Special Points on the Product of Two Modular Curves

Content Provider	Semantic Scholar
Author	Edixhoven, Bas
Copyright Year	1998
Abstract	It is well known that the j-invariant establishes a bijection between C and the set of isomorphism classes of elliptic curves over C, see for example [10]. The endomorphism ring of an elliptic curve E over C is either Z or an order in an imaginary quadratic extension of Q; in the second case E is said to be a CM elliptic curve (CM meaning complex multiplication). A complex number x is said to be CM if the corresponding elliptic curve over C is CM. A point (x1, x2) in C 2 is defined to be CM if both x1 and x2 are CM. The aim of this article is to determine all irreducible algebraic curves C in C containing infinitely many CM points. In other words, we want to determine all irreducible polynomials f in C[x1, x2] that vanish at infinitely many CM points. The motivation for doing this comes from a conjecture of Frans Oort (see [7, Chapter IV, §1] for a precise statement), saying roughly that the irreducible components of the Zariski closure of any set of CM points in any Shimura variety are sub Shimura varieties. For the irreducible components of dimension zero this is trivially true. For those of dimension one Oort’s conjecture was in fact stated earlier by Yves André as a problem in [2, Chapter X, §1]. We view C as the Shimura variety which is the moduli space of pairs of elliptic curves. Then the irreducible sub Shimura varieties of dimension one are the following: C × {x2} with x2 a CM point, {x1} ×C with x1 a CM point, or the image in C , under the usual map, of the modular curve Y0(n) for some integer n ≥ 1. Recall that, for n ≥ 1, Y0(n) is the modular curve classifying elliptic curves with a cyclic subgroup of order n, or, equivalently, cyclic isogenies of degree n between elliptic curves. The usual map from Y0(n) to C 2 sends an isogeny to its source and target, i.e., φ:E1 → E2 is sent
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Alternate Webpage(s)	http://arxiv.org/pdf/alg-geom/9610011v1.pdf
Language	English
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Subject Keyword	Arabic numeral 0 Cardiomyopathies Class Classification Closure Elliptic curve cryptography Estradiol Imaginary time Integer (number) Irreducibility Irreducible polynomial Linear algebra Multiplication Subgroup A Nepoviruses
Content Type	Text
Resource Type	Article