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Newton Polygons of Cyclic Covers of the Projective Line Branched at Three Points

Li, Wanlin and Mantovan, Elena and Pries, Rachel and Tang, Yunqing (2019) Newton Polygons of Cyclic Covers of the Projective Line Branched at Three Points. In: Research Directions in Number Theory - Women in Numbers IV. Association for Women in Mathematics Series. No.19. Springer , Cham, pp. 115-132. ISBN 978-3-030-19477-2.

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We review the Shimura–Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce multiple new examples of Newton polygons that occur for Jacobians of smooth curves in characteristic p. Under certain congruence conditions on p, these include: the supersingular Newton polygon for each genus g with 4 ≤ g ≤ 11; nine non-supersingular Newton polygons with p-rank 0 with 4 ≤ g ≤ 11; and, for all g ≥ 5, the Newton polygon with p-rank g − 5 having slopes 1∕5 and 4∕5.

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Additional Information:© 2019 The Author(s) and The Association for Women in Mathematics. First Online: 02 August 2019. This project began at the Women in Numbers 4 workshop at the Banff International Research Station. Pries was partially supported by NSF grant DMS-15-02227. We thank the referee for the valuable feedback and comments.
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Subject Keywords:Curve; Cyclic cover; Jacobian; Abelian variety; Moduli space; Reduction; Supersingular; Newton polygon; p-rank; Dieudonné module; p-divisible group; Complex multiplication; Shimura–Taniyama method
Series Name:Association for Women in Mathematics Series
Issue or Number:19
Classification Code:MSC10: Primary: 11G20; 11M38; 14G10; 14H40; 14K22; Secondary: 11G10; 14H10; 14H30; 14H40
Record Number:CaltechAUTHORS:20190801-153624143
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97611
Deposited By: Tony Diaz
Deposited On:02 Aug 2019 14:41
Last Modified:16 Nov 2021 17:33

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