The Zeta Function and Weil Conjecture for Elliptic Curves

Abstract

For this thesis we will compute the zeta functions and prove the Weil Conjecture for the elliptic curves While we compute the zeta functions of elliptic curves it is important to determine the number of the points on E over F_{q^s} We will show that the degree of an endomorphism on E is equal to the number of its kernel We consider the n-torsion subgroup which is isomorphic to Zn ? Zn And we will show that det(?_n) ≡ deg(?) (mod n) Then we can determine the points on elliptic curves Finally we will prove the Weil conjecture for elliptic curves and show that Z(q^{?k}) can be defined in a similar way to the Riemann zeta function

Date of Award	2019
Original language	English
Supervisor	Jen-Chieh Hsiao (Supervisor)