2-Selmer parity for hyperelliptic curves in quadratic extensions

Morgan, A. (2023) 2-Selmer parity for hyperelliptic curves in quadratic extensions. Proceedings of the London Mathematical Society, 127(5), pp. 1507-1576. (doi: 10.1112/plms.12565)

Abstract

We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. 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. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels–Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen–Stoll. We establish the local formula in many instances and show that in remaining cases, it follows from standard global conjectures.

• Item Type: Articles
• Additional Information: Parts of this work were completed while the author was supported by the Engineering and Physical Sciences Research Council (EPSRC) grants EP/M016846/1 `Arithmetic of hyperelliptic curves', and EP/V006541/1 `Selmer groups, Arithmetic Statistics and Parity Conjectures'.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Morgan, Dr Adam
• Authors: Morgan, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Proceedings of the London Mathematical Society
• Publisher: Wiley for the London Mathematical Society
• ISSN: 0024-6115
• ISSN (Online): 1460-244X
• Published Online: 30 September 2023
• Copyright Holders: Copyright: © 2023 The Authors
• First Published: First published in Proceedings of the London Mathematical Society 127(5): 1507-1576
• Publisher Policy: Reproduced under a Creative Commons licence

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
310324		Adam Morgan Postdoc fellowship	Adam Morgan	Engineering and Physical Sciences Research Council (EPSRC)	EP/V006541/1	M&S - Mathematics