We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove it over quadratic extensions of the base field, providing essentially the first examples of the 2-parity conjecture in dimension greater than one. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest and works for positive characteristic and characteristic zero local fields alike. Particularly surprising is the appearance in the formula of terms that govern whether or not the Cassels-Tate pairing associated to the Jacobian is alternating, which first appeared in a paper of Poonen and Stoll. We prove the formula in many instances and show that in all cases it follows from standard global conjectures.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:682363 |
| Date | January 2015 |
| Creators | Morgan, Adam John |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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