Realizing Artin-Schreier covers of curves with minimal Newton polygons in positive characteristic

Booher, Jeremy; Pries, Rachel

doi:10.1016/j.jnt.2020.04.010

Cited by 10 publications

(9 citation statements)

References 9 publications

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“…In the case of an Artin-Schreier cover, we are able to recover a result of Booher and Pries (see [3,Corollary 4.3]). In fact, we obtain a much stronger result: In [3], Booher and Pries show that there exists an Artin-Schreier cover X ′ → X such that the q-adic Newton polygon of X ′ has slope set 0, . .…”

Section: Resultssupporting

confidence: 74%

Newton Polygons of Sums on Curves I: Local-to-Global Theorems

Kramer-Miller¹,

Upton²

2021

Preprint

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The purpose of this article is to study Newton polygons of certain abelian L-functions on curves. Let X be a smooth affine curve over a finite field F q and let ρ : π 1 (X) → C × p be a finite character of order p n . By previous work of the first author, the Newton polygon NP(ρ) lies above a 'Hodge polygon' HP(ρ), which is defined using local ramification invariants of ρ. In this article we study the touching between these two polygons. We prove that NP(ρ) and HP(ρ) share a vertex if and only if a corresponding vertex is shared between the Newton and Hodge polygons of 'local' L-functions associated to each ramified point of ρ. As a consequence, we determine a necessary and sufficient condition for the coincidence of NP(ρ) and HP(ρ).

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Section: Resultssupporting

confidence: 74%

Newton Polygons of Sums on Curves I: Local-to-Global Theorems

Kramer-Miller¹,

Upton²

2021

Preprint

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“…This is a clear advantage over earlier techniques. For the general statement of Theorem 4.1, we also need to use recent work of Booher-Pries (see [3]). This work shows the lower bound in [10] is realized when certain congruence conditions between p and the Swan conductors hold.…”

Section: Methods Of Proofmentioning

confidence: 99%

“…Proof. See [3,Corollary 4.3]. The main idea is as follows: work of Blache-Ferard computes the Newton polygon of a generic Z/pZ-cover of P 1 ramified only at ∞ when the Swan conductor is less than 3p.…”

Section: Z/pz-covers Of Curvesmentioning

confidence: 99%

Some unlikely intersections between the Torelli locus and Newton strata in 𝒜 g

Kramer-Miller¹

2021

Journal De Théorie Des Nombres De Bordeaux

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L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 33 (2021), 237-250 Some unlikely intersections between the Torelli locus and Newton strata in A g par Joe KRAMER-MILLER Résumé. Soit p un nombre premier impair. Quels sont les polygones de Newton possibles pour les courbes en caractéristique p ? Autrement dit, quelles sont les strates de Newton qui s'intersectent avec le lieu de Torelli dans A g ? Nous étudions les polygones de Newton de certaines courbes équipées d'une action du groupe fini Z/pZ. Plusieurs de ces courbes fournissent des exemples d'intersections improbables entre le lieu de Torelli et la stratification de Newton dans A g . Voici un exemple qui présente un intérêt particulier : en fixant un genre g > 1, nous montrons que pour tout k tel que 2g 3 1) . Cela confirme une conjecture d'Oort selon laquelle l'amalgamation des polygones de Newton de deux courbes est aussi le polygone de Newton d'une courbe. Nous construisons aussi quelques familles de courbes {C g } g≥1 de genre g, dont les polygones asymptotiques de Newton sont intéressants. Par exemple, nous construisons une famille de courbes dont le polygone asymptotique de Newton est minoré par y = x 2 4g . Les outils principaux de l'article sont un résultat « polygone de Newton est situé au-dessus du polygone de Hodge » pour les courbes équipées d'une action de Z/pZ, dû à l'auteur, et un travail récent de Booher-Pries qui montre que cette borne de Hodge est atteinte.

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“…We believe the bound from Theorem 1.1 should only be generically attained if 𝑁 | 𝑝 − 1 and there are some congruence relations between p and the Swan conductors. When 𝜌 factors through an Artin-Schreier cover, this is known by combining work of the author [15] with work of Booher and Pries [5]. The next step would be to study the case arising from a cyclic cover whose degree divides 𝑝( 𝑝 − 1) (or even allowing higher powers of p).…”

Section: Further Remarksmentioning

confidence: 99%

“…Since the map 𝐴 → 𝐵 is étale and both rings are p-adically complete, we may extend both 𝜎 and 𝜈 to maps 𝜎, 𝜈 : 𝐵 → 𝐵. This extends to a p-Frobenius endomorphism 𝜈 𝑄 of E 𝑄 , which makes the diagrams (5)…”

Section: Global Frobenius and 𝑈 𝑝 Operatorsmentioning

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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Realizing Artin-Schreier covers of curves with minimal Newton polygons in positive characteristic

Cited by 10 publications

References 9 publications

Newton Polygons of Sums on Curves I: Local-to-Global Theorems

Newton Polygons of Sums on Curves I: Local-to-Global Theorems

Some unlikely intersections between the Torelli locus and Newton strata in 𝒜 g

p-Adic estimates of abelian ArtinL-functions on curves

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