An efficient algorithm to compute an ellipticcurve from a corresponding function fieldautomorphic form

Abstract

Elliptic modular forms of weight 2 and elliptic modular curves are strongly related. In the rank-2 Drinfeld module situation, we have still modular curves that can be described analytically through Drinfeld modular forms. Gekeler and Reversat prove how the results of Drinfeld can be used to construct the analytic uniformization of the elliptic curve attached to a given automorphic form. Longhi, building on ideas of Darmon, defines a multiplicative integral that theoretically allows to find the corresponding Tate parameter. In this thesis we develop and present a polynomial time algorithm to compute the integral proposed by Longhi. Also we devised a method to find a rational equation of the corresponding representative for the isogeny class.