| | | 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 |
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