Some unlikely intersections between the Torelli locus and Newton strata in 𝒜 g
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 237-250.

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 /p-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) 32k(p-1), there exists a curve of genus g whose Newton polygon has slopes {0,1} g-k(p-1) {1 2} 2k(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 } g1 , 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 /p-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 /p. 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 3-2p(p-1) 32k(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) {1 2} 2k(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 } g1 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 /p, dû à l’auteur, et un travail récent de Booher–Pries qui montre que cette borne de Hodge est atteinte.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1159
Classification: 11G20, 11M38, 14K10
Keywords: Newton polygons of curves, Artin–Schreier curves, Torelli locus
Kramer-Miller, Joe 1

1 University of California, Irvine Department of Mathematics 510 V Rowland Hall Irvine CA, 92697
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Kramer-Miller, Joe. 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. http://archive.numdam.org/articles/10.5802/jtnb.1159/

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