Algebraic Geometry Seminar

In this talk I will discuss generalizations of the recent AGM algorithm for determining the zeta function of an elliptic curve over F_2ⁿ. This class of algorithms work for elliptic curves over proper extensions of the prime field in any characteristic p.

First I will describe the parametrization of elliptic curves by a class of curves X₀(p^r). The latter curves are course moduli spaces for elliptic curves with p^r-isogeny structure. I make this explicit for the the case p^r = 8 which gives rise to the AGM algorithm. By computing the pullback of an explicit isogeny between these parametrized modular curves we should how to obtain a representation of the endomorphism ring in the base ring of the curve.

As a second step I will explain how the elliptic curve E/F_q can be replaced by a representative Heegner point on a modular curve X₀(p^r), and how this is connected to the endomorphism ring structure of E/F_q, and to a Galois cycle of isogenies

E0 = E - > E1 - > ... - > En = E0.

Using explicit models for X₀(p^r), and the image of an embedding X₀(p^r+1) - > X₀(p^r) ×X₀(p^r), we can solve generically for the moduli for E_i+1 given the moduli for E_i. Using the moduli over F_q as approximation to a p-adic neighborhood on X₀(p^r), we show that the iteration of the p-adic power series converges to a Galois cycle of Heegner points on X₀(p^r).

Combining the first and second steps of this construction yields a solution to the problem of determining the zeta function of the elliptic curve from the p-adic lift of the Heegner points. The resulting algorithms, denoted AGM-X₀(p^r), generalize the AGM algorithm and provide an efficient solution to the determination of the zeta function of an elliptic curve over any proper extension F_q of a small prime field F_p.