On length spectra of lattices [Elektronische Ressource] / von Thomas A. Willging

On Length Spectra of LatticesZur Erlangung des akademischen Grades einesDoktors der Naturwissenschaftenvon der Fakulta¨t fu¨r Mathematik desKIT(Karlsruher Institut fu¨r Technologie)genehmigteDissertationvonThomas A. Willgingaus StuttgartTag der mundlichen Prufung: 16. November 2010¨ ¨Referent: PD Dr. Stefan Ku¨hnleinKorreferent: Prof. Dr. Frank HerrlichPrefaceThe theory of quadratic forms is a subject in number theory of the purest sort goingback to Fermat, Euler, Lagrange, Gauß and Minkowski to mention but a few. By aquadratic form Q in dimension n we mean a functionnXQ(x) = a xxij i ji,j=1T nof degree 2 with x := (x ,...,x ) ∈R and coefficients a =a ∈R.1 n ij jiAn old natural question is to ask which integers are represented overZ by a given formwith integer coefficients and moreover in how many ways they are represented. In someinstances these questions can be answered but there is no general answer. The centralidea is the so called local-global principle, that means we first check if an integer isrepresented locally over allZ and overR.pAs is well known, the theoryof positivedefinite quadraticformsprovidesan alternativeapproach for studying lattices. There is a one-to-one correspondence between congru-ence classes of lattices and equivalence classes of quadratic forms. Therefore otherclassical questions as for instance the sphere packing problem or the kissing numberproblem can be considered in this context too.