Bruin, N. and Flynn, E. V. (2006) Exhibiting Sha[2] on hyperelliptic jacobians. Journal of Number Theory, 118 . pp. 266291. ISSN 0022314X
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Abstract
We discuss approaches to computing in the ShafarevichTate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the BrauerManin obstruction on these surfaces can be used to compute members of the ShafarevichTate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the SharevichTate group. Finally we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32.
Item Type:  Article 

Uncontrolled Keywords:  Higher Genus Curves, Jacobians, Visualisation, BrauerManin obstruction, ShafarevichTate Group 
Subjects:  A  C > Algebraic geometry H  N > Number theory 
Research Groups:  Number Theory Group 
ID Code:  799 
Deposited By:  E. Victor Flynn 
Deposited On:  01 Sep 2009 07:34 
Last Modified:  29 May 2015 18:29 
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Exhibiting Sha[2] on hyperelliptic jacobians. (deposited 12 Jul 2006)
 Exhibiting Sha[2] on hyperelliptic jacobians. (deposited 01 Sep 2009 07:34) [Currently Displayed]
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