Klein Quartic Curve and its Modularity

Abstract

The local Zeta function of a variety encodes important information about the variety. From the works of Weil, Deligne, Dwork, and others, many things are known about the local Zeta function of a smooth projective variety. In this article, we find the local Zeta function for the Klein Quartic curve, $x^{3}y+y^{3}z+z^{3}x=0$, by realizing it as a quotient of degree 7 Fermat curve. We conclude by giving the associated modular forms via Galois representations.