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 :
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V∞(E) = Lie(E) = Hom(ℝ,E) is of dimensionℝ two,
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Tp(E) = Hom(ℚp/ℤp, E) free ℤp-module of rank two,
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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.
The online version of the original chapter can be found at http://dx.doi.org/10.1007/978-3-540-46916-2_3
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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_7
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DOI: https://doi.org/10.1007/978-3-540-46916-2_7
Publisher Name: Springer, Berlin, Heidelberg
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