# 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

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topMasser, 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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