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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 article, it was shown that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2.
Abstract: We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than the square of n. We deduce that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2. In order to show these results we prove also that if G is a connected, simply connected, semi-simple algebraic group defined over the field of rational numbers, then the G-representation variety of the fundamental group of a compact Riemann surface of genus n has rational singularities if and only if the G-character variety has rational singularities.

5 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of constructing large subsets of the natural numbers while avoiding 3-term geometric progressions over real and imaginary quadratic number fields.
Abstract: In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number elds. We rst construct high-density subsets of the algebraic integers of an imaginary quadratic number eld that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number eld, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets greedily, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

5 citations

Journal ArticleDOI
TL;DR: Hellmann as mentioned in this paper gave a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are semi-linear with respect to the ring morphism sending u to u^b (b > 1 is an integer).
Abstract: This note is an appendix to a preprint by E. Hellmann. We give a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are semi-linear with respect to the ring morphism sending u to u^b (b > 1 is an integer) and acting on elements of k through a fixed automorphism. Some of these phi-modules are involved in the classification of finite flat group schemes over ring of integers of p-adic fields.

5 citations

Journal ArticleDOI
TL;DR: In this article, an adjusted trace map TQ(n)/K with the property that TQn/K(O) = OK (here Q denotes the n cyclotomic field and O(n) its ring of integers) is defined.
Abstract: After first determining criteria for wild ramification of L/K (which can only happen at primes above 2), the above result is obtained for n = 2 (e ≥ 3) by computing TL/K(OL) explicitly, and is then extended to the general case. This approach does not rely on Leopoldt’s Theorem, in contrast to the techniques used in [6]. The explicit nature of the calculations used to compute I(L/K) leads to the definition of an “adjusted trace map” TQ(n)/K with the property that TQ(n)/K(O) = OK (here Q denotes the n cyclotomic field and O(n) its ring of integers). Using this map, we restate Leopoldt’s Theorem and show that its proof can be reduced to the (easier) cyclotomic case.

5 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the group of universal norms of a formal group corresponding to an elliptic curve of one of the three main types defined over a quasilocal field is trivial.
Abstract: In this paper we prove that the group of universal norms of a formal group corresponding to an elliptic curve of one of the three main types defined over a quasilocal field [11] is trivial. Applications are also indicated.Bibliography: 12 items.

5 citations


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