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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, the Artin-Hasse exponentials are replaced by a functor WF(- which is essentially the functor of q-typical curves in a (twisted) Lubin-Tate formal group law over A, where A is a discrete valuation ring that admits a Frobenius-like endomorphism a (we require o(a) =_ a q mod m for all a E A where m is the maximal idea of A).
Abstract: For any ring R let A(R) denote the multiplicative group of power series of the form I + a It + * * * with coefficients in R. The Artin-Hasse exponential mappings are homomorphisms Wp, ,(k) -+ A( Wp,, (k)), which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism E: Wp .(-) -> W ,,,0( Wp(-)), where Wp is the functor of infinite-length Witt vectors associated to the prime p. In this paper we present ramified versions of both Wp,4(-) and E, with Wp .(-) replaced by a functor WF(-), which is essentially the functor of q-typical curves in a (twisted) Lubin-Tate formal group law over A, where A is a discrete valuation ring that admits a Frobenius-like endomorphism a (we require o(a) =_ a q mod m for all a E A, where m is the maximal idea of A). These ramified-Witt-vector functors W!F,(-) do indeed have the property that, if k = A/rm is perfect, A is complete, and I/k is a finite extension of k, then Wqo.(l) is the ring of integers of the unique unramified extension LIK covering I/k.

12 citations

Journal ArticleDOI
Sangtae Jeong1
TL;DR: In this paper, it was shown that digit expansion of shift operators becomes an orthonormal basis for the space of continuous functions on F q [ [ [ T ], including a closed-form expression for expansion coefficients, and this is also true for p-adic integers, excluding the coefficient formula.

12 citations

Journal ArticleDOI
TL;DR: In this article, a Dennis trace map mod n, from K_1(A,Z/n) to the Hochschild homology group with coefficients HH_ 1(A; Z/n).
Abstract: Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and the number of roots of Mirimanoff polynomials.

12 citations

Journal ArticleDOI
TL;DR: In this article, the authors construct a functor from a semi-infinite reduced chamber gallery (alcove walk) to a maximal split torus in a finite extension of the ring of integers of a finite algebra.
Abstract: Let ${\mathfrak o}$ be the ring of integers in a finite extension $K$ of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let $T$ be a maximal split torus in $G$. Let ${\mathcal H}(G,I_0)$ be the pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. Given a semiinfinite reduced chamber gallery (alcove walk) $C^{({\bullet})}$ in the $T$-stable apartment, a period $\phi\in N(T)$ of $C^{({\bullet})}$ of length $r$ and a homomorphism $\tau:{\mathbb Z}_p^{\times}\to T$ compatible with $\phi$, we construct a functor from the category ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ of finite length ${\mathcal H}(G,I_0)$-modules to \'{e}tale $(\varphi^r,\Gamma)$-modules over Fontaine's ring ${\mathcal O}_{\mathcal E}$. If $G={\rm GL}_{d+1}({\mathbb Q}_p)$ there are essentially two choices of ($C^{({\bullet})}$, $\phi$, $\tau$) with $r=1$, both leading to a functor from ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ to \'{e}tale $(\varphi,\Gamma)$-modules and hence to ${\rm Gal}_{{\mathbb Q}_p}$-representations. Both induce a bijection between the set of absolutely simple supersingular ${\mathcal H}(G,I_0)\otimes_{\mathfrak o} k$-modules of dimension $d+1$ and the set of irreducible representations of ${\rm Gal}_{{\mathbb Q}_p}$ over $k$ of dimension $d+1$. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of $G$ over $K$. For $d=1$ we recover Colmez' functor (when restricted to ${\mathfrak o}$-torsion ${\rm GL}_{2}({\mathbb Q}_p)$-representations generated by their pro-$p$-Iwahori invariants)

12 citations

Journal ArticleDOI
Wieb Bosma1
TL;DR: It is shown how the use of a certain integral basis for cyclotomic fields enables one to perform the basic operations in their ring of integers efficiently.
Abstract: It is shown how the use of a certain integral basis for cyclotomic fields enables one to perform the basic operations in their ring of integers efficiently. In particular, from the representation with respect to this basis, one obtains immediately the smallest possible cyclotomic field in which a given sum of roots of unity lies. This is of particular interest when computing with the ordinary representations of a finite group.

12 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202250
2021117
2020121
2019111
201896