scholarly journals Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II

2018 ◽  
Vol 30 (6) ◽  
pp. 1487-1520 ◽  
Author(s):  
Sungmun Cho
Keyword(s):  
Mass Formula ◽  
Local Density ◽  
Group Scheme ◽  
Field Extension ◽  
Integral Group ◽  
Group Schemes ◽  
Quadratic Extensions ◽  
Local Densities ◽  
Characteristic 2 ◽  
Residue Characteristic

Abstract This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions {E/F} , where F is a finite unramified field extension of {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice {(L,H)} , when a base field is unramified over {\mathbb{Q}} at a prime {(2)} .

scholarly journals Group schemes and local densities of quadratic lattices in residue characteristic 2

2014 ◽  
Vol 151 (5) ◽  
pp. 793-827 ◽  
Author(s):  
Sungmun Cho
Keyword(s):  
Mass Formula ◽  
Local Density ◽  
Finite Extension ◽  
Group Scheme ◽  
Generic Fiber ◽  
Group Schemes ◽  
Local Densities ◽  
Quadratic Lattices ◽  
Totally Real

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.

Finite group schemes of p-rank ≤ 1

2017 ◽  
Vol 166 (2) ◽  
pp. 297-323
Author(s):  
HAO CHANG ◽  
ROLF FARNSTEINER
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Group Scheme ◽  
Closed Field ◽  
Difficult Case ◽  
Modular Representation Theory ◽  
Group Schemes ◽  
Restricted Lie Algebras ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

scholarly journals Witt kernels of bi-quadratic extensions in characteristic 2: Addendum

2004 ◽  
Vol 70 (2) ◽  
pp. 351-351
Author(s):  
Hamza Ahmad
Keyword(s):  
Quadratic Extensions ◽  
Characteristic 2

In a recent referee's report reviewing [2], it was pointed out to me that Theorem 2.2 in [1] was already known in the literature, and is (originally) due to Mammone and Moresi in [3]. Also in [3], the authors establish the excellence of inseparable quadratic extensions in a shorter way than I presented in [1]. As the paper [3] was unknown to me at the time of submitting [1], I hope by this note to acknowledge and credit [3] for the result(s).

scholarly journals Witt kernels of bi-quadratic extensions in characteristic 2

2004 ◽  
Vol 69 (3) ◽  
pp. 433-440 ◽  
Author(s):  
Hamza Ahmad
Keyword(s):  
Quadratic Extension ◽  
Quadratic Extensions ◽  
Witt Equivalence ◽  
Characteristic 2

Let κ be a field of characteristic 2. The author's previous results (Arch. Math. (1994)) are used to prove the excellence of quadratic extensions of κ. This in turn is used to determine the Witt kernel of a quadratic extension up to Witt equivalence. An example is given to show that Witt equivalence cannot be strengthened to isometry.

scholarly journals Local densities of diagonal integral ternary quadratic forms at odd primes

2021 ◽  
pp. 1-29
Author(s):  
Edna Jones
Keyword(s):  
Quadratic Forms ◽  
Mass Formula ◽  
Exponential Sums ◽  
Gauss Sums ◽  
Ternary Quadratic Forms ◽  
Local Densities

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel’s mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

scholarly journals Higgs bundles and fundamental group schemes

2019 ◽  
Vol 19 (3) ◽  
pp. 381-388
Author(s):  
Indranil Biswas ◽  
Ugo Bruzzo ◽  
Sudarshan Gurjar
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Higgs Bundles ◽  
Chern Classes ◽  
Group Schemes ◽  
Tannakian Category

Abstract Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

scholarly journals Local conspecific density does not influence reproductive output in a secondary cavity-nesting songbird

The Auk ◽  
2020 ◽  
Vol 137 (2) ◽  
Author(s):  
Jeffrey P Hoover ◽  
Nicole M Davros ◽  
Wendy M Schelsky ◽  
Jeffrey D Brawn
Keyword(s):  
Body Condition ◽  
Population Biology ◽  
Life Histories ◽  
Reproductive Output ◽  
Local Density ◽  
Food Limitation ◽  
Primary Objective ◽  
Cavity Nesting ◽  
Nestling Provisioning ◽  
Local Densities

Abstract Density dependence is a conceptual cornerstone of avian population biology and, in territorial songbirds, past research has emphasized interactions among food limitation, density, and reproduction. Documenting the importance of density effects is central to understanding how selective forces shape life histories and population dynamics. During the 2008–2011 breeding seasons, we nearly doubled overall conspecific breeding densities on study sites, and manipulated nest box spacing to increase local breeding densities (defined as the number of pairs breeding within 200 m of a pair’s nest) of a secondary cavity-nesting songbird, the Prothonotary Warbler (Protonotaria citrea). Our primary objective was to test for effects of food limitation, as mediated by conspecific local densities, on measures of productivity. We monitored breeding pairs and recorded the total number of fledglings produced along with several components of reproductive output (clutch size, hatching success, nestling survival, and probability of attempting a second brood), rates of nestling provisioning, and nestling body condition prior to fledging. We predicted that if the availability of food were affected by local densities, then one or more of these parameters measuring reproduction would be affected negatively. We did not detect an effect of local density on total reproductive output or its components despite our vast range of local densities (1–27 pairs; i.e. 0.16–2.23 pairs ha–1). Further, we also did not detect differences in nestling provisioning rates and nestling body condition relative to local density. By breeding in a productive ecosystem rich in food resources, these warblers appear to avoid reduced reproductive output when breeding in high densities. Whereas density-dependent food limitation may commonly reduce reproductive output in many species, the ecological circumstances underlying when it does not occur merit further investigation and may provide new insights into what is driving territoriality and what are the primary factors affecting individual fitness.

scholarly journals The different and differentials of local fields with imperfect residue fields

1997 ◽  
Vol 40 (2) ◽  
pp. 353-365 ◽  
Author(s):  
Bart de Smit
Keyword(s):  
Galois Extension ◽  
Correction Term ◽  
Residue Field ◽  
Field Extension ◽  
Local Fields ◽  
Valuation Rings ◽  
Complete Field ◽  
Residue Fields ◽  
Residue Characteristic ◽  
Module Of Differentials

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.

scholarly journals Tannakization in derived algebraic geometry

2014 ◽  
Vol 14 (3) ◽  
pp. 642-700 ◽  
Author(s):  
Isamu Iwanari
Keyword(s):  
Algebraic Geometry ◽  
Galois Group ◽  
Monoidal Category ◽  
Group Scheme ◽  
Affine Group ◽  
Stable Category ◽  
Group Schemes ◽  
Symmetric Monoidal Category ◽  
Derived Algebraic Geometry ◽  
Mixed Motives

AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

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