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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Generalized Jacobians and explicit descents

Author: Brendan Creutz
Journal: Math. Comp. 89 (2020), 1365-1394
MSC (2010): Primary 11G10, 11G30, 14605
Published electronically: November 15, 2019
MathSciNet review: 4063321
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Abstract: We develop a cohomological description of explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $ n$ dividing the degree of some reduced, effective, and base point free divisor $ \frak {m}$ on a curve $ C$, we show that multiplication by $ n$ on the generalized Jacobian $ J_\frak {m}$ factors through an isogeny $ \varphi :A_\frak {m} \to J_\frak {m}$ whose kernel is dual to the Galois module of divisor classes $ D$ such that $ nD$ is linearly equivalent to some multiple of $ \frak {m}$. By geometric class field theory, this corresponds to an abelian covering of $ C_{\overline {k}} := C \times _{\mathrm {Spec}{k}} \mathrm {Spec}(\overline {k})$ of exponent $ n$ unramified outside $ \frak {m}$. We show that the $ n$-coverings of $ C$ parameterized by explicit descents are the maximal unramified subcoverings of the $ k$-forms of this ramified covering. We present applications to the computation of Mordell-Weil ranks of nonhyperelliptic curves.

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Additional Information

Brendan Creutz
Affiliation: School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Received by editor(s): November 6, 2018
Received by editor(s) in revised form: August 13, 2019, and September 1, 2019
Published electronically: November 15, 2019
Article copyright: © Copyright 2019 American Mathematical Society