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.

Binary GCD Like Algorithms for Some Complex Quadratic Rings

Abstract: On the lines of the binary gcd algorithm for rational integers, algorithms for computing the gcd are presented for the ring of integers in \(\mathbb{Q}(\sqrt{d})\) where d ∈ { − 2, − 7, − 11, − 19}. Thus a binary gcd like algorithm is presented for a unique factorization domain which is not Euclidean (case d=-19). Together with the earlier known binary gcd like algorithms for the ring of integers in \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\), one now has binary gcd like algorithms for all complex quadratic Euclidean domains. The running time of our algorithms is O(n 2) in each ring. While there exists an O(n 2) algorithm for computing the gcd in quadratic number rings by Erich Kaltofen and Heinrich Rolletschek, it has large constants hidden under the big-oh notation and it is not practical for medium sized inputs. On the other hand our algorithms are quite fast and very simple to implement.

Galois Codescent For Motivic Tame Kernels

Abstract: Let $L/F$ be a finite Galois extension of number fields with an arbitrary Galois group $G$. We give an explicit description of the kernel of the natural map on motivic tame kernels $H^2_{\mathcal{M}}(o_L, {\bf Z}(i))_{G} {\rightarrow} H^2_{\mathcal{M}}(o_F, {\bf Z}(i))$. Using the link between motivic cohomology and $K$-theory, we deduce genus formulae for all even $K$-groups $K_{2i-2}(o_F)$ of the ring of integers. As a by-product, we also obtain lower bounds for the order of the kernel and cokernel of the functorial map $H^2_{\mathcal{M}}(F, {\bf Z}(i)) \rightarrow H^2_{\mathcal{M}}( L, {\bf Z}(i) )^{G}$.

Remarks on Automorphy of Residually Dihedral Representations

Abstract: We prove automorphy lifting results for geometric representations $\rho:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, such that the residual representation $\bar{\rho}$ is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F(\zeta_p)/F$. Such representations fail the Taylor-Wiles hypothesis and the patching techniques to prove automorphy do not work. We apply this to automorphy of elliptic curves $E$ over $F$, when $E$ has no $F$ rational 7-isogeny and such that the image of $G_F$ acting on $E[7]$ normalizes a split Cartan subgroup of $GL_2(\mathbb{F}_7)$.

Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve

Abstract: We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.

On a Problem of MacDowell and Specker

Abstract: Let T extend the theory P of Peano arithmetic, and suppose 1= T. Form from W a model 91= , in analogy to the way in which the ordered ring of integers is formed from the standard model of arithmetic. Let P' and T' be the corresponding analogues of P and T respectively. Now consider the group Gv = . In [5] MacDowell and Specker set out to determine the structure of such groups. (The precise statement in [5] refers to the ring of integers rather than the ordered ring. However, as pointed out to us by J. Knight, since Lagrange's Theorem that a positive integer is the sum of four squares is provable in the analogue of P' for rings (see, for example, the proof in [7, p. 102]), the set of positive elements is definable in the ring, and consequently, so is the ordering. Thus, for the present purpose it makes no difference which of the two structures is used. Of course, one needs the ordering to discuss end extensions, as considered in [5]. On the other hand, one should be aware that in Pr' one cannot define an ordering, where the theory Pr' is the theory of the group of integers with distinguished element 1, . The constant 1 is needed so that divisibility mod n can be expressed. We will return to this point later.) In ? 1 we shall outline the results in this direction obtained in [5]. Lipshitz and Nadel, unaware that a similar question had been posed and investigated in [5] (though, of course aware that [5] contained the celebrated results on end extensions) set out to characterize those models of Pr = Presburger Arithmetic (the complete theory of ) which can be expanded to models of P. They were able to give a complete characterization for countable models in [4], which we describe in ?2. The characterization in [4] does not lift, at least in any obvious way, to uncountable models. Nevertheless, we have been able to lift enough of it that, assuming the continuum hypothesis, a complete answer to the original question of MacDowell and Specker for the group structure for P itself is given in ?3. We consider arbitrary T P in ?4. We wish to thank W. Knight for his translation of [5] and J. Knight for her many helpful observations which have greatly aided us in the preparation of this paper.