在本論文中，我們將計算橢圓曲線的zeta函數，並證明橢圓曲線的Weil猜想。
當我們計算橢圓曲線E的zeta函數時，最重要的是在F_{q^s}上橢圓曲線點的個數。首先證明橢圓曲線上自同態的degree等於其核的個數。之後利用n-扭轉子群與Zn ⊕ Zn同構，我們將證明det(αn) ≡ deg(α) (mod n)。透過用Frobenius endomorphism代入前面所證明的結果，即可計算出橢圓曲線在不同體(F_{q^s})上點的個數。最後，我們將證明橢圓曲線的Weil猜想，並證明Z(q^{-k})可以用與Riemann zeta函數類似的方式定義。

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(α_n) ≡ deg(α) (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.

[1] N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd ed. Springer-Verlag. (1984), 109-114.

[2] L. C. Washington, Elliptic Curves : Number Theory and Cryptography, Chapman & Hall.(2003).

[3] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag. (1986).