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- W2548484739 abstract "This thesis consists of two parts. In part one ofthis thesis, we study the relationship between the Artin conductorand the minimal discriminant of a hyperelliptic curve defined overthe fraction field K of a discrete valuation ring. The Artinconductor and the minimal discriminant are two measures ofdegeneracy of the singular fiber in a family of hyperellipticcurves. In the case of elliptic curves, the Ogg-Saito formula showsthat (the negative of) the Artin conductor equals the minimaldiscriminant. In the case of genus 2 curves, Liu showed thatequality no longer holds in general, but the two invariants arerelated by an inequality. We extend Liu's inequality tohyperelliptic curves of arbitrary genus, assuming rationality ofthe Weierstrass points over K. In part two of this thesis, wecompute the sizes of component groups and Tamagawa numbers of Neronmodels of Jacobians using matrix tree theorems from combinatorics.Raynaud gave a description of the component group of the specialfiber of the Neron model of a Jacobian, in terms of themultiplicities and intersection numbers of components in thespecial fiber of a regular model of the underlying curve. Bosch andLiu used this description, along with some Galois cohomologycomputations to provide similar descriptions of Tamagawa numbers.We use various versions of the matrix tree theorem to makeRaynaud's and Bosch and Liu's descriptions more explicit in termsof the combinatorics of the dual graph and the action of theabsolute Galois group of the residue field on it. We then derivesome consequences of these explicit descriptions. First, we use theexplicit formula to provide a new geometric condition on the curvefor obtaining a uniform bound on the size of the component group ofits Jacobian. Then we prove a certain periodicity property of thecomponent group of a Jacobian under contraction of connectingchains of specified lengths in the dual graph. As a thirdapplication, we obtain an alternate proof of one of the key stepsin Halle and Nicaise's proof of the rationality of the Neroncomponent series for Jacobians." @default.
- W2548484739 created "2016-11-11" @default.
- W2548484739 creator A5034227164 @default.
- W2548484739 date "2016-01-01" @default.
- W2548484739 modified "2023-09-24" @default.
- W2548484739 title "Invariants linked to models of curves over discrete valuation rings" @default.
- W2548484739 hasPublicationYear "2016" @default.
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