Bicyclotomic polynomials and impossible intersections

David Masser[1]; Umberto Zannier[2]

  • [1] Mathematisches Institut Universität Basel, Rheinsprung 21 4051 Basel, Switzerland
  • [2] Scuola Normale, Piazza Cavalieri 7 56126 Pisa, Italy

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 3, page 635-659
  • ISSN: 1246-7405

Abstract

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In a recent paper we proved that there are at most finitely many complex numbers t ≠ 0 , 1 such that the points ( 2 , 2 ( 2 - t ) ) and ( 3 , 6 ( 3 - t ) ) are both torsion on the Legendre elliptic curve defined by y 2 = x ( x - 1 ) ( x - t ) . In a sequel we gave a generalization to any two points with coordinates algebraic over the field Q ( t ) and even over C ( t ) . Here we reconsider the special case ( u , u ( u - 1 ) ( u - t ) ) and ( v , v ( v - 1 ) ( v - t ) ) with complex numbers u and v .

How to cite

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Masser, David, and Zannier, Umberto. "Bicyclotomic polynomials and impossible intersections." Journal de Théorie des Nombres de Bordeaux 25.3 (2013): 635-659. <http://eudml.org/doc/275776>.

@article{Masser2013,
abstract = {In a recent paper we proved that there are at most finitely many complex numbers $t \ne 0,1$ such that the points $(2,\sqrt\{2(2-t)\})$ and $(3, \sqrt\{6(3-t)\})$ are both torsion on the Legendre elliptic curve defined by $y^2=x(x-1)(x-t)$. In a sequel we gave a generalization to any two points with coordinates algebraic over the field $\{\bf Q\}(t)$ and even over $\{\bf C\}(t)$. Here we reconsider the special case $(u,\sqrt\{u(u-1)(u-t)\})$ and $(v, \sqrt\{v(v-1)(v-t)\})$ with complex numbers $u$ and $v$.},
affiliation = {Mathematisches Institut Universität Basel, Rheinsprung 21 4051 Basel, Switzerland; Scuola Normale, Piazza Cavalieri 7 56126 Pisa, Italy},
author = {Masser, David, Zannier, Umberto},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {cyclotomic polynomials; transcendence degree; elliptic curves; elliptic threefolds},
language = {eng},
month = {11},
number = {3},
pages = {635-659},
publisher = {Société Arithmétique de Bordeaux},
title = {Bicyclotomic polynomials and impossible intersections},
url = {http://eudml.org/doc/275776},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Masser, David
AU - Zannier, Umberto
TI - Bicyclotomic polynomials and impossible intersections
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/11//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 3
SP - 635
EP - 659
AB - In a recent paper we proved that there are at most finitely many complex numbers $t \ne 0,1$ such that the points $(2,\sqrt{2(2-t)})$ and $(3, \sqrt{6(3-t)})$ are both torsion on the Legendre elliptic curve defined by $y^2=x(x-1)(x-t)$. In a sequel we gave a generalization to any two points with coordinates algebraic over the field ${\bf Q}(t)$ and even over ${\bf C}(t)$. Here we reconsider the special case $(u,\sqrt{u(u-1)(u-t)})$ and $(v, \sqrt{v(v-1)(v-t)})$ with complex numbers $u$ and $v$.
LA - eng
KW - cyclotomic polynomials; transcendence degree; elliptic curves; elliptic threefolds
UR - http://eudml.org/doc/275776
ER -

References

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  7. D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves, Math. Annalen 352 (2012), 453–484. Zbl1306.11047MR2874963
  8. J. Pila, Integer points on the dilation of a subanalytic surface, Quart. J. Math. 55 (2004), 207–223. Zbl1111.32004MR2068319
  9. J. Pila, Counting rational points on a certain exponential-algebraic surface, Annales Institut Fourier 60 (2010), 489–514. Zbl1210.11074MR2667784
  10. M. Raynaud, Courbes sur une variété abélienne et points de torsion, Inventiones Math. 71 (1983), 207–233. Zbl0564.14020MR688265
  11. J.B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. Zbl0122.05001MR137689
  12. J.H. Silverman, The arithmetic of elliptic curves, Springer-Verlag (1986). Zbl1194.11005MR817210
  13. A. Weil, Foundations of algebraic geometry, American Math. Soc. Colloquium Pub. XXIX (1946). Zbl0168.18701MR23093
  14. U. Zannier, Some problems of unlikely intersections in arithmetic and geometry, Annals of Math. Studies, 181, Princeton (2012). Zbl1246.14003MR2918151

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