Isogenies and Endomorphism Rings of Abelian Varieties of Low Dimension

Download or read book Isogenies and Endomorphism Rings of Abelian Varieties of Low Dimension written by and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work we regard methods for computing isogenies between abelian varieties over finite fields. We develop new theoretical results leading to a new algorithm which for supersingular elliptic curves over F_p improves the previous approaches notably and which is as fast as algorithms for ordinary elliptic curves. In order to achieve this, we use restricted endomorphism rings to establish a connection to an ideal class group. We also show that isogenies under this reduction are defined over F_p. These results yield a simple description of F_p-rational supersingular isogeny graphs in an ordered level-structure, which provides the basis for the new computational method of feasible bi-directional search. MAGMA implementations reveal computational results which validate the preceding complexity analysis. Additionally, we also investigate Jacobians of hyperelliptic curves of genus two. Especially supersingular abelian varieties prove to be more difficult since approaches of the ordinary case cannot be generalized directly to this setting. engl.