Topic
Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
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TL;DR: In this paper, it was shown that if char(k) = 0 then the number of congruence subgroups of index at mostn in a global field is infinite.
Abstract: Letk be a global field,O its ring of integers,G an almost simple, simply connected, connected algebraic subgroups ofGL
m
, defined overk and Γ=G(O) which is assumed to be infinite. Let σ
n
(Г) (resp. γ
n
(Г) be the number of all (resp. congruence) subgroups of index at mostn in Γ. We show:(a) If char(k)=0 then:
113 citations
TL;DR: In this paper, it was shown that toric varieties can be defined over the field with one element and a motivic interpretation of the image of the J-homomorphism defined by Adams was given.
Abstract: We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-homomorphism defined by Adams. ~ ~ ~ ~
112 citations
TL;DR: The main result of as mentioned in this paper is that the groups PSL(2;Od) are the same as the groups in the ring of integers in Q(p id) for any positive integer d.
Abstract: Let d be a square free positive integer and Od the ring of integers in Q( p id). The main result of this paper is that the groups PSL(2;Od) are
110 citations
TL;DR: In this article, the authors defined the smallest degree of any polynomial function over the ring of integers modulom, such that for all 0-1 assignments, the degree is 0 if the number of input ones is a multiple ofn and is one otherwise.
Abstract: Define the MOD
m
-degree of a boolean functionF to be the smallest degree of any polynomialP, over the ring of integers modulom, such that for all 0–1 assignments
$$\vec x$$
,
$$F(\vec x) = 0$$
iff
$$P(\vec x) = 0$$
. We obtain the unexpected result that the MOD
m
-degree of the OR ofN variables is
$$O(\sqrt[\tau ]{N})$$
, wherer is the number of distinct prime factors ofm. This is optimal in the case of representation by symmetric polynomials. The MOD
n
function is 0 if the number of input ones is a multiple ofn and is one otherwise. We show that the MOD
m
-degree of both the MOD
n
and
$$
eg MOD_n$$
functions isN
Ω(1) exactly when there is a prime dividingn but notm. The MOD
m
-degree of the MOD
m
function is 1; we show that the MOD
m
-degree of
$$
eg MOD_m$$
isN
Ω(1) ifm is not a power of a prime,O(1) otherwise. A corollary is that there exists an oracle relative to which the MOD
m
P classes (such as ⊕P) have this structure: MOD
m
P is closed under complementation and union iffm is a prime power, and MOD
n
P is a subset of MOD
m
P iff all primes dividingn also dividem.
109 citations
TL;DR: In this paper, it was shown that the existence of Fv-points on a smooth projective quadric over a p-adic field K is not sufficient to ensure that Y has an F-point.
Abstract: Let F = K(X) be the function field of a smooth projective curve over a p-adic field K. To each rank one discrete valuation of F one may associate the completion Fv. Given an F-variety Y which is a homogeneous space of a connected reductive group G over F, one may wonder whether the existence of Fv-points on Y for each v is enough to ensure that Y has an F-point. In this paper we prove such a result in two cases : (i) Y is a smooth projective quadric and p is odd. (ii) The group G is the extension of a reductive group over the ring of integers of K, and Y is a principal homogeneous space of G. An essential use is made of recent patching results of Harbater, Hartmann and Krashen. There is a connection to injectivity properties of the Rost invariant and a result of Kato.
101 citations