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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 Eichler class number formula was generalized to arbitrary Z-orders in a quaternion algebra over a finite prime field Fp and the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over Fp were investigated.
Abstract: Let F be a totally real field with ring of integers OF , and D be a totally definite quaternion algebra over F . A well-known formula established by Eichler and then extended by Korner computes the class number of any OF -order in D. In this paper we generalize the Eichler class number formula so that it works for arbitrary Z-orders in D. Our motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a finite prime field Fp. We give explicit formulas for the number of these isomorphism classes for all primes p.
10 citations
TL;DR: This algorithm relies on the Chinese Remainder Theorem, with powers of cyclotomic polynomials presented as the moduli, and can be implemented over any ring and its implementation does not depend on the ring constants.
10 citations
TL;DR: In this article, an abc-conjecture for the field of all algebraic numbers was proposed based on the definition of the radical and the height of an algebraic number.
Abstract: The abc–conjecture for the ring of integers states that, for every e > 0 and every triple of
relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1 + e
with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors
of m. In the present paper we propose an abc–conjecture for the field of all algebraic numbers. It is based
on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number.
From this abc–conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic
numbers, and we discuss from this point of view known results on solutions of Fermat's equation in
fields of small degrees over ℚ.
10 citations
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TL;DR: In this article, it was shown that a hyperelliptic curve has a Weierstrass model over the ring of integers of a number field with height effectively bounded only in terms of the genus of the curve.
Abstract: Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.
10 citations
TL;DR: Following work of Shapiro, Mills, Catlin and Noe on iterations of Euler's φ-function, analogous results on iteration of the function φe, when restricted to a particular subset of the positive integers are developed.
Abstract: A unit x in a commutative ring R with identity is called exceptional if 1−x is also a unit in R. For any integer n ≥ 2, define φe(n) to be the number of exceptional units in the ring of integers modulo n. Following work of Shapiro, Mills, Catlin and Noe on iterations of Euler’s φ-function, we develop analogous results on iterations of the function φe, when restricted to a particular subset of the positive integers.
10 citations