Newton polygons arising from special families of cyclic covers of the projective line

Li, Wanlin; Mantovan, Elena; Pries, Rachel; Tang, Yunqing

doi:10.1007/s40993-018-0149-3

Cited by 16 publications

(22 citation statements)

References 33 publications

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“…Using this analysis, they give an inductive process to construct arithmetic families of curves with prescribed Newton polygon. Combining this with their previous results on special subvarieties of Shimura varieties (see [10]) gives many interesting examples.…”

Section: The Torelli Locussupporting

confidence: 56%

Some unlikely intersections between the Torelli locus and Newton strata in $\mathcal{A}_g$

Kramer-Miller

2020

Preprint

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Let p be an odd prime. What are the possible Newton polygons for a curve in characteristic p? Equivalently, which Newton strata intersect the Torelli locus in A g ? In this note, we study the Newton polygons of certain curves with Z/pZ-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in A g . Here is one example of particular interest: fix a genus g. We show that for any k with 2g 3 − 2p(p−1) 3 ≥ 2k(p − 1), there exists a curve of genus g whose Newton polygon has slopes {0, 1) . This provides evidence for Oort's conjecture that the amalgamation of the Newton polygons of two curves is again the Newton polygon of a curve. We also construct families of curves {C g } g≥1 , where C g is a curve of genus g, whose Newton polygons have interesting asymptotic properties. For example, we construct a family of curves whose Newton polygons are asymptotically bounded below by the graph y = x 2 4g . The proof uses a Newton-over-Hodge result for Z/pZ-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.

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Section: The Torelli Locussupporting

confidence: 56%

Some unlikely intersections between the Torelli locus and Newton strata in $\mathcal{A}_g$

Kramer-Miller

2020

Preprint

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“…The issue is that a generic tame cyclic cover of degree N is not ordinary, even if X is ordinary [6]. Even when 𝑋 = P 1 , the study of Newton polygons for tame cyclic covers is already a complicated topic (e.g., [16]).…”

Section: Further Remarksmentioning

confidence: 99%

p-Adic estimates of abelian ArtinL-functions on curves

Kramer-Miller

2022

Forum of Mathematics, Sigma

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The purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. Let X be a smooth proper curve over a finite field $\mathbb {F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $\rho :\pi _1^{et}(V) \to \mathbb {C}^{\times }$ . In this article, we prove a lower bound on the Newton polygon of the L-function $L(\rho ,s)$ . The estimate depends on monodromy invariants of $\rho $ : the Swan conductor and the local exponents. Under certain nondegeneracy assumptions, this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne’s prediction that the irregular Hodge filtration would force p-adic bounds on L-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

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“…Thus the knowledge of I a p is needed to determine the level of these automorphic forms. In [LMPT19], the authors used a formula for the number of irreducible components in the supersingular locus of unitary Shimura varieties to prove results on the image of the Torelli map. This requires information on the volume of I a p .…”

Section: Theorem B (See Corollary 523) There Exists a Bijection Betwe...mentioning

confidence: 99%

Stabilizers of irreducible components of affine Deligne--Lusztig varieties

He,

Zhou,

Zhu

2021

Preprint

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We study the J b (F )-action on the set of top-dimensional irreducible components of affine Deligne-Lusztig varieties in the affine Grassmannian. We show that the stabilizer of any such component is a parahoric subgroup of J b (F ) of maximal volume, verifying a conjecture of X. Zhu. As an application, we give a description of the set of top-dimensional irreducible components in the basic locus of Shimura varieties.

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Newton polygons arising from special families of cyclic covers of the projective line

Cited by 16 publications

References 33 publications

Some unlikely intersections between the Torelli locus and Newton strata in $\mathcal{A}_g$

Some unlikely intersections between the Torelli locus and Newton strata in $\mathcal{A}_g$

p-Adic estimates of abelian ArtinL-functions on curves

Stabilizers of irreducible components of affine Deligne--Lusztig varieties

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