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

Joe Kramer-Miller

doi:10.5802/jtnb.1159

Some unlikely intersections between the Torelli locus and Newton strata in

𝒜_{g}

Joe Kramer-Miller ¹

¹ University of California, Irvine Department of Mathematics 510 V Rowland Hall Irvine CA, 92697

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 237-250.

Abstract
Résumé

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 $ℤ / p ℤ$ -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 $\frac{2 g}{3} - \frac{2 p (p - 1)}{3} \geq 2 k (p - 1)$ , there exists a curve of genus $g$ whose Newton polygon has slopes ${0, 1}^{g - k (p - 1)} ⊔ {\frac{1}{2}}^{2 k (p - 1)}$ . 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 ${C_{g}}_{g \geq 1}$ , 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 $y = \frac{x^{2}}{4 g}$ . The proof uses a Newton-over-Hodge result for $ℤ / p ℤ$ -covers of curves due to the author, in addition to recent work of Booher–Pries on the realization of this Hodge bound.

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 $ℤ / p ℤ$ . 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 $\frac{2 g}{3} - \frac{2 p (p - 1)}{3} \geq 2 k (p - 1)$ , il existe une courbe $C$ de genre $g$ telle que les pentes de Newton de $C$ sont ${0, 1}^{g - k (p - 1)} ⊔ {\frac{1}{2}}^{2 k (p - 1)}$ . 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 ${C_{g}}_{g \geq 1}$ 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 $y = \frac{x^{2}}{4 g}$ . 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 $ℤ / p ℤ$ , dû à l’auteur, et un travail récent de Booher–Pries qui montre que cette borne de Hodge est atteinte.

Received: 2020-07-01
Revised: 2020-10-04
Accepted: 2020-12-13
Published online: 2021-05-21

DOI: 10.5802/jtnb.1159

Classification: 11G20, 11M38, 14K10
Keywords: Newton polygons of curves, Artin–Schreier curves, Torelli locus

Author's affiliations:

Joe Kramer-Miller ¹

¹ University of California, Irvine Department of Mathematics 510 V Rowland Hall Irvine CA, 92697

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Some unlikely intersections between the {Torelli} locus and {Newton} strata in $\protect \mathcal{A}_g$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {237--250},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Joe Kramer-Miller. Some unlikely intersections between the Torelli locus and Newton strata in $\protect \mathcal{A}_g$. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 237-250. doi : 10.5802/jtnb.1159. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1159/

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