On the independence of Heegner points
Başlık çevirisi mevcut değil.
- Tez No: 400919
- Danışmanlar: DR. JOSEPH H. SILVERMAN
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2012
- Dil: İngilizce
- Üniversite: Brown University
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 82
Özet
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Özet (Çeviri)
In this thesis we describe the construction of a set of algebraic points, called Heegner points, on elliptic curves. Then we investigate under which constraints Heegner points are independent. We give a sucient condition on the class numbers of distinct quadratic imaginary elds so that on a given CM elliptic curve over Q with xed modular parametrisation, the Heegner points associated to (the maximal orders of) quadratic imaginary elds are linearly independent. This result extends the results of Rosen and Silverman from non-CM elliptic curves to CM ones. We will also show how to generalize this result from Heegner points associated to ring of integers of distinct quadratic imaginary elds to the case of Heegner points associated to orders of a xed conductor of quadratic imaginary elds. We will look at independence of Heegner points arising from Shimura curve parametrisations, p-adic uniformisations, and also Stark-Heegner points. Finally we build tools to investigate the independence of the Heegner points on higher dimensional abelian varieties when Heegner points can be constructed. In the rst chapter we give some basics in the theory of elliptic curves and construction of Heegner points for di erent parametrisations of elliptic curves, along with class eld theory, theory of CM and some representation theory results. Following that in the second chapter which is the introduction chapter, we talk about the literature related to Heegner points. In the third chapter we explain properties of Heegner points in detail and we talk about our independence criteria for a set of Heegner points on CM elliptic curves. Later we use our method to analyze independence of Heegner points arising from di erent parametrisations of the elliptic curves. In the fourth chapter we derive relations between the size of a nite submodule and subgroups of the general linear group and symplectic group that act on a submodule in an abelian way. We believe these linear algebra results can be useful to examine the independence of analog of Heegner points on some higher dimensional abelian varieties.
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