Zeta functions of local orders [Elektronische Ressource] / vorgelegt von Siamak Firouzian Bandpey

Zeta FunctionsofLocal OrdersDISSERTATION ZUR ERLANGUNG DES DOKTORGERADES DER-˜NATURWISSENSCHAFTEN (DR. RER. NAT) DER FAKULTAT˜NWF I-MATHEMATIK DER UNIVERSITAT REGENSBURGvorgelegt vonSiamak Firouzian BandpeyRegensburg, Januar 2006Promotionsgesuch eingereicht am: 10. Januar 2006Die Arbeit wurde angeleitet von Prof. Dr.JannsenPrufungsaussc˜ huss: Vorsitzender : Prof. Dr. Finster1.Gutachter : Prof. Dr. Jannsen2.Gutachter : Prof. Dr. Schmidtweiter Prufer˜ : Prof. Dr. GoetteContentsIntroduction 41 Some background from commutative algebra and algebraicgeometry 102 Zeta functions of orders: deflnition and basic properties 243 A formula for the zeta function and the functional equation 354 Comparison with Galkin’s zeta function 465 A concrete example 496 The rational unibranch case I 597 Two more examples 678 The rational unibranch case II 749 On the Riemann hypothesis 79Notation 90Bibliography 93IntroductionThe zeta-functions associated with algebraic curves over flnite flelds encodemany arithmetic properties of the curves. In the non-singular case the the-ory is well-known. It is analogous to the theory of zeta-functions for numberflelds and culminates in the Hasse-Weil theorem about the Riemann hypoth-esis for curves. In the singular case, which will be the main topic of thisthesis, the theory is more di–cult and less explored. First of all, one doesnot deal with Dedekind rings anymore, but with orders, i.e., certain subringofthem.