Topic
Rational point
About: Rational point is a research topic. Over the lifetime, 1327 publications have been published within this topic receiving 22198 citations.
Papers published on a yearly basis
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Papers
TL;DR: When N is a prime number, there are no Q-rational N-isogenies beyond those exhibited in the above table as discussed by the authors, which is the case for the case when N = 11, 19, 27, 43, 67, 163 and for the two noncuspidal rational points on Xo(14 ).
Abstract: In this table, g is the genus of Xo(N), and v the number of noncuspidal rational points of Xo(N) (which is, in effect, the number of rational N-isogenies classified up to "twist"). For an excellent readable account of isogenies and their related diophantine problems, see Ogg's [25, 26]. The first column of the table corresponds to the genus 0 cases; for each of these values of N rational parametrizat ions of Xo(N) are known [10]. For each integer N, and each order R ~ Q(1/-ZN) such that R contains ~ and has class number one, there is a Q-rat ional N-isogeny. This accounts for one noncuspidal rational point on Xo(N ) for N = 11, 19, 27, 43, 67, 163 and for the two noncuspidal rational points on Xo(14 ). For a discussion of the cases: N = l l , 15, 17, 21 see ([43], pp.78-80) and for the peculiar N = 37, see ([22], w 5). The object of this paper is to show that when N is a prime number there are no Q-rat ional N-isogenies beyond those exhibited in the above table. To prove this (in the light of known results concerning Xo(N)(Q) for the twelve prime numbers N appearing in the table [19]) it suffices to show:
830 citations
TL;DR: In this paper, it was shown that for any ζ > 0, the inequality has only a finite number of solutions in relatively prime integers h, q. The main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of possible solutions.
Abstract: It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequalityhas only a finite number of solutions in relatively prime integers h, q. Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.
603 citations
556 citations
TL;DR: This article is an expository description of quadratic rational maps from the Riemann sphere to itself and some examples of these maps are given.
Abstract: This article is an expository description of quadratic rational maps from the Riemann sphere to itself
384 citations
TL;DR: The number of projectively inequivalent nonsingular plane cubic curves over a finite field F q with a fixed number of points defined over F q is determined by counting elliptic curves over Fq together with a rational point which is annihilated by 3 up to a certain equivalence relation.
361 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 4 |
| 2024 | 1 |
| 2023 | 11 |
| 2022 | 25 |
| 2021 | 18 |
| 2020 | 25 |