewhat higher WTP for a more extensive beach size and coral restoration. The analysis allowed quantification of maritime transportation passengers’ preferences for sustainable resource management. The findings provide useful insights for maritime transportation and coastal management agencies wishing to finance conservation initiatives. Furthermore, the study offers valuable methodological contributions and policy relevant information to incorporate ecosystem service valuations into the decision-making process beyond the study areas.

Lifting elementary Abelian covers of curves

Given a Galois cover of curves f over a field of characteristic p, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is f. In this paper, we consider the case where the Galois groups are elementary abelian p-groups. We prove a combinatorial criterion for lifting an elementary abelian p-cover, dependent on the branch loci of lifts of its p-cyclic subcovers. We also study how branch points of a lift coalesce on the special fiber. Finally, for p=2, we analyze lifts for several families of (Z/2)3-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.

Branching rules for the restriction of regular representations of GL2(o/℘ r ) to SL2(o/℘ r )

Let o be a compact discrete valuation ring with maximal ideal ℘ such that the finite residue field o / ℘ has characteristic p. For r ≥ 2 and p = 2 , we obtain the branching rules for the restriction of a regular representation of GL 2 ( o / ℘ r ) to SL 2 ( o / ℘ r ) . These results have different behavior than that of the known case of p ≠ 2.

Erratum to Discriminants and Artin conductors (J. reine angew. Math. 712 (2016), 107–121)

Abstract The original article computes the Deligne discriminant for a degeneration of curves over a discrete valuation ring. One of the computations was only correctly performed in the case when the total space of the degeneration is normal, which should be added as a hypothesis.

Factorization in rings of integer-valued rational functions

For a domain D, the ring Int ( D ) of integer-valued polynomials over D is atomic if D satisfies the ascending chain condition on principal ideals. However, even for a discrete valuation domain V, the ring Int R ( V ) of integer-valued rational functions over V is antimatter. We introduce a family of atomic rings of integer-valued rational functions and study various factorization properties in these rings.

Zero-cycles in families of rationally connected varieties

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Kollár (Publ. Res. Inst. Math. Sci. 40(3):689–708, 2004).

Sums of squares in function fields over henselian discretely valued fields

Let n∈N and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F/K be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n. In the case of function fields of hyperelliptic curves of genus g, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2n(g+1). We show that this bound is optimal. Moreover, in the case where n=1, we show that if F/K is a hyperelliptic function field such that the order of this quotient group is 2g+1, then F is nonreal.

Descent of splendid Rickard equivalences in alternating groups

We show that any block of an alternating group with an abelian defect group over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent. This provides new evidence for a refined version of Broué's abelian defect group conjecture proposed by Kessar and Linckelmann.

Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Let D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and 1<n1≤⋯≤nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < n_1 \\le \\cdots \\le n_k$$\\end{document}, there exists an integer-valued polynomial on D, that is, an element of Int(D)={f∈K[X]∣f(D)⊆D}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D) = \\{ f \\in K[X] \\mid f(D) \\subseteq D \\}$$\\end{document}, which has precisely k essentially different factorizations into irreducible elements of Int(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D)$$\\end{document} whose lengths are exactly n1,…,nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1, \\ldots , n_k$$\\end{document}. Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.

Morita equivalence classes of 2‐blocks with abelian defect groups of rank 4

AbstractWe classify all 2‐blocks with abelian defect groups of rank 4 up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application is that Broué's abelian defect group conjecture holds for all blocks under consideration here.

Construction of even-level representations of [formula omitted] with residue field of characteristic two

We construct the finite-dimensional continuous complex representations of SL2 over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. We also prove that the complex group algebras of SL2 over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras C[SL2(Z/2rZ)] and C[SL2(F2[t]/(tr))] are not isomorphic for any r≥4.

On the relative Gersten conjecture for Milnor K-theory in the smooth case

We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the generic point of the special fibre. This complements results of Gillet-Levine for K-theory, Geisser for motivic cohomology and Schmidt-Strunk and the author for étale cohomology.

Completed prismatic F-crystals and crystalline Zp-local systems

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $\mathbf {Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.