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Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 3, Pages 5–14 (Mi izv483)  

This article is cited in 2 scientific papers (total in 2 papers)

On the number of rational points on certain elliptic curves

E. Bombieria, U. Zannierb

a Institute for Advanced Study, School of Mathematics
b University Iuav of Venice

Abstract: Let $E$ be an elliptic curve defined over the rationals, with rational 2-torsion. We prove a uniform bound for the number of rational points on $E$ of height $\leqslant B$ of the form $#\{P\in E({\mathbb Q})\colon H(P)\leqslant B\}\leqslant c(\varepsilon)(\max(H(E),B))^\varepsilon$, valid for every fixed $\varepsilon>0$ and a suitable positive computable constant $c(\varepsilon)$. We give an application of this result to the counting of quadruples $(p_1,p_2,p_3,p_4)$ of distinct primes that do not exceed $X$ and satisfy $p_i^2\Delta_{jk}-p_j^2\Delta_{ik}+p_k^2\Delta_{ij}=0$ for all $1\leqslant i<j<k\leqslant 4$, where $\Delta_{ij}$ are given integers. This is applied by Konyagin (in the paper [3], which is published simultaneously with the present one) to a problem on the large sieve by squares.

DOI: https://doi.org/10.4213/im483

Full text: PDF file (833 kB)
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English version:
Izvestiya: Mathematics, 2004, 68:3, 437–445

Bibliographic databases:

UDC: 512.752
MSC: 11G05, 14G40, 14H52, 14N10
Received: 15.08.2003

Citation: E. Bombieri, U. Zannier, “On the number of rational points on certain elliptic curves”, Izv. RAN. Ser. Mat., 68:3 (2004), 5–14; Izv. Math., 68:3 (2004), 437–445

Citation in format AMSBIB
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\by E.~Bombieri, U.~Zannier
\paper On the number of rational points on certain elliptic curves
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\vol 68
\issue 3
\pages 5--14
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\jour Izv. Math.
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\vol 68
\issue 3
\pages 437--445
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. V. Konyagin, “Problems on the set of squarefree numbers”, Izv. Math., 68:3 (2004), 493–520  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Draziotis K.A., “On the Number of Integer Points on the Elliptic Curve $y^2 = x^3+Ax$”, Int J Number Theory, 7:3 (2011), 611–621  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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