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Register a new userLorentz groups of cyclotomic extensions
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Vadim Schechtman
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In this (mostly historical) note we show how a unified Kummer-Artin-Schreier sequence from [W], [SOS] may be recovered from the relativistic velocity addition law.
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We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukudas theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension.
Let $k$ be a field of characteristic $q$, $cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p e q$ be a prime number and $cactocac$ a finite geometrically textsc{Galois} and etale cover defined over $k$ with function field $K:=k(cac)$. Let $(tau,B)$ be the $K/k$-trace of $A/K$. We give an upper bound for the $bbz_p$-corank of the textsc{Selmer} group $text{Sel}_p(Atimes_KK)$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $bbz$-rank of the textsc{Lang-Neron} group $A(K)/tauB(k)$. In the case of a geometric tower of curves whose textsc{Galois} group is isomorphic to $bbz_p$, we give sufficient conditions for the textsc{Lang-Neron} group of $A$ to be uniformly bounded along the tower.
Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $mathbb{F}_q$. A general result of our study is that $(a,b)leq 3$ for all $a,b in mathbb{Z}$ if $p> (sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a eq b$ and $a,b in {1,dots,e-1}$. The main idea we use is to transform equations over $mathbb{F}_q$ into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.
Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite fields.
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $infty$-categories) for additive $infty$-categories, (2) define the notion of nilpotent extensions for suitable $infty$-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas-Goodwillie-McCarthy theorem for stable $infty$-categories which are not monogenically generated (such as the stable $infty$-category of Voevodskys motives or the stable $infty$-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarkos notion of weight structures which provides a ring-with-many-objects analog of a connective $mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work of Hoyois-Krishna for homotopy $K$-theory, and establish new cases of Blancs lattice conjecture.
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