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Document Type:Latin Dissertation
Language of Document:English
Record Number:54623
Doc. No:TL24577
Call number:‭3254072‬
Main Entry:Shahed Islam Sharif
Title & Author:Construction of curves with prescribed *period and *indexShahed Islam Sharif
College:University of California, Berkeley
Date:2006
Degree:Ph.D.
student score:2006
Page No:47
Abstract:Given a curve C over a field K, the period of C is the g.c.d. of the degrees of Gal(K ̄/K)-invariant invertible sheaves on C, while the index of C is the g.c.d. of the degrees of Gal(K ̄/K)-invariant divisors. S. Lichtenbaum proved divisibility conditions between the genus, period and index of C. Using rigid analytic methods, we show that all genera, periods and indices which satisfy Lichtenbaum's conditions occur if K is a local field with characteristic not equal to 2. We also show that there are genus 1 curves over [special characters omitted] with any odd period n and maximal possible index---in this case, index n2. We do this by considering an obstruction map, defined between two cohomology groups, which describes the quotient of the index by the period. Over general number fields, we exploit the same obstruction map to find genus 1 curves with any period and index satisfying Lichtenbaum's conditions. Finally, we show that for any genus, period, and index satisfying Lichtenbaum's conditions and such that 4 does not divide the index, there exists a number field and a curve over that number field with the given genus, period, and index.
Subject:Pure sciences; Arithmetic geometry; Curves; Index; Period; Mathematics; 0405:Mathematics
Added Entry:B. Poonen
Added Entry:University of California, Berkeley