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Division Points

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Part of the Lecture Notes in Mathematics book series (LNM,volume 326)

Abstract

Division points on a complex torus play a role analogous to roots of unity on the circle. In particular, there are plenty of them, and if the torus is given as cubic with rational coefficients, their coordinates are algebraic numbers, hence their algebraic interest. By taking suitable limits of groups of division points, some canonical p-adic spaces are attached to the curve, which, formally at least, are similar to the tangent space at the origin :

  • V(E) = Lie(E) = Hom(ℝ,E) is of dimension two,

  • Tp(E) = Hom(ℚp/ℤp, E) free ℤp-module of rank two,

  • Vp(E) = \( T_p (E)\mathop \otimes \limits_{\mathbb{Z}_p } \mathbb{Q}_p \) is of \( dimension_{\mathbb{Q}_p } \) two.

Apart from the basic definitions and properties, we have also indicated some applications. In particular, if L is a sublattice of an imaginary quadratic field, we have proved in two ways that its invariant j(L) is an algebraic integer. The first one is the classical analytical one, whereas the second one (Tate) uses division points through ℓ-adic representations of Tate’s curves.

Keywords

  • Meromorphic Function
  • Elliptic Curve
  • Elliptic Curf
  • Galois Extension
  • Endomorphism Ring

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-540-46916-2_7

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© 1973 Springer-Verlag Berlin Heidelberg

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(1973). Division Points. In: Elliptic Curves. Lecture Notes in Mathematics, vol 326. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-46916-2_3

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  • DOI: https://doi.org/10.1007/978-3-540-46916-2_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06309-4

  • Online ISBN: 978-3-540-46916-2

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