Johanna Tano: Classification of Elliptic Curves with l-Torsion over Finite Fields

Abstract: We develop theory and algorithms for the classification of elliptic curves over finite fields. Using geometric and arithmetic invariants, we study Weierstrass equations up to isomorphism, and for ordinary curves we show that the l-adic height of the endomorphism ring relative to Frobenius determines the rank of the rational l-torsion subgroup. We interpret this description through the theory of isogeny volcanoes, obtaining both structural insight and an efficient algorithm for computing this rank. We can then count Fq-rational points of order l up to isomorphism by classifying their orbits under automorphisms. As an application, we compute traces of Hecke operators up to an explicit correction term.