The Prime Number Theorem for L-functions of Elliptic Curves with CM

An elliptic curve over a field K is a nonsingular plane projective curve E of degree 3 to-gether with an exceptional point O ∈ E. It can be proved that such an object corresponds to an equation of the form Y 2Z + a1XY Z + a3Y Z 2 = X3 + a2X 2Z + a4XZ 2 + a6Z 3 known as the model of the elliptic curve, which will depend on char(K), the characteristic of K. In this work we will consider an elliptic curve E whose model looks like Y 2 = X3 + 16Z3 char(K) 6 = 2, 3 Y 2Z + Y Z2 = X3 char(K) = 2 Y 2Z = X3 char(K) = 3. If we take K = Fpr, with p a prime number and r ≥ 1 an integer, the corresponding model for E can be seen as equation over a finite field; so, it makes sense to count its solutions