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Showing 1 to 20 of 78 for “"elliptic curves"”.
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Elliptic Curves
<p>The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points <em>E</em>(<strong><em>Q</em></strong>) of an elliptic curve <em>E</em> form a group, which is a finitely-generated abelian group by Mordell’s …
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Elliptic Curves Over Finite Fields
… thesis provides a self-contained introduction to elliptic curves accessible to advanced undergraduates and graduate students in mathematics, with emphasis on the the theory of elliptic curves over finite fields. In Chapter 1, affine and projective planes are introduced. Chapter 2 introduces the …
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Elliptic curves and their cryptographic applications
<p>"This thesis is a basic overview of elliptic curves and their applications to Cryptography. We begin with basic definitions and a demonstration that, given an elliptic curve addition, the points of an elliptic curve form a mathematical group. We then proceed to delve further into the …
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Power Integral Points on Elliptic Curves
… divisibility sequences (Bn) formed by Mordell elliptic curves ED : y2 = x3+D. For the curve-point pair (E−2, P), where E−2 : y2 = x3 −2, and P = (3, 5) is a nontorsion point, we prove that no term Bn is a perfect 5th power, and we give the explicit bound p � 137 for any term in the associated …
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Data on elliptic curves with high rank
… thesis, a map that takes points from a given Elliptic Curve in Weierstrass form to points on the corresponding curve in Hasse form was derived. This was implemented in the Maple programming language in order to allow the transformation of points into Hasse form so that an embedding into the …
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Counting elliptic curves of bounded Faltings height
Because many invariants and properties of elliptic curves are difficult to understand directly, the study of arithmetic statistics instead looks at what happens "on average", using heights to make this notion rigorous. Previous work has primarily used the naive height, which can be calculated …
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RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES
Let E_m be the family of elliptic curves given by y^2=x^3-x+m^2, which has rank 2 when regarded as an elliptic curve over Q(m). (Here Q represents the field of rational numbers.) Brown and Myers show that a certain quadratic polynomial m(t) has the property that E_m(t) contains an additional …
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Average lang-trotter conjecture for 2 elliptic curves
Let E 1 and E 2 be two elliptic curves without complex multiplication over the rationals. For primes p of good reduction, let a p ( E 1 ) and a p ( E 2 ) be the traces of the Frobenius morphism of E 1 and E 2 respectively. By Hasse's theorem, we know that a p ( E i ), i = 1,2, are integers and …
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The fundamental critical points of modular elliptic curves
In their paper, "Arithmetic of Weil Curves", Mazur and Swinnerton-Dyer prove that the number of fundamental critical points of the normalized weight two modular form associated with an elliptic curve is an upper bound on the analytic rank of the curve. Their calculation of this quantity for all …
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Elliptic curves with complex multiplication and {u039B}-structures
This thesis examines the relationship between elliptic curves with complex multiplication and Lambda structures. Our main result is to show that the moduli stack of elliptic curves with complex multiplication, and the universal elliptic curve with complex multiplication over it, both admit Lambda …
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Elliptic curves with complex multiplication and {u039B}-structures
This thesis examines the relationship between elliptic curves with complex multiplication and Lambda structures. Our main result is to show that the moduli stack of elliptic curves with complex multiplication, and the universal elliptic curve with complex multiplication over it, both admit Lambda …
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Families of Rank Zero Twists of Elliptic Curves
… to generalize this result to a bigger class of elliptic curves. We are going to generalize Serf's result to the family of elliptic curves with full 2-torsion, i.e. elliptic curves of the form $E:y\sp2 = (x-e\sb1)(x-e\sb2)(x-e\sb3)$ with $e\sb{i}\in\doubz.$ We can also assume that $e\sb1 = 0$ and …
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The Birch and Swinnerton-Dyer Conjecture for elliptic curves
… in order to provide a characterisation for elliptic curves over rational numbers. We investigate the group structure of rational points on elliptic curves, and show that this group is finitely generated by the Mordell-Weil Theorem. The Shafarevich-Tate group is introduced by way of an …
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L-functions of twisted elliptic curves over function fields
… studied, both theoretically and computationally, elliptic curves and their L-functions over number fields, in particular over the rational numbers. Much less work has been done over function fields, especially computationally, where the underlying geometry of the function field plays an intimate …
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Moduli for pairs of elliptic curves with isomorphic N-torsion
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.
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P-adic L-functions for elliptic curves over CM fields
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 1983
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