Special Values of Zeta-Functions for Proper Regular Arithmetic Surfaces

Author: Siebel, Daniel A.

Year: 2019

Degree: Dissertation (Ph.D.)

Advisor: Flach, Matthias

Committee Members: Mantovan, Elena; Flach, Matthias; Graber, Thomas B.; Amir-Khosravi, Zavosh

Option: Mathematics

DOI: 10.7907/YMHN-2T74

Abstract

We explicate Flach's and Morin's special value conjectures in [8] for proper regular arithmetic surfaces π : X → Spec Z and provide explicit formulas for the conjectural vanishing orders and leading Taylor coefficients of the associated arithmetic zeta-functions. In particular, we prove compatibility with the Birch and Swinnerton-Dyer conjecture, which has so far only been known for projective smooth X. Further, we derive a direct sum decomposition of Rπ*Z(n) into motivic degree components.

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