Abstract Infrastructures of Unit Rank Two (abstract only)

Infrastructures of Unit Rank Two Felix Fontein, University of Zürich, felix.fontein@math.uzh.ch On our poster, we want to give information on the infrastructure of a global field of unit rank two. The infrastructure of a global field is the set of all minima of a fractional ideal, together with the neighbor relation and the baby step operations [HMPLR87, Fon08c]. In the case of unit rank one, it is both used for computation of fundamental units [Buc85a] and for cryptography [SSW96, JSS07]. One main emphasis lies on visualization, both of the set of minima together with baby steps (in the sense of J. Buchmann in [Buc85a]) and the generalized Voronŏı algorithm. The generalized Voronŏı algorithm was first described by J. Buchmann in [Buc85a, Buc85b] for number fields. In the case of purely cubic function fields, it has been introduced by Y. Lee, R. Scheidler and C. Yarrish in [LSY03]. Some screenshots of the current experimental version of our program can be seen on the second page; the shown cases are purely cubic function fields over F5. We also plan to present a live version on our laptop during the poster session. Depending on our progress, we also plan to include new results on the unit rank two case [Fon08a], which are related to the interpretation of certain unit rank one infrastructures as cyclic groups as in [Fon08b]. References [Buc85a] Johannes Buchmann. A generalization of Voronŏı’s unit algorithm. I. J. Number Theory, 20(2):177–191, 1985. [Buc85b] Johannes Buchmann. A generalization of Voronŏı’s unit algorithm. II. J. Number Theory, 20(2):192–209, 1985. [Fon08a] Felix Fontein. The infrastructure of a global field of arbitrary unit rank. In preparation. [Fon08b] Felix Fontein. Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures, 2008. To appear in Advances in Mathematics of Communications. [Fon08c] Felix Fontein. The infrastructure of a global field of unit rank one, 2008. In preparation. [HMPLR87] Y. Hellegouarch, D. L. McQuillan, and R. Paysant-Le Roux. Unités de certains sousanneaux des corps de fonctions algébriques. Acta Arith., 48(1):9–47, 1987. [JSS07] M. J. Jacobson, R. Scheidler, and A. Stein. Cryptographic protocols on real hyperelliptic curves. Adv. Math. Commun., 1(2):197–221, 2007.