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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, the exterior powers of a non-Archimedean local field and a fixed uniformizer of the ring of integers were investigated and three main results were established.
Abstract: Let $ \CO $ be the ring of integers of a non-Archimedean local field and $ \pi $ a fixed uniformizer of $ \CO $. We establish three main results. The first one states that the exterior powers of a $ \pi $-divisible $ \CO $-module scheme of dimension at most 1 over a field exist and commute with algebraic field extensions. The second one states that the exterior powers of a $p$-divisible group of dimension at most 1 over arbitrary base exist and commute with arbitrary base change. The third one states that when $ \CO $ has characteristic zero, then the exterior powers of $ \pi $-divisible groups with scalar $ \CO $-action and dimension at most 1 over a locally Noetherian base scheme exist and commute with arbitrary base change. We also calculate the height and dimension of the exterior powers in terms of the height of the given $p$-divisible group or $ \pi $-divisible $ \CO $-module scheme.
17 citations
TL;DR: In this paper, a necessary and sufficient condition for the integer ring OL to be free over the associated order was given, and when OL is free, a free basis explicitly was constructed.
17 citations
01 Aug 2020
TL;DR: In this article, totally positive definite quadratic forms over the ring of integers were studied and several new results about their properties were proved. But these results were not applicable to the case of integers.
Abstract: We study totally positive definite quadratic forms over the ring of integers ; we prove several new results about their properties.
17 citations
17 Jul 1989
TL;DR: A modification of the Goldwasser-Kilian-Atkin primality test, which, when given an input n, outputs either prime or composite, along with a certificate of correctness which may be verified in polynomial time.
Abstract: We present a modification of the Goldwasser-Kilian-Atkin primality test, which, when given an input n, outputs either prime or composite, along with a certificate of correctness which may be verified in polynomial time. Atkin's method computes the order of an elliptic curve whose endomorphism ring is isomorphic to the ring of integers of a given imaginary quadratic field Q(√—D). Once an appropriate order is found, the parameters of the curve are computed as a function of a root modulo n of the Hilbert class equation for the Hilbert class field of Q(√—D). The modification we propose determines instead a root of the Watson class equation for Q(√—D) and applies a transformation to get a root of the corresponding Hilbert equation. This is a substantial improvement, in that the Watson equations have much smaller coefficients than do the Hilbert equations.
17 citations
TL;DR: In this article, the authors considered the problem of deciding whether f 1 and f 2 have the same eigenvalues mod p m (where p is a fixed prime of K over p) for Hecke operators T l at all primes l ∤ N p.
17 citations