Browsing by Author "Caranay, Perlas"

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    Computing Isogeny Volcanoes of Rank Two Drinfeld Modules
    (2018-01-19) Caranay, Perlas; Greenberg, Matthew; Scheidler, Renate; Bauer, Kristine; Bauer, Mark; Dimitrov, Vassil Simeonov; Chen, Imin
    Elliptic curves have long been widely studied mathematical objects. They do not only feature prominently in established areas of mathematics such as number theory, algebraic geometry, and topology, but have recently gained practical importance due to applications in coding theory and cryptography. More recently, Drinfeld modules have received increased attention due to their surprising similarity to elliptic curves - objects to which they bear little superficial resemblance. However, little is yet known about real-world applications of Drinfeld modules. Elliptic curves come in two kinds -- ordinary and supersingular. Endomorphism rings of ordinary and supersingular elliptic curves, made up of isogenies, are very different. An analogous dichotomy holds for Drinfeld modules. Recently, major progress has been achieved by researchers in explicitly computing endomorphism rings of elliptic curves using isogeny volcanoes, but very little if anything of this kind has yet been done for Drinfeld modules. Our aim here is to study the theoretical and computational aspects of isogeny volcanoes of rank two Drinfeld modules defined over finite fields and determine how to explicitly compute these mathematical structures. We establish theoretical properties of isogeny volcanoes in the Drinfeld module case. Then we design, analyze, and implement algorithms for computing (1) j-invariants and Drinfeld modular polynomials, (2) isogeny volcanoes, and (3) endomorphism rings and explicit isogenies of rank two Drinfeld modules.