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 $ℤ/pℤ$-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 $\frac{2g}{3}-\frac{2p\left(p-1\right)}{3}\ge 2k\left(p-1\right)$, there exists a curve of genus $g$ whose Newton polygon has slopes ${\left\{0,1\right\}}^{g-k\left(p-1\right)}\bigsqcup {\left\{\frac{1}{2}\right\}}^{2k\left(p-1\right)}$. 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 ${\left\{{C}_{g}\right\}}_{g\ge 1}$, 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 $y=\frac{{x}^{2}}{4g}$. 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.

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 $\frac{2g}{3}-\frac{2p\left(p-1\right)}{3}\ge 2k\left(p-1\right)$, il existe une courbe $C$ de genre $g$ telle que les pentes de Newton de $C$ sont ${\left\{0,1\right\}}^{g-k\left(p-1\right)}\bigsqcup {\left\{\frac{1}{2}\right\}}^{2k\left(p-1\right)}$. 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 ${\left\{{C}_{g}\right\}}_{g\ge 1}$ 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 $y=\frac{{x}^{2}}{4g}$. 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.

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

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

[1] Jeffrey D. Achter; Rachel Pries Generic Newton polygons for curves of given $p$-rank, Algebraic curves and finite fields (Radon Series on Computational and Applied Mathematics), Volume 16, Walter de Gruyter, 2014, pp. 1-21 | MR | Zbl

[2] Régis Blache; Éric Férard Newton stratification for polynomials: the open stratum, J. Number Theory, Volume 123 (2007) no. 2, pp. 456-472 | DOI | MR | Zbl

[3] Jeremy Booher; Rachel Pries Realizing Artin–Schreier covers of curves with minimal Newton polygons in positive characteristic, J. Number Theory, Volume 214 (2020), pp. 240-250 | DOI | MR | Zbl

[4] Irene I. Bouw The $p$-rank of ramified covers of curves, Compos. Math., Volume 126 (2001) no. 3, pp. 295-322 | DOI | MR | Zbl

[5] Gunther Cornelissen; Frans Oort Problems from the workshop on Automorphisms of Curves (Leiden, August, 2004), Rend. Semin. Mat. Univ. Padova, Volume 113 (2005), pp. 129-177 | Numdam | MR | Zbl

[6] Richard Crew Etale $p$-covers in characteristic $p$, Compos. Math., Volume 52 (1984) no. 1, pp. 31-45 | Numdam | MR | Zbl

[7] Pierre Deligne; Bernard Malgrange; Jean-Pierre Ramis Singularités irrégulières. Correspondance et documents, Documents Mathématiques, 5, Société Mathématique de France, 2007, xii+188 pages | MR | Zbl

[8] Gerard van der Geer; Marcel van der Vlugt On the existence of supersingular curves of given genus, J. Reine Angew. Math., Volume 458 (1995), pp. 53-61 | MR | Zbl

[9] Nicholas Katz Slope filtration of $F$-crystals, Groupes formels, représentations galoisiennes et cohomologie des variétés de caractéristique positive. I (Astérisque), Volume 63, Société Mathématique de France, 1979, pp. 113-164 | MR | Zbl

[10] Joe Kramer-Miller $p$-adic estimates of exponential sums on curves (to appear in Algebra Number Theory) | MR | Zbl

[11] Wanlin Li; Elena Mantovan; Rachel Pries; Yunqing Tang Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties (2018) (https://arxiv.org/abs/1811.00604)

[12] Wanlin Li; Elena Mantovan; Rachel Pries; Yunqing Tang Newton polygons arising from special families of cyclic covers of the projective line, Res. Number Theory, Volume 5 (2019) no. 1, 12, 31 pages | DOI | MR | Zbl

[13] Chunlei Liu; Dasheng Wei The $L$-functions of Witt coverings, Math. Z., Volume 255 (2007) no. 1, pp. 95-115 | DOI | MR | Zbl

[14] Frans Oort Newton polygon strata in the moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999) (Progress in Mathematics), Volume 195, Birkhäuser, 2001, pp. 417-440 | DOI | MR | Zbl

[15] Rachel Pries Current results on Newton polygons of curves, Open problems in arithmetic algebraic geometry (Advanced Lectures in Mathematics (ALM)), Volume 46, International Press, 2019, pp. 179-207 | MR | Zbl

[16] Philippe Robba Index of $p$-adic differential operators. III. Application to twisted exponential sums, Cohomologie $p$-adique (Astérisque), Volume 119-120, Société Mathématique de France, 1984, pp. 191-266 | Numdam | MR | Zbl

[17] Hui June Zhu $p$-adic variation of $L$ functions of one variable exponential sums. I, Am. J. Math., Volume 125 (2003) no. 3, pp. 669-690 | DOI | MR | Zbl

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