Mohamed Mohamed Alaa Eldin Mostafa <br/><br/>
Elliptic Surfaces with positive Mordell-Weil rank and Quadratic twists of elliptic curves /  
 أسطح ناقصية ذات رتبة موجبة و الإنحرافات التربيعية للمنحنيات الناقصية 
Mohamed Mohamed Alaa Eldin Mostafa ; Supervised Nabil L. Youssef , Mohammad M. Sadek 
 - Cairo :   Mohamed Mohamed Alaa Eldin Mostafa ,   2017 
 - 51 P. :   charts ;   25cm 
<br/><br/>Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics 
<br/><br/> In 1922, Mordell proved that the group of rational points , of an  elliptic curve , is a finitely generated abelian group. While the finite  part is well understood, the infinite part is much more mysterious. In this  thesis we aim to deepen our understanding of the arithmetic of the group  of rational points of elliptic curves by discussing certain arithmetic  questions on elliptic curves. We investigate the existence of high length  geometric progressions on elliptic and hyperelliptic curves. Moreover,  we study ranks of quadratic twists of pairs of elliptic curves 
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<br/><br/><label>Subjects--Index Terms: </label>Elliptic curves Elliptic surfaces Geometric progression