PERIODIC POINTS OF PRYM EIGENFORMS SAM FREEDMAN Abstract. A point of a Veech surface is periodicif it has a ﬁnite

2025-05-02 0 0 817.35KB 26 页 10玖币

侵权投诉

PERIODIC POINTS OF PRYM EIGENFORMS

SAM FREEDMAN

Abstract.

A point of a Veech surface is periodic if it has a ﬁnite

orbit under the surface’s aﬃne automorphism group. We show that

the periodic points of Prym eigenforms in the minimal strata of

translation surfaces in genera 2, 3 and 4 are the ﬁxed points of the

Prym involution. This answers a question of Apisa–Wright and

gives a geometric proof of Möller’s classiﬁcation of periodic points

of Veech surfaces in the minimal stratum in genus 2.

1. Introduction

Aperiodic point of a Veech surface

is a point

that has ﬁnite

orbit under the aﬃne automorphism group

Aﬀ+

(

). Examples of

periodic points are the singular points of

and the ﬁxed points of

the hyperelliptic involution. Chowdhury–Everett–Freedman–Lee [5]

produced an algorithm that inputs a (non-square-tiled) Veech surface

and outputs its set of periodic points. They used this algorithm to

conjecture that the periodic points of certain Veech surfaces

(

w, e

Prym eigenforms in the minimal stratum in genus 3, are the ﬁxed points

of an involution. In this article we prove that conjecture as well as give

a uniform classiﬁcation of periodic points of Prym eigenforms in the

minimal strata of genus

translation surfaces Ω

(2)

Ω

(4) and

ΩM4(6):

Theorem 1.1.

The periodic points of a nonarithmetic Prym eigenform

in the stratum Ω

g−

2) for 2

≤g≤

4are the ﬁxed points of its

Prym involution.

For an overview of the proof, see §4.1.Theorem 1.1 answers Problem

1.5 in Apisa–Wright [4] for Prym eigenforms in minimal strata. The

genus 2 case gives a new proof of the classiﬁcation of periodic points on

nonarithmetic Veech surfaces in Ω

(2), due originally to Möller [17]:

Corollary 1.2.

The periodic points of a nonarithmetic Veech surface

in ΩM2(2) are the Weierstrass points.

Date: November 2022.

Key words and phrases. Periodic points, Prym eigenforms, Veech surfaces, Teich-

müller dynamics.

arXiv:2210.13503v1 [math.GT] 24 Oct 2022

2 SAM FREEDMAN

Figure 1.1. The L-shaped, S-shaped and X-shaped

Prym eigenforms considered in this paper.

By the classiﬁcation of the connected components of the Prym eigen-

form loci due to McMullen [15] and Lanneau–Nguyen [10] [11], to prove

Theorem 1.1 it suﬃces to determine the periodic points of one pro-

totypical eigenform

(

w, e

)in each connected component of the loci.

Here

w, e ∈Z

are particular choices of integer prototypes; see Figure 1.1

for examples in each genus. Although each prototype

(

w, e

)could a

priori require an individual treatment, when the discriminant

D:=





e2+ 4w, g ∈2,4

e2+ 8w, g = 3

is suﬃciently large we give a uniform argument using explicit aﬃne

automorphisms called butterﬂy moves. McMullen [15] ﬁrst introduced

these move to classify the number of connected components of diﬀerent

Teichmüller curves in the strata Ω

(2). We use butterﬂy moves as

concrete elements of

Aﬀ+

(

w, e

)) to rule out large regions of the

surface from containing periodic points.

Applications of Theorem 1.1.

Periodic points are central in Teich-

müller dynamics because they represent nongeneric behavior for the

)-action on strata of translation surfaces. Here, nongeneric

means that the orbit closure of a surface with a marked periodic point

PERIODIC POINTS OF PRYM EIGENFORMS 3

(

M, p

)in a stratum of marked translation surfaces has smaller than

expected dimension. We give three applications here:

•Counting holomorphic sections of surface bundles over

Teichmüller curves

: Veech surfaces generate certain surface

bundles over (covers of) Teichmüller curves. (See, e.g., Apisa

[2].) Shinomiya [18] proved that a holomorphic section of such

bundles is uniquely determined by a choice of periodic point

on the generating Veech surface. Combining Shinomiya’s result

with Theorem 1.1, we can restrict the number of holomorphic

sections of those bundles generated by Prym eigenforms:

Corollary 1.3.

The surface bundles generated by Prym eigen-

forms in the minimal strata in genera 2

≤g≤

4have 10

−

distinct holomorphic sections.

•Solving the ﬁnite-blocking problem

: Two points

and

on a translation surface

are ﬁnitely blocked if there is a ﬁnite

set

B⊂M

such that all straight line paths from

contain

a point of

. Apisa–Wright [4, Theorem 3.6] showed that if

and

are ﬁnitely blocked, then either both

and

are periodic

points, or

πQmin

(

) =

πQmin

(

)where

πQmin

M→Qmin

is a

certain degree 1 or 2 covering of a half-translation surface (c.f.

Apisa–Wright [4, Lemma 3.3]). We can then follow the proof of

Apisa–Saavedra–Zhang [3, Corollary 1.6] to show

Corollary 1.4.

The pairs of points on a Prym eigenform in

the minimal stratum in genus 2, 3 or 4 that are ﬁnitely blocked

from each other are a nonsingular point and its image under the

Prym involution.

•Evidence for higher-rank orbit closures

: Apisa [2] ﬁnished

the classiﬁcation of periodic points on primitive genus 2 transla-

tion surfaces that Möller [17] initiated. By ﬁnding degenerations

to lower genus eigenforms with marked points in the bound-

ary, Apisa showed that the gothic locus that Eskin–McMullen–

Mukamel–Wright [7] constructed is the unique nonarithmetic

rank two aﬃne invariant subvariety in Ω

(6). It seems in-

teresting whether one can combine Theorem 1.1 with similar

degeneration arguments to classify periodic points on Prym

eigenforms in nonminimal strata and restrict the existence of

other higher-rank orbit closures.

Previous Work.

Gutkin–Hubert–Schmidt [8] ﬁrst showed that (nonar-

ithmetic) closed

)-orbits have a ﬁnite number of periodic points

4 SAM FREEDMAN

(see Chowdhury–Everett–Freedman–Lee [5] for another proof). Eskin–

Filip–Wright [6] then showed ﬁniteness for all nonarithmetic aﬃne

invariant subvarieties.

Authors have computed this ﬁnite set of periodic points for other

Veech surfaces: Möller [17] for Veech surfaces in genus 2, Apisa [2] for

nonarithmetic eigenform loci in genus 2, Apisa [1] for components of

strata of translation surfaces, Apisa–Saavedra–Zhang [3] for regular

-gons and double (2

+ 1)-gons, and B. Wright [20] for Veech–Ward

surfaces.

Remark 1.5.All known periodic points on Veech surfaces are the ﬁxed

points of an involution. It would be interesting to see if this holds for the

Veech surfaces in the gothic loci due to Eskin–Möller–Mukamel–Wright

[7].

The philosophy of the algorithm in Chowdhury–Everett–Freedman–

Lee [5] is that the Rational Height Lemma (see Lemma 2.1) applied

with three “independent” parabolic directions is enough to determine

the periodic points of the surface. Our argument in genus two uses

three such directions, yet our arguments in genera four and ﬁve use

slightly more. Whether there is another argument using three directions

in these latter cases, and formally proving this heuristic, seems worthy

of future study.

Outline of Paper.

In §2, we give background on ﬂat geometry, Prym

eigenforms, butterﬂy moves, and periodic points.

As discussed above, the work of McMullen [15] and Lanneau–Nguyen

[10,11] allows us to consider speciﬁc choices of Prym eigenforms. We

choose certain prototypes

(

w, e

)that have evident horizontal and

vertical cylinder decompositions, each with a corresponding global

multi-twist

and

respectively. In §3 we use Lemma 3.2 to classify

the points of M(w, e)having ﬁnite orbit under the subgroup hTH, TVi.

This reduces the problem to considering the periodic points on certain

horizontal and vertical boundary saddle connections of

(

w, e

), which

we carry out in §4.2,§4.3 and §4.4 for genera 2, 3 and 4 respectively.

For this, we use butterﬂy moves

(see Figure 2.1 for an example)

to produce new cylinder directions on

(

w, e

)with which we can rule

out points from being periodic. In fact the butterﬂy move

yields

a new prototypical eigenform

(

w, e

)

Bq·M

(

w, e

), onto which

we can transfer the candidate periodic points. One diﬃculty is that

some prototypes (

w, e

)determine eigenforms

(

w, e

)that are not

again rectilinear. But in Appendix A we show that, in each connected

component of Prym eigenforms, we can ﬁnd a good prototype

(

w, e

)

such that

(

w, e

)is rectilinear. Under the assumption that

(

w, e

)

PERIODIC POINTS OF PRYM EIGENFORMS 5

is good, we can analyze its candidate points on the new prototype

M0(w, e)to ﬁnish the classiﬁcation.

Acknowledgements. Part of this work took place at Univerté Paris-

Saclay under a Fondation Mathématique Jacques Hadamard Junior

Scientiﬁc Visibility Grant. We thank Paul Apisa, Jeremy Kahn, Samuel

Lelièvre and Duc-Manh Nguyen for helpful conversations, as well as

Vincent Delecroix and Julian Rüth for their collaboration and support

in working with Flatsurf. We thank Ethan Dlugie, Joseph Hlavinka,

Siddarth Kannan and Jordan Katz for comments and discussions on

earlier drafts.

2. Background

2.1.

Flat Geometry.

General surveys on translation surfaces are

Zorich [21], Wright [19] and Massart [12].

Asaddle connection

on a translation surface is a straight-line locally

geodesic segment connecting two cone points and having no cone points

in its interior. Each saddle connection has an associated vector in

deﬁned up to ±1called its holonomy vector.

Acylinder

is the isometric image of a right Euclidean cylinder hav-

ing a union of saddle connections for each boundary component. Cylin-

ders have a foliation by homotopic closed trajectories of the straight-line

ﬂow that are parallel to the boundary saddle connections, determining

adirection for the cylinder. The cylinder is simple if each boundary

component consists of a single saddle connection. For example, the

top square horizontal cylinder of Figure 4.1 is simple, yet the bottom

rectangular horizontal cylinder is not. The total length of its boundary

components is its circumference

(

), and the perpendicular distance

between them is its height

(

). The ratio

(

)

(

)

(

)is

its modulus. A cylinder decomposition of

is a collection of parallel

cylinders whose union is M.

2.2.

Veech Surfaces and Multi-Twists.

A reference for this section

is Hubert–Schmidt [9].

An aﬃne automorphism of

is a self-diﬀeomorphism that, in

charts from the translation atlas of

, has the form

x7→ Ax

with

A∈SL

)and

b∈R2

; we write

Aﬀ+

(

)for the group of

orientation-preserving aﬃne automorphisms. One checks that the ma-

trix

is independent of choice of coordinates, giving a well-deﬁned

map derivative map

Aﬀ+

(

)

→SL

). The Veech group

(

)

is the image of

Aﬀ+

(

)in

). For example, the Veech group of

the square torus is SL(2,Z).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

PERIODICPOINTSOFPRYMEIGENFORMSSAMFREEDMANAbstract.ApointofaVeechsurfaceisperiodicifithasaniteorbitunderthesurface'saneautomorphismgroup.WeshowthattheperiodicpointsofPrymeigenformsintheminimalstrataoftranslationsurfacesingenera2,3and4arethexedpointsofthePryminvolution.ThisanswersaquestionofApisaW...

展开>> 收起<<

PERIODIC POINTS OF PRYM EIGENFORMS SAM FREEDMAN Abstract. A point of a Veech surface is periodicif it has a ﬁnite.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

PERIODIC POINTS OF PRYM EIGENFORMS SAM FREEDMAN Abstract. A point of a Veech surface is periodicif it has a ﬁnite

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: