text: So, the ﬂight time ﬂuctuations are ν times those in a normal medium. For a typical nonlinear medium, it has been estimat√ed in Ref. [22] that the root mean square of the − ∼ fractional ﬂight time ﬂuctuation δ = δ2 is of the order of 10 8 when d 10 µm, which is rms potentially observable. Our result shows that the detection of the eﬀect would become easier with an anisotropic metamaterial that mimics a cosmic string. In principle, an observable light cone ﬂuctuation within the current experiment precision would be obtained if the analogue cosmic string could be fabricated with a large enough ν in laboratory. (cid:29) When the trajectory of the probe light is far from the analogue string, i.e. ρ d, we have ≈ n (χ(2))2  δ2 B − 0 (cid:88) (cid:90) 16π2(n2 n2 )2n4 d4  P B P ∞ − (χ(2))2 [ν/2] (cid:48) 1 − ν sin νπ 1 1 0 dζ − . (26) 8π2n3 n4 ρ4 sin4 mπ 2π cosh νζ cos νπ cosh4 ζ B P m=1 ν 0 2 (cid:29) Therefore, when ρ d, the corrections to the ﬂight time variance due to the presence − of an analogue cosmic string (the second term in Eq. (26)) is proportional to ρ 4, which is a higher-order correction compared with that in a normal isotropic medium which is − proportional to d 4. For an integer ν, the result can be further calculated as − ≈ n (χ(2))2 − (χ(2))2 (ν4 + 10ν2 11) δ2 B − 0 0 , (27) 16π2(n2 n2 )2n4 d4 720π2n3 n4 ρ4 P B P B P where we have used the relation (cid:88) − ν 1 1 1 − = (ν4 + 10ν2 11) . (28) sin4 mπ 45 m=1 ν It is interesting to note that the correction term is negative and so the ﬂight time ﬂuctu- ations of the probe light is weakened as compared with those in a normal medium. The discussions above suggest that the ﬂight time ﬂuctuations for a probe light can be ampliﬁed by a factor of ν when its trajectory is close to the analogue cosmic string. Now 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: So,

text: the

text: ﬂight

text: time

text: ﬂuctuations

text: are

text: ν

text: times

text: those

text: in

text: a

text: normal

text: medium.

text: For

text: a

text: typical

text: nonlinear

text: medium,

text: it

text: has

text: been

text: estimat√ed

text: in

text: Ref.

text: that

text: the

text: root

text: mean

text: square

text: of

text: the

text: fractional

text: ﬂight

text: time

text: ﬂuctuation

text: δ

text: δ2

text: is

text: of

text: the

text: order

text: of

text: when

text: d

text: µm,

text: which

text: is

text: rms

text: potentially

text: observable.

text: Our

text: result

text: shows

text: that

text: the

text: detection

text: of

text: the

text: eﬀect

text: would

text: become

text: easier

text: with

text: an

text: anisotropic

text: metamaterial

text: that

text: mimics

text: a

text: cosmic

text: string.

text: In

text: principle,

text: an

text: observable

text: light

text: cone

text: ﬂuctuation

text: within

text: the

text: current

text: experiment

text: precision

text: would

text: be

text: obtained

text: if

text: the

text: analogue

text: cosmic

text: string

text: could

text: be

text: fabricated

text: with

text: a

text: large

text: enough

text: ν

text: in

text: laboratory.

text: When

text: the

text: trajectory

text: of

text: the

text: probe

text: light

text: is

text: far

text: from

text: the

text: analogue

text: string,

text: i.e.

text: ρ

text: d,

text: we

text: have

text: n

text: δ2

text: B

text: P

text: B

text: P

text: ν

text: sin

text: νπ

text: dζ

text: ρ4

text: mπ

text: 2π

text: cosh

text: νζ

text: cos

text: νπ

text: ζ

text: B

text: P

text: ν

text: Therefore,

text: when

text: ρ

text: d,

text: the

text: corrections

text: to

text: the

text: ﬂight

text: time

text: variance

text: due

text: to

text: the

text: presence

text: of

text: an

text: analogue

text: cosmic

text: string

text: (the

text: second

text: term

text: in

text: Eq.

text: is

text: proportional

text: to

text: ρ

text: which

text: is

text: a

text: higher-order

text: correction

text: compared

text: with

text: that

text: in

text: a

text: normal

text: isotropic

text: medium

text: which

text: is

text: proportional

text: to

text: d

text: For

text: an

text: integer

text: ν,

text: the

text: result

text: can

text: be

text: further

text: calculated

text: as

text: n

text: δ2

text: B

text: ρ4

text: P

text: B

text: P

text: B

text: P

text: where

text: we

text: have

text: used

text: the

text: relation

text: ν

text: mπ

text: ν

text: It

text: is

text: interesting

text: to

text: note

text: that

text: the

text: correction

text: term

text: is

text: negative

text: and

text: so

text: the

text: ﬂight

text: time

text: ﬂuctu-

text: ations

text: of

text: the

text: probe

text: light

text: is

text: weakened

text: as

text: compared

text: with

text: those

text: in

text: a

text: normal

text: medium.

text: The

text: discussions

text: above

text: suggest

text: that

text: the

text: ﬂight

text: time

text: ﬂuctuations

text: for

text: a

text: probe

text: light

text: can

text: be

text: ampliﬁed

text: by

text: a

text: factor

text: of

text: ν

text: when

text: its

text: trajectory

text: is

text: close

text: to

text: the

text: analogue

text: cosmic

text: string.

text: Now

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: IV. SUMMARY In this paper, we have investigated the lightcone ﬂuctuations in an anisotropic meta- material of an analogue cosmic string. In the absence of the probe ﬁeld, the medium the ﬂuctuating background ﬁeld propagates in can be treated as a linear medium that mimics the cosmic string spacetime in terms of the classical propagation of light. That is, in the frequency regime of the background ﬁeld, the second order susceptibility can be neglected. However, when the probe ﬁeld is present, the nonlinear eﬀect has to be taken account of, which leads to ﬂight time ﬂuctuations due to an eﬀective ﬂuctuating refractive index. More- over, the second order susceptibility for the probe light is supposed to be position dependent along the path of the probe light, which eﬀectively suppresses the contributions from high frequency ﬂuctuating vacuum modes as required in our model. For a probe light close to the analogue string, it has been shown that the ﬂight time ﬂuctuations of the probe light can be ampliﬁed compared with those in a normal medium and become observable if an analogue cosmic string with a large enough ν could be fabricated in laboratory. So, our study here may seem to oﬀer a possible way to manipulate lightcone ﬂuctuations, presumably the most direct eﬀect of quantum gravity, in analogue systems, and make it potentially observable. Acknowledgments.— We would like to thank Hui Liu and Chong Sheng for valuable discussions. This work was supported in part by the NSFC under Grants No. 11435006, No. 11690034, No. 11375092, and No. 11447022; the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ15A050001. [1] W. Pauli, Helv. Phys. Acta Suppl. 4, 69 (1956). [2] S. Deser, Rev. Mod. Phys. 29, 417 (1957). [3] B. S. DeWitt, Phys. Rev. Lett. 13, 114 (1964). [4] C. J. Isham, A. Salam, and J. A. Strathdee, Phys. Rev. D 3, 1805 (1971). [5] C. J. Isham, A. Salam, and J. A. Strathdee, Phys. Rev. D 5, 2548 (1972). [6] B. L. Hu and K. Shiokawa, Phys. Rev. D 57, 3474 (1998). [7] Y. J. Ng and H. van Dam, Phys. Lett. B 477, 429 (2000). [8] G. Amelino-Camelia, Phys. Rev. D 62, 024015 (2000). [9] J. R. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Gen. Relativ. Gravit. 32, 127 (2000). 13 ##LTLine## ##LTLine## ##LTLine##

text: IV.

label: section

text: SUMMARY

text: In

label: paragraph

text: this

text: paper,

text: we

text: have

text: investigated

text: the

text: lightcone

text: ﬂuctuations

text: in

text: an

text: anisotropic

text: meta-

text: material

text: of

text: an

text: analogue

text: cosmic

text: string.

text: In

text: the

text: absence

text: of

text: the

text: probe

text: ﬁeld,

text: the

text: medium

text: the

text: ﬂuctuating

text: background

text: ﬁeld

text: propagates

text: in

text: can

text: be

text: treated

text: as

text: a

text: linear

text: medium

text: that

text: mimics

text: the

text: cosmic

text: string

text: spacetime

text: in

text: terms

text: of

text: the

text: classical

text: propagation

text: of

text: light.

text: That

text: is,

text: in

text: the

text: frequency

text: regime

text: of

text: the

text: background

text: ﬁeld,

text: the

text: second

text: order

text: susceptibility

text: can

text: be

text: neglected.

text: However,

text: when

text: the

text: probe

text: ﬁeld

text: is

text: present,

text: the

text: nonlinear

text: eﬀect

text: has

text: to

text: be

text: taken

text: account

text: of,

text: which

text: leads

text: to

text: ﬂight

text: time

text: ﬂuctuations

text: due

text: to

text: an

text: eﬀective

text: ﬂuctuating

text: refractive

text: index.

text: More-

text: over,

text: the

text: second

text: order

text: susceptibility

text: for

text: the

text: probe

text: light

text: is

text: supposed

text: to

text: be

text: position

text: dependent

text: along

text: the

text: path

text: of

text: the

text: probe

text: light,

text: which

text: eﬀectively

text: suppresses

text: the

text: contributions

text: from

text: high

text: frequency

text: ﬂuctuating

text: vacuum

text: modes

text: as

text: required

text: in

text: our

text: model.

text: For

text: a

text: probe

text: light

text: close

text: to

text: the

text: analogue

text: string,

text: it

text: has

text: been

text: shown

text: that

text: the

text: ﬂight

text: time

text: ﬂuctuations

text: of

text: the

text: probe

text: light

text: can

text: be

text: ampliﬁed

text: compared

text: with

text: those

text: in

text: a

text: normal

text: medium

text: and

text: become

text: observable

text: if

text: an

text: analogue

text: cosmic

text: string

text: with

text: a

text: large

text: enough

text: ν

text: could

text: be

text: fabricated

text: in

text: laboratory.

text: So,

text: our

text: study

text: here

text: may

text: seem

text: to

text: oﬀer

text: a

text: possible

text: way

text: to

text: manipulate

text: lightcone

text: ﬂuctuations,

text: presumably

text: the

text: most

text: direct

text: eﬀect

text: of

text: quantum

text: gravity,

text: in

text: analogue

text: systems,

text: and

text: make

text: it

text: potentially

text: observable.

text: Acknowledgments.—

text: We

text: would

text: like

text: to

text: thank

text: Hui

text: Liu

text: and

text: Chong

text: Sheng

text: for

text: valuable

text: discussions.

text: This

text: work

text: was

text: supported

text: in

text: part

text: by

text: the

text: NSFC

text: under

text: Grants

text: No.

text: No.

text: No.

text: and

text: No.

text: the

text: Zhejiang

text: Provincial

text: Natural

text: Science

text: Foundation

text: of

text: China

text: under

text: Grant

text: No.

text: W.

text: Pauli,

text: Helv.

text: Phys.

text: Acta

text: Suppl.

text: S.

text: Deser,

text: Rev.

text: Mod.

text: Phys.

text: B.

text: S.

text: DeWitt,

text: Phys.

text: Rev.

text: Lett.

text: C.

text: J.

text: Isham,

text: A.

text: Salam,

text: and

text: J.

text: A.

text: Strathdee,

text: Phys.

text: Rev.

text: D

text: C.

text: J.

text: Isham,

text: A.

text: Salam,

text: and

text: J.

text: A.

text: Strathdee,

text: Phys.

text: Rev.

text: D

text: B.

text: L.

text: Hu

text: and

text: K.

text: Shiokawa,

text: Phys.

text: Rev.

text: D

text: Y.

text: J.

text: Ng

text: and

text: H.

text: van

text: Dam,

text: Phys.

text: Lett.

text: B

text: G.

text: Amelino-Camelia,

text: Phys.

text: Rev.

text: D

text: J.

text: R.

text: Ellis,

text: N.

text: E.

text: Mavromatos,

text: and

text: D.

text: V.

text: Nanopoulos,

text: Gen.

text: Relativ.

text: Gravit.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: FIG. 1. A probe light ray (dot-dashed line) propagates in the direction parallel to the analogue string located at the z-axis, from point (t , ρ, φ, z ) to point (t , ρ, φ, z ). 1 1 2 2 This integration is along the path of the probe pulse, i.e. z = v t = t/n . In the present P P paper, we assume that the background ﬁeld E is the ﬂuctuating vacuum electromagnetic 0 ﬁeld in the analogue medium of a cosmic string. The ﬂuctuations of E will cause the 0 ﬂuctuations of the ﬂight time t. The relative ﬂight time variance takes the form [21, 22] (cid:90) (cid:90) (cid:16) (cid:17) (cid:16) (cid:17) (cid:104) (cid:105) − (cid:104) (cid:105) t2 t 2 1 z2 z2 (cid:48) (cid:104) (cid:48) (cid:48) (cid:105) δ2 = (cid:104) (cid:105) = dz dz χ(2) + χ(2) χ(2) + χ(2) Ei(t, (cid:126)x)Ej(t , (cid:126)x ) , t 2 4n4 d2 ρρi ρiρ ρρj ρjρ 0 0 P z1 z1 (11) (cid:104) (cid:105) where denotes the expectation value over the vacuum state, and we have assumed that (cid:104) (cid:105) Ej = 0. Here, the vacuum expectation is a summation over the contributions of vacuum 0 ﬁeld modes in all frequencies, while the derivation of the wave equation (8) is based on the assumption that the frequency of the background ﬁeld is much smaller than that of the probe ﬁeld. So, contributions from high frequency modes should be suppressed. Fortunately, this can be realized by engineering a smoothly varying second order susceptibility in the frequency regime of the probe light in the direction parallel to the analogue string. To see this, let 6 ##LTFigure## ##LTLine## ##LTLine##

text: FIG.

text: A

label: caption

text: probe

text: light

text: ray

text: (dot-dashed

text: line)

text: propagates

text: in

text: the

text: direction

text: parallel

text: to

text: the

text: analogue

text: string

text: located

text: at

text: the

text: z-axis,

text: from

text: point

text: (t

text: ρ,

text: φ,

text: z

text: to

text: point

text: (t

text: ρ,

text: φ,

text: z

text: This

text: integration

text: is

text: along

text: the

text: path

text: of

text: the

text: probe

text: pulse,

text: i.e.

text: z

text: v

text: t

text: t/n

text: In

text: the

text: present

text: P

text: P

text: paper,

text: we

text: assume

text: that

text: the

text: background

text: ﬁeld

text: E

text: is

text: the

text: ﬂuctuating

text: vacuum

text: electromagnetic

text: ﬁeld

text: in

text: the

text: analogue

text: medium

text: of

text: a

text: cosmic

text: string.

text: The

text: ﬂuctuations

text: of

text: E

text: will

text: cause

text: the

text: ﬂuctuations

text: of

text: the

text: ﬂight

text: time

text: t.

text: The

text: relative

text: ﬂight

text: time

text: variance

text: takes

text: the

text: form

text: t

text: δ2

text: dz

text: dz

text: Ei(t,

text: t

text: ρρi

text: ρiρ

text: ρρj

text: ρjρ

text: P

text: where

text: denotes

text: the

text: expectation

text: value

text: over

text: the

text: vacuum

text: state,

text: and

text: we

text: have

text: assumed

text: that

text: Ej

text: Here,

text: the

text: vacuum

text: expectation

text: is

text: a

text: summation

text: over

text: the

text: contributions

text: of

text: vacuum

text: ﬁeld

text: modes

text: in

text: all

text: frequencies,

text: while

text: the

text: derivation

text: of

text: the

text: wave

text: equation

text: is

text: based

text: on

text: the

text: assumption

text: that

text: the

text: frequency

text: of

text: the

text: background

text: ﬁeld

text: is

text: much

text: smaller

text: than

text: that

text: of

text: the

text: probe

text: ﬁeld.

text: So,

text: contributions

text: from

text: high

text: frequency

text: modes

text: should

text: be

text: suppressed.

text: Fortunately,

text: this

text: can

text: be

text: realized

text: by

text: engineering

text: a

text: smoothly

text: varying

text: second

text: order

text: susceptibility

text: in

text: the

text: frequency

text: regime

text: of

text: the

text: probe

text: light

text: in

text: the

text: direction

text: parallel

text: to

text: the

text: analogue

text: string.

text: To

text: see

text: this,

text: let

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 1 Limitation of SDMA in Ultra-Dense Small Cell Networks Junyu Liu, Min Sheng, Jiandong Li State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an, Shaanxi, 710071, China Email: junyuliu@xidian.edu.cn, {msheng, jdli}@mail.xidian.edu.cn Abstract— Beneﬁtting from multi-user gain brought by multi- the superiority of SU-BF in UDN1. antenna techniques, space division multiple access (SDMA) is II. SYSTEM MODEL capable of signiﬁcantly enhancing spatial throughput (ST) in wireless networks. Nevertheless, we show in this letter that, even We consider a downlink small cell network, where the when SDMA is applied, ST would diminish to be zero in ultra- locations of N -antenna BSs (with constant transmit power a dense networks (UDN), where small cell base stations (BSs) are P) and single-antenna users in a two-dimension plane are fully densiﬁed. More importantly, we compare the performance modeled as two independent homogeneous Poisson Point of SDMA, single-user beamforming (SU-BF) (one user is served N in each cell) and full SDMA (the number of served users equals Processes (PPPs) NΠBS = {BSi} (i ∈ ) of density λ and the number of equipped antennas). Surprisingly, it is shown that ΠU = {Uj} (j ∈ ) of density λU, respectively. If r is the SU-BF achieves the highest ST and critical density, beyond which 2-dimension distance of cellular link, the distance between the √ ST starts to degrade, in UDN. The results in this work could shed antennas of BSs and the associated users is d = r2 + ∆h2, light on the fundamental limitation of SDMA in UDN. where ∆h > 0 denotes the antenna height difference (AHD) between BSs and users − f (x) =2πλxe πλx2. (x ≥ 0) (1) I. INTRODUCTION r While network densiﬁcation is promising to improve net- Denote NU (NU ≤ Na) as the number of active users within one cell. Meanwhile, a dense user deployment is considered, work capacity, its limitation has been manifest in recent research [1]–[3]. Depending on practical system settings, it i.e., λ (cid:29) λU, such that each full-buffer BS could select at is shown that area spectral efﬁciency would decrease as base most Na users to serve using SDMA. Channel gain consists of a pathloss component and a small- station (BS) density grows in downlink cellular networks [2]. scale fading component. In particular, given transmission dis- Worsestill, network capacity is shown to diminish to zero when single-antenna BSs are over deployed [3]. To rejuvenate tance d, a generalized multi-slope pathloss model (MSPM) is adopted to characterize differentiated signal power attenuation the potential of network densiﬁcation, space division multiple rates within different regions, i.e., access (SDMA), which is capable of simultaneously serving multiple users within one cell over identical spectrum re- (cid:16) (cid:17) sources, is shown to be of great potential [4], [5]. Especially, it l {α }N−1 ; d = K d−αn, R ≤ d < R (2) is shown that single-user beamforming (SU-BF), a special case N n n=0 n n n+1 of SDMA, could improve network capacity by tens of folds in ultra-dense networks (UDN), compared to the single-antenna (cid:81)where α−n denotes the pathloss exponent, K0 = 1, Kn = regime [5]. However, only single user could be served within n Rαi αi−1 (n ≥ 1), and 0 = R < R < · · · < i=1 i 0 1 one cell using SU-BF. Therefore, it is imperative to study RN = ∞. Note that αi ≤ αj (i < j) and αN−1 > 2 for whether the multi-user gain of SDMA could be harvested to practical concerns. Rayleigh fading is used to model small- further improve network capacity in UDN. scale fading for mathematical tractability. The rationality of Rayleigh fading assumption in UDN has been veriﬁed via In this work, we investigate the performance of SDMA in experimental results in [7]. ultra-dense downlink small cell networks. As more than one SDMA with zero-forcing precoding is applied, where per- user is served within one cell using SDMA, spatial resources fect channel state information (CSI) from the BSs to the serv- could be fully exploited. However, it is unexpectedly shown ing downlink users could be obtained to design the precoders. that network spatial throughput (ST) would monotonously According to [4], [8], [9], under Rayleigh fading channels, the decrease with the number of served users within one cell when channel power gain caused by small-scaling fading of both BS density is sufﬁciently large, which radically contradicts direct link and interfering link follows gamma distribution with the sparse deployment cases. Moreover, it is proved that when zero-forcing precoding is applied by the multi-antenna applying SU-BF could as well maximize the critical density, beyond which network capacity starts to diminish. Therefore, 1Different from [3], which investigates the limitation of network densiﬁca- the results reveal the fundamental limits of SDMA and conﬁrm tion, we intend to unveil the limitation of SDMA in UDN in this work. ##LTLine##

text: Limitation

label: title

text: of

text: SDMA

text: in

text: Ultra-Dense

text: Small

text: Cell

text: Networks

text: Junyu

label: author

text: Liu,

text: Min

text: Sheng,

text: Jiandong

text: Li

text: State

text: Key

text: Laboratory

text: of

text: Integrated

text: Service

text: Networks,

text: Xidian

text: University,

text: Xi’an,

text: Shaanxi,

text: China

text: Email:

text: junyuliu@xidian.edu.cn,

text: {msheng,

text: jdli}@mail.xidian.edu.cn

text: Abstract—

text: Beneﬁtting

label: abstract

text: from

text: multi-user

text: gain

text: brought

text: by

text: multi-

text: the

text: superiority

text: of

text: SU-BF

text: in

text: antenna

text: techniques,

text: space

text: division

text: multiple

text: access

text: (SDMA)

text: is

text: II.

text: SYSTEM

text: MODEL

text: capable

text: of

text: signiﬁcantly

text: enhancing

text: spatial

text: throughput

text: (ST)

text: in

text: wireless

text: networks.

text: Nevertheless,

text: we

text: show

text: in

text: this

text: letter

text: that,

text: even

text: We

text: consider

text: a

text: downlink

text: small

text: cell

text: network,

text: where

text: the

text: when

text: SDMA

text: is

text: applied,

text: ST

text: would

text: diminish

text: to

text: be

text: zero

text: in

text: ultra-

text: locations

text: of

text: N

text: -antenna

text: BSs

text: (with

text: constant

text: transmit

text: power

text: a

text: dense

text: networks

text: (UDN),

text: where

text: small

text: cell

text: base

text: stations

text: (BSs)

text: are

text: P)

text: and

text: single-antenna

text: users

text: in

text: a

text: two-dimension

text: plane

text: are

text: fully

text: densiﬁed.

text: More

text: importantly,

text: we

text: compare

text: the

text: performance

text: modeled

text: as

text: two

text: independent

text: homogeneous

text: Poisson

text: Point

text: of

text: SDMA,

text: single-user

text: beamforming

text: (SU-BF)

text: (one

text: user

text: is

text: served

text: N

text: in

text: each

text: cell)

text: and

text: full

text: SDMA

text: (the

text: number

text: of

text: served

text: users

text: equals

text: Processes

text: (PPPs)

text: NΠBS

text: {BSi}

text: (i

text: of

text: density

text: λ

text: and

text: the

text: number

text: of

text: equipped

text: antennas).

text: Surprisingly,

text: it

text: is

text: shown

text: that

text: ΠU

text: {Uj}

text: (j

text: of

text: density

text: λU,

text: respectively.

text: If

text: r

text: is

text: the

text: SU-BF

text: achieves

text: the

text: highest

text: ST

text: and

text: critical

text: density,

text: beyond

text: which

text: distance

text: of

text: cellular

text: link,

text: the

text: distance

text: between

text: the

text: ST

text: starts

text: to

text: degrade,

text: in

text: UDN.

text: The

text: results

text: in

text: this

text: work

text: could

text: shed

text: antennas

text: of

text: BSs

text: and

text: the

text: associated

text: users

text: is

text: d

text: light

text: on

text: the

text: fundamental

text: limitation

text: of

text: SDMA

text: in

text: UDN.

text: where

text: ∆h

text: denotes

text: the

text: antenna

text: height

text: difference

text: (AHD)

text: between

text: BSs

text: and

text: users

text: f

text: (x)

text: (x

text: I.

text: INTRODUCTION

text: r

text: While

text: network

text: densiﬁcation

text: is

text: promising

text: to

text: improve

text: net-

text: Denote

text: NU

text: (NU

text: Na)

text: as

text: the

text: number

text: of

text: active

text: users

text: within

text: one

text: cell.

text: Meanwhile,

text: a

text: dense

text: user

text: deployment

text: is

text: considered,

text: work

text: capacity,

text: its

text: limitation

text: has

text: been

text: manifest

text: in

text: recent

text: research

text: Depending

text: on

text: practical

text: system

text: settings,

text: it

text: i.e.,

text: λ

text: λU,

text: such

text: that

text: each

text: full-buffer

text: BS

text: could

text: select

text: at

text: is

text: shown

text: that

text: area

text: spectral

text: efﬁciency

text: would

text: decrease

text: as

text: base

text: most

text: users

text: to

text: serve

text: using

text: SDMA.

text: Channel

text: gain

text: consists

text: of

text: a

text: pathloss

text: component

text: and

text: a

text: small-

text: station

text: (BS)

text: density

text: grows

text: in

text: downlink

text: cellular

text: networks

text: scale

text: fading

text: component.

text: In

text: particular,

text: given

text: transmission

text: dis-

text: Worsestill,

text: network

text: capacity

text: is

text: shown

text: to

text: diminish

text: to

text: zero

text: when

text: single-antenna

text: BSs

text: are

text: over

text: deployed

text: To

text: rejuvenate

text: tance

text: d,

text: a

text: generalized

text: multi-slope

text: pathloss

text: model

text: (MSPM)

text: is

text: adopted

text: to

text: characterize

text: differentiated

text: signal

text: power

text: attenuation

text: the

text: potential

text: of

text: network

text: densiﬁcation,

text: space

text: division

text: multiple

text: rates

text: within

text: different

text: regions,

text: i.e.,

text: access

text: (SDMA),

text: which

text: is

text: capable

text: of

text: simultaneously

text: serving

text: multiple

text: users

text: within

text: one

text: cell

text: over

text: identical

text: spectrum

text: re-

text: sources,

text: is

text: shown

text: to

text: be

text: of

text: great

text: potential

text: Especially,

text: it

text: l

text: {α

text: d

text: K

text: d−αn,

text: R

text: d

text: R

text: is

text: shown

text: that

text: single-user

text: beamforming

text: (SU-BF),

text: a

text: special

text: case

text: N

text: n

text: n

text: n

text: of

text: SDMA,

text: could

text: improve

text: network

text: capacity

text: by

text: tens

text: of

text: folds

text: in

text: ultra-dense

text: networks

text: (UDN),

text: compared

text: to

text: the

text: single-antenna

text: α−n

text: denotes

text: the

text: pathloss

text: exponent,

text: Kn

text: regime

text: However,

text: only

text: single

text: user

text: could

text: be

text: served

text: within

text: n

text: Rαi

text: αi−1

text: (n

text: and

text: R

text: R

text: i

text: one

text: cell

text: using

text: SU-BF.

text: Therefore,

text: it

text: is

text: imperative

text: to

text: study

text: RN

text: Note

text: that

text: αi

text: αj

text: (i

text: j)

text: and

text: αN−1

text: for

text: whether

text: the

text: multi-user

text: gain

text: of

text: SDMA

text: could

text: be

text: harvested

text: to

text: practical

text: concerns.

text: Rayleigh

text: fading

text: is

text: used

text: to

text: model

text: small-

text: further

text: improve

text: network

text: capacity

text: in

text: UDN.

text: scale

text: fading

text: for

text: mathematical

text: tractability.

text: The

text: rationality

text: of

text: Rayleigh

text: fading

text: assumption

text: in

text: UDN

text: has

text: been

text: veriﬁed

text: via

text: In

text: this

text: work,

text: we

text: investigate

text: the

text: performance

text: of

text: SDMA

text: in

text: experimental

text: results

text: in

text: ultra-dense

text: downlink

text: small

text: cell

text: networks.

text: As

text: more

text: than

text: one

text: SDMA

text: with

text: zero-forcing

text: precoding

text: is

text: applied,

text: where

text: per-

text: user

text: is

text: served

text: within

text: one

text: cell

text: using

text: SDMA,

text: spatial

text: resources

text: fect

text: channel

text: state

text: information

text: (CSI)

text: from

text: the

text: BSs

text: to

text: the

text: serv-

text: could

text: be

text: fully

text: exploited.

text: However,

text: it

text: is

text: unexpectedly

text: shown

text: ing

text: downlink

text: users

text: could

text: be

text: obtained

text: to

text: design

text: the

text: precoders.

text: that

text: network

text: spatial

text: throughput

text: (ST)

text: would

text: monotonously

text: According

text: to

text: under

text: Rayleigh

text: fading

text: channels,

text: the

text: decrease

text: with

text: the

text: number

text: of

text: served

text: users

text: within

text: one

text: cell

text: when

text: channel

text: power

text: gain

text: caused

text: by

text: small-scaling

text: fading

text: of

text: both

text: BS

text: density

text: is

text: sufﬁciently

text: large,

text: which

text: radically

text: contradicts

text: direct

text: link

text: and

text: interfering

text: link

text: follows

text: gamma

text: distribution

text: with

text: the

text: sparse

text: deployment

text: cases.

text: Moreover,

text: it

text: is

text: proved

text: that

text: when

text: zero-forcing

text: precoding

text: is

text: applied

text: by

text: the

text: multi-antenna

text: applying

text: SU-BF

text: could

text: as

text: well

text: maximize

text: the

text: critical

text: density,

text: beyond

text: which

text: network

text: capacity

text: starts

text: to

text: diminish.

text: Therefore,

text: from

text: which

text: investigates

text: the

text: limitation

text: of

text: network

text: densiﬁca-

text: the

text: results

text: reveal

text: the

text: fundamental

text: limits

text: of

text: SDMA

text: and

text: conﬁrm

text: tion,

text: we

text: intend

text: to

text: unveil

text: the

text: limitation

text: of

text: SDMA

text: in

text: UDN

text: in

text: this

text: work.

text: ##LTLine##


text: 2 JIZHAN HONG closed ﬁelds are documented in the book [Fried and Jarden, 2008]. Ex- amples of pseudo-algebraically closed ﬁelds include separably closed ﬁelds, pseudo-ﬁnite ﬁelds and some Hilbertian ﬁelds. Global ﬁelds and non-trivial purely transcendental extensions over any ﬁeld are Hilber- tian ﬁelds. It was proved in [Jarden, 1972] that if K is a countable Hilbertian ﬁeld, then with respect to the Haar measure, for almost ∈ all tuples (σ , . . . , σ ) Gal(K)e, the ﬁxed ﬁeld of the tuple is a 1 e pseudo-algebraically closed ﬁeld. A lot is known on the model theory of pseudo-algebraically closed ﬁelds. A characterization of the elementary equivalence relation among pseudo-algebraically closed ﬁelds in terms of Galois groups was given in [Cherlin et al., 1982]. The model the- ory of pseudo-algebraically closed ﬁelds turns out to depend very much on the Galois groups. Using an important Embedding Lemma [Jarden and Kiehne, 1975], it was possible to prove that certain class of pseudo- algebraically closed ﬁelds admits quantiﬁer elimination in a suitably nice language. While the theory of pseudo-algebraically closed ﬁelds itself in the same language does not admit quantiﬁer elimination due to some Galois theoretic obstructions, it was however shown that the theory itself admits a new kind of ‘quantiﬁer elimination’ using the so-called Galois stratiﬁcations. Upon a close study of the so-called ‘co- model theory’ over these pseudo-algebraically closed ﬁelds, in [Cherlin et al., 1982], it was also proved that the theory of perfect Frobenius ﬁelds admits quantiﬁer elimination in a so-called ‘Galois formalism’. In this article, we are interested in the model theory of pseudo- algebraically closed ﬁelds endowed with a non-trivial valuation. It was proved in [Koll´ar, 2007] that if K is pseudo-algebraically closed, V is any valuation on the algebraic closure Kalg of K, and X is a geomet- rically integral K-variety, then X(K) is dense in X(Kalg) with respect to the topology induced by V . This provides a critical tool in the study of pseudo-algebraically closed valued ﬁelds. It enables us to prove the Valuation Theoretic Embedding Lemma and the quantiﬁer elimination results in this article. The strategy for proving the quantiﬁer elimination results in this ar- ticle turns out to be almost similar to the ﬁeld theoretic case; one only needs to take a little care of the valuations. The main result in this ar- ticle is in Section . Since that section is not very long, we will not give another summary here. But roughly speaking, it says that the theory of non-trivially valued Frobenius ﬁelds with a ﬁxed exponent of imperfec- tion has quantiﬁer elimination in a suﬃciently nice language. Similar to the ﬁeld theoretic case, the same Galois theoretic obstructions render the quantiﬁer elimination for the theory of pseudo-algebraically closed

text: JIZHAN

text: HONG

text: closed

text: ﬁelds

text: are

text: documented

text: in

text: the

text: book

text: [Fried

text: and

text: Jarden,

text: Ex-

text: amples

text: of

text: pseudo-algebraically

text: closed

text: ﬁelds

text: include

text: separably

text: closed

text: ﬁelds,

text: pseudo-ﬁnite

text: ﬁelds

text: and

text: some

text: Hilbertian

text: ﬁelds.

text: Global

text: ﬁelds

text: and

text: non-trivial

text: purely

text: transcendental

text: extensions

text: over

text: any

text: ﬁeld

text: are

text: Hilber-

text: tian

text: ﬁelds.

text: It

text: was

text: proved

text: in

text: [Jarden,

text: that

text: if

text: K

text: is

text: a

text: countable

text: Hilbertian

text: ﬁeld,

text: then

text: with

text: respect

text: to

text: the

text: Haar

text: measure,

text: for

text: almost

text: all

text: tuples

text: (σ

text: σ

text: Gal(K)e,

text: the

text: ﬁxed

text: ﬁeld

text: of

text: the

text: tuple

text: is

text: a

text: e

text: pseudo-algebraically

text: closed

text: ﬁeld.

text: A

text: lot

text: is

text: known

text: on

text: the

text: model

text: theory

text: of

text: pseudo-algebraically

text: closed

text: ﬁelds.

text: A

text: characterization

text: of

text: the

text: elementary

text: equivalence

text: relation

text: among

text: pseudo-algebraically

text: closed

text: ﬁelds

text: in

text: terms

text: of

text: Galois

text: groups

text: was

text: given

text: in

text: [Cherlin

text: et

text: al.,

text: The

text: model

text: the-

text: ory

text: of

text: pseudo-algebraically

text: closed

text: ﬁelds

text: turns

text: out

text: to

text: depend

text: very

text: much

text: on

text: the

text: Galois

text: groups.

text: Using

text: an

text: important

text: Embedding

text: Lemma

text: [Jarden

text: and

text: Kiehne,

text: it

text: was

text: possible

text: to

text: prove

text: that

text: certain

text: class

text: of

text: pseudo-

text: algebraically

text: closed

text: ﬁelds

text: admits

text: quantiﬁer

text: elimination

text: in

text: a

text: suitably

text: nice

text: language.

text: While

text: the

text: theory

text: of

text: pseudo-algebraically

text: closed

text: ﬁelds

text: itself

text: in

text: the

text: same

text: language

text: does

text: not

text: admit

text: quantiﬁer

text: elimination

text: due

text: to

text: some

text: Galois

text: theoretic

text: obstructions,

text: it

text: was

text: however

text: shown

text: that

text: the

text: theory

text: itself

text: admits

text: a

text: new

text: kind

text: of

text: ‘quantiﬁer

text: elimination’

text: using

text: the

text: so-called

text: Galois

text: stratiﬁcations.

text: Upon

text: a

text: close

text: study

text: of

text: the

text: so-called

text: ‘co-

text: model

text: theory’

text: over

text: these

text: pseudo-algebraically

text: closed

text: ﬁelds,

text: in

text: [Cherlin

text: et

text: al.,

text: it

text: was

text: also

text: proved

text: that

text: the

text: theory

text: of

text: perfect

text: Frobenius

text: ﬁelds

text: admits

text: quantiﬁer

text: elimination

text: in

text: a

text: so-called

text: ‘Galois

text: formalism’.

text: In

text: this

text: article,

text: we

text: are

text: interested

text: in

text: the

text: model

text: theory

text: of

text: pseudo-

text: algebraically

text: closed

text: ﬁelds

text: endowed

text: with

text: a

text: non-trivial

text: valuation.

text: It

text: was

text: proved

text: in

text: [Koll´ar,

text: that

text: if

text: K

text: is

text: pseudo-algebraically

text: closed,

text: V

text: is

text: any

text: valuation

text: on

text: the

text: algebraic

text: closure

text: Kalg

text: of

text: K,

text: and

text: X

text: is

text: a

text: geomet-

text: rically

text: integral

text: K-variety,

text: then

text: X(K)

text: is

text: dense

text: in

text: X(Kalg)

text: with

text: respect

text: to

text: the

text: topology

text: induced

text: by

text: V

text: This

text: provides

text: a

text: critical

text: tool

text: in

text: the

text: study

text: of

text: pseudo-algebraically

text: closed

text: valued

text: ﬁelds.

text: It

text: enables

text: us

text: to

text: prove

text: the

text: Valuation

text: Theoretic

text: Embedding

text: Lemma

text: and

text: the

text: quantiﬁer

text: elimination

text: results

text: in

text: this

text: article.

text: The

text: strategy

text: for

text: proving

text: the

text: quantiﬁer

text: elimination

text: results

text: in

text: this

text: ar-

text: ticle

text: turns

text: out

text: to

text: be

text: almost

text: similar

text: to

text: the

text: ﬁeld

text: theoretic

text: case;

text: one

text: only

text: needs

text: to

text: take

text: a

text: little

text: care

text: of

text: the

text: valuations.

text: The

text: main

text: result

text: in

text: this

text: ar-

text: ticle

text: is

text: in

text: Section

text: Since

text: that

text: section

text: is

text: not

text: very

text: long,

text: we

text: will

text: not

text: give

text: another

text: summary

text: here.

text: But

text: roughly

text: speaking,

text: it

text: says

text: that

text: the

text: theory

text: of

text: non-trivially

text: valued

text: Frobenius

text: ﬁelds

text: with

text: a

text: ﬁxed

text: exponent

text: of

text: imperfec-

text: tion

text: has

text: quantiﬁer

text: elimination

text: in

text: a

text: suﬃciently

text: nice

text: language.

text: Similar

text: to

text: the

text: ﬁeld

text: theoretic

text: case,

text: the

text: same

text: Galois

text: theoretic

text: obstructions

text: render

text: the

text: quantiﬁer

text: elimination

text: for

text: the

text: theory

text: of

text: pseudo-algebraically

text: closed


text: QE ON SOME PACVFS 17 Thus by Equation (3.4) and Equation (3.5), we get ∈ a , . . . , a Kp(a , . . . , a − , φ(y ), . . . , φ(y )). 1 n 1 n m 1 m But a , . . . , a are p-independent over Kp, so 1 n a , . . . , a − , φ(y ), . . . , φ(y ) 1 n m 1 m are also p-independent over Kp. Thus φ satisﬁes all the desired prop- (cid:3) erties. Combining Lemma 3.8 and the proof of Lemma 3.7, we can improve the later to preserve the p-independence relation. Lemma 3.9. Suppose that K is a PAC ﬁeld with a non-trivial valua- tion V , (L, W ) is a valued ﬁeld extension of (K, V ) where L/K is regu- lar. Suppose that E is a subﬁeld of L contained in a K-subalgebra R of | | L, where R is generated by not more than E elements over K. Sup- ∈ ∪{∞} pose furthermore that the exponent of imperfection of K is d ω , { } ⊆ | | ≤ {| | } y ∈ R are p-independent in L, where M min E , d . Sup- m m M | | pose that (K, V ) is E +-saturated in L . Then there is a ring ho- → div momorphism φ : R K satisfying the following conditions simultane- ously: (1) φ restricts to the identity map on K; ∩ (2) φ restricts to an L -embedding of (E, W E) into (K, V ); { } div (3) φ(y ) ∈ are p-independent in K. j j J Proof. The strategy of the proof is very much the same as the one for Lemma 3.7. Using the notation from that proof, assume that y = m g ((cid:126)x). Instead of showing the set of formulas Σ is ﬁnitely realizable in m K, we show that ∪ { } Σ g (X(cid:126) ) are p-independent ∈ m m M (cid:3) is ﬁnitely realizable in K, which is guaranteed by Lemma 3.8. 4. Existence of various pseudo-algebraically closed fields In the manuscript [Cherlin et al., 1982], the elementary theory of regularly closed ﬁelds were studied, where a ﬁeld is called a regularly closed ﬁeld exactly when it is pseudo-algebraically closed. The follow- ing fact was proved in [Cherlin et al., 1982].

text: QE

text: ON

text: SOME

text: PACVFS

text: Thus

text: by

text: Equation

text: and

text: Equation

text: we

text: get

text: a

text: a

text: Kp(a

text: a

text: φ(y

text: φ(y

text: n

text: n

text: m

text: m

text: But

text: a

text: a

text: are

text: p-independent

text: over

text: Kp,

text: so

text: n

text: a

text: a

text: φ(y

text: φ(y

text: n

text: m

text: m

text: are

text: also

text: p-independent

text: over

text: Kp.

text: Thus

text: φ

text: satisﬁes

text: all

text: the

text: desired

text: prop-

text: erties.

text: Combining

text: Lemma

text: and

text: the

text: proof

text: of

text: Lemma

text: we

text: can

text: improve

text: the

text: later

text: to

text: preserve

text: the

text: p-independence

text: relation.

text: Lemma

text: Suppose

text: that

text: K

text: is

text: a

text: PAC

text: ﬁeld

text: with

text: a

text: non-trivial

text: valua-

text: tion

text: V

text: (L,

text: W

text: is

text: a

text: valued

text: ﬁeld

text: extension

text: of

text: (K,

text: V

text: where

text: L/K

text: is

text: regu-

text: lar.

text: Suppose

text: that

text: E

text: is

text: a

text: subﬁeld

text: of

text: L

text: contained

text: in

text: a

text: K-subalgebra

text: R

text: of

text: L,

text: where

text: R

text: is

text: generated

text: by

text: not

text: more

text: than

text: E

text: elements

text: over

text: K.

text: Sup-

text: pose

text: furthermore

text: that

text: the

text: exponent

text: of

text: imperfection

text: of

text: K

text: is

text: d

text: ω

text: y

text: R

text: are

text: p-independent

text: in

text: L,

text: where

text: M

text: min

text: E

text: d

text: Sup-

text: m

text: m

text: M

text: pose

text: that

text: (K,

text: V

text: is

text: E

text: +-saturated

text: in

text: L

text: Then

text: there

text: is

text: a

text: ring

text: ho-

text: div

text: momorphism

text: φ

text: R

text: K

text: satisfying

text: the

text: following

text: conditions

text: simultane-

text: ously:

text: φ

text: restricts

text: to

text: the

text: identity

text: map

text: on

text: K;

text: φ

text: restricts

text: to

text: an

text: L

text: -embedding

text: of

text: (E,

text: W

text: E)

text: into

text: (K,

text: V

text: div

text: φ(y

text: are

text: p-independent

text: in

text: K.

text: j

text: j

text: J

text: Proof.

text: The

text: strategy

text: of

text: the

text: proof

text: is

text: very

text: much

text: the

text: same

text: as

text: the

text: one

text: for

text: Lemma

text: Using

text: the

text: notation

text: from

text: that

text: proof,

text: assume

text: that

text: y

text: m

text: g

text: Instead

text: of

text: showing

text: the

text: set

text: of

text: formulas

text: Σ

text: is

text: ﬁnitely

text: realizable

text: in

text: m

text: K,

text: we

text: show

text: that

text: Σ

text: g

text: are

text: p-independent

text: m

text: m

text: M

text: is

text: ﬁnitely

text: realizable

text: in

text: K,

text: which

text: is

text: guaranteed

text: by

text: Lemma

text: Existence

text: of

text: various

text: pseudo-algebraically

text: closed

text: fields

text: In

text: the

text: manuscript

text: [Cherlin

text: et

text: al.,

text: the

text: elementary

text: theory

text: of

text: regularly

text: closed

text: ﬁelds

text: were

text: studied,

text: where

text: a

text: ﬁeld

text: is

text: called

text: a

text: regularly

text: closed

text: ﬁeld

text: exactly

text: when

text: it

text: is

text: pseudo-algebraically

text: closed.

text: The

text: follow-

text: ing

text: fact

text: was

text: proved

text: in

text: [Cherlin

text: et

text: al.,


text: QE ON SOME PACVFS 21 theory of pseudo-algebraically closed ﬁelds with non-trivial valuations itself as a total, does not admit quantiﬁer elimination. Theorem 5.5. The following statements are true. (1) Let PACVF be the theory, in L , of pseudo-algebraically θ,div closed non-trivially valued ﬁelds. Then PACVF does NOT ad- mit quantiﬁer elimination. (2) Let PACVF be the theory, in L , of pseudo-algebraically p,d θ,p,div closed non-trivially valued ﬁelds, of characteristic p, with expo- ∈ ∪ {∞} nent of imperfection d (d ω ). Then PACVF does p,d NOT admit quantiﬁer elimination either. Proof. It is enough to show that PACVF is not substructure com- p,d plete. The obstruction occurs already on the ﬁeld theoretic level, not necessary to involve the valuations. The K be a countable separably closed ﬁeld of characteristic p with exponent of imperfection d. Let t be transcendental element over K. Then K(t) is a countable Hilbertian ﬁeld and the extension K(t)/K is regular. Let V be any non-trivial valuation on K(t) By Fact 2.26 and Fact 2.27, there exists a ﬁeld E /K(t) which is 1-free PAC and there 1 exists a ﬁeld E /K(t) which is 2-free PAC. Let V and V be any two 2 1 2 extensions of V to E and E respectively. Since E /K and E /K are 1 2 1 2 both separable and K is separably closed, the L -substructure on θ,p,div K in E and the L -substructure on K in E agree. Thus 1 θ,p,div 2 (cid:54)≡ (E , V ) (E , V ), 1 1 (K,V ) 2 2 (cid:3) because they have diﬀerent free absolute Galois groups. In view of the theorem above, in order to obtain quantiﬁer elimina- tion, we need to impose conditions on the Galois groups of the models. This leads us to obtain the results in Section . 6. The Embedding Lemma The so-called Embedding Lemma (Lemma 20.2.2 in [Fried and Jar- den, 2008]) plays an important role in the analysis of the model theory of PAC ﬁelds. Analogously, we also have a valuation theoretic ver- sion of the Embedding Lemma, which plays a similar important role in the model theoretic analysis of the pseudo-algebraically closed valued ﬁelds.

text: QE

text: ON

text: SOME

text: PACVFS

text: theory

text: of

text: pseudo-algebraically

text: closed

text: ﬁelds

text: with

text: non-trivial

text: valuations

text: itself

text: as

text: a

text: total,

text: does

text: not

text: admit

text: quantiﬁer

text: elimination.

text: Theorem

text: The

text: following

text: statements

text: are

text: true.

label: list

text: Let

text: PACVF

text: be

text: the

text: theory,

text: in

text: L

text: of

text: pseudo-algebraically

text: θ,div

text: closed

text: non-trivially

text: valued

text: ﬁelds.

text: Then

text: PACVF

text: does

text: NOT

text: ad-

text: mit

text: quantiﬁer

text: elimination.

text: Let

text: PACVF

text: be

text: the

text: theory,

text: in

text: L

text: of

text: pseudo-algebraically

text: p,d

text: θ,p,div

text: closed

text: non-trivially

text: valued

text: ﬁelds,

text: of

text: characteristic

text: p,

text: with

text: expo-

text: nent

text: of

text: imperfection

text: d

text: (d

text: ω

text: Then

text: PACVF

text: does

text: p,d

text: NOT

text: admit

text: quantiﬁer

text: elimination

text: either.

text: Proof.

text: It

text: is

text: enough

text: to

text: show

text: that

text: PACVF

text: is

text: not

text: substructure

text: com-

text: p,d

text: plete.

text: The

text: obstruction

text: occurs

text: already

text: on

text: the

text: ﬁeld

text: theoretic

text: level,

text: not

text: necessary

text: to

text: involve

text: the

text: valuations.

text: The

text: K

text: be

text: a

text: countable

text: separably

text: closed

text: ﬁeld

text: of

text: characteristic

text: p

text: with

text: exponent

text: of

text: imperfection

text: d.

text: Let

text: t

text: be

text: transcendental

text: element

text: over

text: K.

text: Then

text: K(t)

text: is

text: a

text: countable

text: Hilbertian

text: ﬁeld

text: and

text: the

text: extension

text: K(t)/K

text: is

text: regular.

text: Let

text: V

text: be

text: any

text: non-trivial

text: valuation

text: on

text: K(t)

text: By

text: Fact

text: and

text: Fact

text: there

text: exists

text: a

text: ﬁeld

text: E

text: /K(t)

text: which

text: is

text: PAC

text: and

text: there

text: exists

text: a

text: ﬁeld

text: E

text: /K(t)

text: which

text: is

text: PAC.

text: Let

text: V

text: and

text: V

text: be

text: any

text: two

text: extensions

text: of

text: V

text: to

text: E

text: and

text: E

text: respectively.

text: Since

text: E

text: /K

text: and

text: E

text: /K

text: are

text: both

text: separable

text: and

text: K

text: is

text: separably

text: closed,

text: the

text: L

text: -substructure

text: on

text: θ,p,div

text: K

text: in

text: E

text: and

text: the

text: L

text: -substructure

text: on

text: K

text: in

text: E

text: agree.

text: Thus

text: θ,p,div

text: (E

text: V

text: (E

text: V

text: (K,V

text: because

text: they

text: have

text: diﬀerent

text: free

text: absolute

text: Galois

text: groups.

text: In

text: view

text: of

text: the

text: theorem

text: above,

text: in

text: order

text: to

text: obtain

text: quantiﬁer

text: elimina-

text: tion,

text: we

text: need

text: to

text: impose

text: conditions

text: on

text: the

text: Galois

text: groups

text: of

text: the

text: models.

text: This

text: leads

text: us

text: to

text: obtain

text: the

text: results

text: in

text: Section

text: The

text: Embedding

text: Lemma

text: The

text: so-called

text: Embedding

text: Lemma

text: (Lemma

text: in

text: [Fried

text: and

text: Jar-

text: den,

text: plays

text: an

text: important

text: role

text: in

text: the

text: analysis

text: of

text: the

text: model

text: theory

text: of

text: PAC

text: ﬁelds.

text: Analogously,

text: we

text: also

text: have

text: a

text: valuation

text: theoretic

text: ver-

text: sion

text: of

text: the

text: Embedding

text: Lemma,

text: which

text: plays

text: a

text: similar

text: important

text: role

text: in

text: the

text: model

text: theoretic

text: analysis

text: of

text: the

text: pseudo-algebraically

text: closed

text: valued

text: ﬁelds.


text: QE ON SOME PACVFS 25 ∪{ } Since (A, V ) is a substructure in the language L I , A θ,p,div G G ﬁnite group A preserves all the 0-place predicates I , thus Im(Gal(E)) = Im(Gal(F )). G First, notice that A is a ﬁeld, for the following reason. For each ∈ × − − − a A , the maximal splitting factor of ax 1 is x a 1. Thus, since − − − ∈ − ∈ θ ( 1, a) = a 1 A, a 1 A. So A is a ﬁeld. 1,0 Second, E/A is separable, for the following reason. As an L - p substructure, A is closed under all the relative p-coordinate functions. ∈ ∈ ∈ If x Ep, then λ (x) = x1/p, so if furthermore x A, then x1/p A. 0,0 ∩ This means Ap = A Ep. Suppose that b , . . . , b are p-independent ∩ 1 n in A, then because Ap = A Ep, none of the b , . . . , b is in Ep. Thus 1 { n } there exists a maximal p-independent subset of b , . . . , b over Ep, { } 1 n which we may assume to be b , . . . , b . If it were true that k < n, ∈ 1 k Then b Ep(b , . . . , b ) gives a quantiﬁer-free L -formula true in k+1 1 k p E which is then also true in A because A is closed under all the rel- ative p-coordinate functions, which in turn implies that b , . . . , b are 1 n p-dependent in A. Thus k = n and b , . . . , b remains p-independent 1 n in E. So E/A is separable. By the same reasoning, F/ε(A) is also separable. ∩ Third, ε could be extended to an L -embedding of (Aalg, V θ,p,div ∈ E Aalg). For any monic separable irreducible polynomial f(X) A[X], if f(X) has a root in E, then because A is closed under the function of splitting coeﬃcients, the maximal splitting factor of f(X) in E is a polynomial in A[X]; but f(X) irreducible over A, so f(X) is its own maximal splitting factor. Thus f(X) splits in E. Since ε(A) is also an L -substructure of F , f(X) also splits in F . By the same reasoning, θ ∈ any monic separable irreducible polynomial g(X) ε(A)[X] that has a root in F also splits both in E and F . Therefore, by Lemma 3.5, there exists an L -isomorphism div ∩ ∩ ∩ → ∩ ∩ ∩ ε˜ : (E Asep, V E Asep) (F Asep, V F Asep), E F ∩ ∩ extending ε. Since both E Asep and F Asep are relatively algebraically closed in E and F respectively and they are isomorphic as ﬁelds, their ∩ respective L -substructures are preserved by ε˜. Since E/(E Asep) ∩ θ and F/(F Asep) remains separable, ε˜ is also an L -isomorphism. This p means that ε˜ is in fact an L -isomorphism. θ∪,p{,div } ∩ ∩ ∩ Let (L, V ) be the L I -substructure (E Asep, V E L θ,p,div G G E Asep), then ε˜ gives an L -embedding of (L, V ) into (F, V ). Let θ,p,div L F M = ε˜(L). Then E/L and F/M are both regular ﬁeld extensions. → Extend ε˜ to a ﬁeld isomorphism Φ : Lsep Msep. Then Φ induces 0 →0 an isomorphism of the absolute Galois groups ϕ : Gal(M) Gal(L). 0

text: QE

text: ON

text: SOME

text: PACVFS

text: Since

text: (A,

text: V

text: is

text: a

text: substructure

text: in

text: the

text: language

text: L

text: I

text: A

text: θ,p,div

text: G

text: G

text: ﬁnite

text: group

text: A

text: preserves

text: all

text: the

text: predicates

text: I

text: thus

text: Im(Gal(E))

text: Im(Gal(F

text: G

text: First,

text: notice

text: that

text: A

text: is

text: a

text: ﬁeld,

text: for

text: the

text: following

text: reason.

text: For

text: each

text: a

text: A

text: the

text: maximal

text: splitting

text: factor

text: of

text: ax

text: is

text: x

text: a

text: Thus,

text: since

text: θ

text: a)

text: a

text: A,

text: a

text: A.

text: So

text: A

text: is

text: a

text: ﬁeld.

text: Second,

text: E/A

text: is

text: separable,

text: for

text: the

text: following

text: reason.

text: As

text: an

text: L

text: p

text: substructure,

text: A

text: is

text: closed

text: under

text: all

text: the

text: relative

text: p-coordinate

text: functions.

text: If

text: x

text: Ep,

text: then

text: λ

text: (x)

text: so

text: if

text: furthermore

text: x

text: A,

text: then

text: A.

text: This

text: means

text: Ap

text: A

text: Ep.

text: Suppose

text: that

text: b

text: b

text: are

text: p-independent

text: n

text: in

text: A,

text: then

text: because

text: Ap

text: A

text: Ep,

text: of

text: the

text: b

text: b

text: is

text: in

text: Ep.

text: Thus

text: n

text: there

text: exists

text: a

text: maximal

text: p-independent

text: subset

text: of

text: b

text: b

text: over

text: Ep,

text: n

text: which

text: we

text: may

text: assume

text: to

text: be

text: b

text: b

text: If

text: it

text: were

text: true

text: that

text: k

text: n,

text: k

text: Then

text: b

text: Ep(b

text: b

text: gives

text: a

text: quantiﬁer-free

text: L

text: -formula

text: true

text: in

text: k

text: p

text: E

text: which

text: is

text: then

text: also

text: true

text: in

text: A

text: because

text: A

text: is

text: closed

text: under

text: all

text: the

text: rel-

text: ative

text: p-coordinate

text: functions,

text: which

text: in

text: turn

text: implies

text: that

text: b

text: b

text: are

text: n

text: p-dependent

text: in

text: A.

text: Thus

text: k

text: n

text: and

text: b

text: b

text: remains

text: p-independent

text: n

text: in

text: E.

text: So

text: E/A

text: is

text: separable.

text: By

text: the

text: same

text: reasoning,

text: F/ε(A)

text: is

text: also

text: separable.

text: Third,

text: ε

text: could

text: be

text: extended

text: to

text: an

text: L

text: -embedding

text: of

text: (Aalg,

text: V

text: θ,p,div

text: E

text: Aalg).

text: For

text: any

text: monic

text: separable

text: irreducible

text: polynomial

text: f(X)

text: A[X],

text: if

text: f(X)

text: has

text: a

text: root

text: in

text: E,

text: then

text: because

text: A

text: is

text: closed

text: under

text: the

text: function

text: of

text: splitting

text: coeﬃcients,

text: the

text: maximal

text: splitting

text: factor

text: of

text: f(X)

text: in

text: E

text: is

text: a

text: polynomial

text: in

text: A[X];

text: but

text: f(X)

text: irreducible

text: over

text: A,

text: so

text: f(X)

text: is

text: its

text: own

text: maximal

text: splitting

text: factor.

text: Thus

text: f(X)

text: splits

text: in

text: E.

text: Since

text: ε(A)

text: is

text: also

text: an

text: L

text: -substructure

text: of

text: F

text: f(X)

text: also

text: splits

text: in

text: F

text: By

text: the

text: same

text: reasoning,

text: θ

text: any

text: monic

text: separable

text: irreducible

text: polynomial

text: g(X)

text: ε(A)[X]

text: that

text: has

text: a

text: root

text: in

text: F

text: also

text: splits

text: both

text: in

text: E

text: and

text: F

text: Therefore,

text: by

text: Lemma

text: there

text: exists

text: an

text: L

text: -isomorphism

text: div

text: ε˜

text: (E

text: Asep,

text: V

text: E

text: Asep)

text: (F

text: Asep,

text: V

text: F

text: Asep),

text: E

text: F

text: extending

text: ε.

text: Since

text: both

text: E

text: Asep

text: and

text: F

text: Asep

text: are

text: relatively

text: algebraically

text: closed

text: in

text: E

text: and

text: F

text: respectively

text: and

text: they

text: are

text: isomorphic

text: as

text: ﬁelds,

text: their

text: respective

text: L

text: -substructures

text: are

text: preserved

text: by

text: ε˜.

text: Since

text: E/(E

text: Asep)

text: θ

text: and

text: F/(F

text: Asep)

text: remains

text: separable,

text: ε˜

text: is

text: also

text: an

text: L

text: -isomorphism.

text: This

text: p

text: means

text: that

text: ε˜

text: is

text: in

text: fact

text: an

text: L

text: -isomorphism.

text: θ∪,p{,div

text: Let

text: (L,

text: V

text: be

text: the

text: L

text: I

text: -substructure

text: (E

text: Asep,

text: V

text: E

text: L

text: θ,p,div

text: G

text: G

text: E

text: Asep),

text: then

text: ε˜

text: gives

text: an

text: L

text: -embedding

text: of

text: (L,

text: V

text: into

text: (F,

text: V

text: Let

text: θ,p,div

text: L

text: F

text: M

text: ε˜(L).

text: Then

text: E/L

text: and

text: F/M

text: are

text: both

text: regular

text: ﬁeld

text: extensions.

text: Extend

text: ε˜

text: to

text: a

text: ﬁeld

text: isomorphism

text: Φ

text: Lsep

text: Msep.

text: Then

text: Φ

text: induces

text: →0

text: an

text: isomorphism

text: of

text: the

text: absolute

text: Galois

text: groups

text: ϕ

text: Gal(M)

text: Gal(L).


text: 30 JIZHAN HONG Proof. The proof is more or less similar to the ﬁeld theoretic case given in [Cherlin et al., 1982], with some care for the valuations. ≡ Suppose that (K, V ) (L, W ). Then obviously K and L have L div ∗ the same exponent of imperfectness. Let f be an isomorphism of an ∗ ∗ elementary extension (L , W ) of (L, W ) onto an elementary extension ∗ ∗ (K , V ) of (K, V ). By Fact 2.10, there exists an L -isomorphism ∗ ∗ → ∗ ∗ div f sep : (L )sep (K )sep. Let f be the restriction of f sep on Abs(L)sep. ∗ ∗ Because L and K are elementary extensions of L and K respectively, ∗ ∗ we have Abs(L ) = Abs(L) and Abs(K ) = Abs(K). Thus f restricts ∩ ∩ to an L -isomorphism from (Abs(L), W Abs(L)) to (Abs(K), V div Abs(K)). Meanwhile, by the proof of Lemma 32 in [Cherlin et al., 1982], we also have ≡ (Gal(K),ˆi) o (Gal(L), ˆj). fˆ For the other direction, suppose that K and L have the same expo- nent of imperfectness and that there is an L -isomorphism r → f : Abs(L)sep Abs(K)sep ∩ that restricts to an L -isomorphism from (Abs(L), W Abs(L)) onto ∩ div (Abs(K), V Abs(K)) such that ≡ (Gal(K),ˆi) o (Gal(L), ˆj), fˆ ≡ then we show that (K, V ) (L, W ). We also follow the proof of L div Proposition 33 in [Cherlin et al., 1982]. We may assume that both ℵ (K, V ) and (L, W ) are both -saturated. We show the existence of 1 a back-and-forth system of triples (g, M, N) satisfying the following  Property (8.1),        K/M and L/N are regular ﬁeld extensions where M and  N are countable subﬁelds, g is an L -isomorphism  → r (8.1)  g : Msep Nsep which restricts to an Ldiv-isomorphism from M onto N, and with the inclusions i and j , M N ≡ (Gal(K),ˆi ) o (Gal(L), ˆj ) holds. M gˆ N Note that ﬁrst (f, Abs(K), Abs(L)) satisﬁes Property (8.1). Now, if (g, M, N) satisﬁes Property (8.1), A a countable subset of L, then we show that there is a triple (g , M , N ) satisfying Property (8.1) 1 1 1 such that M /M and N /N(A) and g extending g. This is guaranteed 1 1 1 by the proof of Proposition 33 of [Cherlin et al., 1982] and our Valuation Theoretic Embedding Lemma 6.1. This back-and-forth system then ≡ (cid:3) proves that (K, V ) (L, W ). L div

text: JIZHAN

text: HONG

text: Proof.

text: The

text: proof

text: is

text: more

text: or

text: less

text: similar

text: to

text: the

text: ﬁeld

text: theoretic

text: case

text: given

text: in

text: [Cherlin

text: et

text: al.,

text: with

text: some

text: care

text: for

text: the

text: valuations.

text: Suppose

text: that

text: (K,

text: V

text: (L,

text: W

text: Then

text: obviously

text: K

text: and

text: L

text: have

text: L

text: div

text: the

text: same

text: exponent

text: of

text: imperfectness.

text: Let

text: f

text: be

text: an

text: isomorphism

text: of

text: an

text: elementary

text: extension

text: (L

text: W

text: of

text: (L,

text: W

text: onto

text: an

text: elementary

text: extension

text: (K

text: V

text: of

text: (K,

text: V

text: By

text: Fact

text: there

text: exists

text: an

text: L

text: -isomorphism

text: div

text: f

text: sep

text: (L

text: )sep

text: (K

text: )sep.

text: Let

text: f

text: be

text: the

text: restriction

text: of

text: f

text: sep

text: on

text: Abs(L)sep.

text: Because

text: L

text: and

text: K

text: are

text: elementary

text: extensions

text: of

text: L

text: and

text: K

text: respectively,

text: we

text: have

text: Abs(L

text: Abs(L)

text: and

text: Abs(K

text: Abs(K).

text: Thus

text: f

text: restricts

text: to

text: an

text: L

text: -isomorphism

text: from

text: (Abs(L),

text: W

text: Abs(L))

text: to

text: (Abs(K),

text: V

text: div

text: Abs(K)).

text: Meanwhile,

text: by

text: the

text: proof

text: of

text: Lemma

text: in

text: [Cherlin

text: et

text: al.,

text: we

text: also

text: have

text: (Gal(K),ˆi)

text: o

text: (Gal(L),

text: ˆj).

text: fˆ

text: For

text: the

text: other

text: direction,

text: suppose

text: that

text: K

text: and

text: L

text: have

text: the

text: same

text: expo-

text: nent

text: of

text: imperfectness

text: and

text: that

text: there

text: is

text: an

text: L

text: -isomorphism

text: r

text: f

text: Abs(L)sep

text: Abs(K)sep

text: that

text: restricts

text: to

text: an

text: L

text: -isomorphism

text: from

text: (Abs(L),

text: W

text: Abs(L))

text: onto

text: div

text: (Abs(K),

text: V

text: Abs(K))

text: such

text: that

text: (Gal(K),ˆi)

text: o

text: (Gal(L),

text: ˆj),

text: fˆ

text: then

text: we

text: show

text: that

text: (K,

text: V

text: (L,

text: W

text: We

text: also

text: follow

text: the

text: proof

text: of

text: L

text: div

text: Proposition

text: in

text: [Cherlin

text: et

text: al.,

text: We

text: may

text: assume

text: that

text: both

text: ℵ

text: (K,

text: V

text: and

text: (L,

text: W

text: are

text: both

text: -saturated.

text: We

text: show

text: the

text: existence

text: of

text: a

text: back-and-forth

text: system

text: of

text: triples

text: (g,

text: M,

text: N)

text: satisfying

text: the

text: following

text: Property

text: K/M

text: and

text: L/N

text: are

text: regular

text: ﬁeld

text: extensions

text: where

text: M

text: and

text: N

text: are

text: countable

text: subﬁelds,

text: g

text: is

text: an

text: L

text: -isomorphism

text: r

text: g

text: Msep

text: Nsep

text: which

text: restricts

text: to

text: an

text: Ldiv-isomorphism

text: from

text: M

text: onto

text: N,

text: and

text: with

text: the

text: inclusions

text: i

text: and

text: j

text: M

text: N

text: (Gal(K),ˆi

text: o

text: (Gal(L),

text: ˆj

text: holds.

text: M

text: gˆ

text: N

text: Note

text: that

text: ﬁrst

text: (f,

text: Abs(K),

text: Abs(L))

text: satisﬁes

text: Property

text: Now,

text: if

text: (g,

text: M,

text: N)

text: satisﬁes

text: Property

text: A

text: a

text: countable

text: subset

text: of

text: L,

text: then

text: we

text: show

text: that

text: there

text: is

text: a

text: triple

text: (g

text: M

text: N

text: satisfying

text: Property

text: such

text: that

text: M

text: /M

text: and

text: N

text: /N(A)

text: and

text: g

text: extending

text: g.

text: This

text: is

text: guaranteed

text: by

text: the

text: proof

text: of

text: Proposition

text: of

text: [Cherlin

text: et

text: al.,

text: and

text: our

text: Valuation

text: Theoretic

text: Embedding

text: Lemma

text: This

text: back-and-forth

text: system

text: then

text: proves

text: that

text: (K,

text: V

text: (L,

text: W

text: L

text: div


text: 32 JIZHAN HONG School of Mathematical Sciences, Huaqiao University, 269 Chenghua North, Quanzhou Fujian, China, 362021 Email address: shuxuef@hqu.edu.cn

text: JIZHAN

text: HONG

text: School

text: of

text: Mathematical

text: Sciences,

text: Huaqiao

text: University,

text: Chenghua

text: North,

text: Quanzhou

text: Fujian,

text: China,

text: Email

text: address:

text: shuxuef@hqu.edu.cn

label: footer


text: 16 JIANGWEI XUE AND CHIA-FU YU O O(cid:48) When p = 3, there are two maximal orders , up to conjugation. We have O× × (cid:39) O(cid:48)× × (cid:39) /O S and /O D , the dihedral group of order 24. F 4 F 12 O Proposition 5.2. (1) The morphism ρ is injective for any order in the O ≥ genus of with r = 1, 8, 16 when p 5. r (2) When p = 2, ρ is surjective. O O (cid:39) O O(cid:48) (3) When p = 3, ρ is an isomorphism for = , and img(ρ) D for = . 4 ∈ O× × Proof. Let u˜ /R be a nontrivial element. According to the classiﬁcation ∈ in [XYY16a], if ord(u˜) is a power of 2, then ord(u˜) = 2 or 4, and [O : R[u˜]] { } OK×u˜ × 1, 2, 4 . By Lemma 5.1, ρ is injective on the Sylow 2-subgroups of /R when (cid:54) ∈ { } p is odd. Similarly, if p = 3 and ord(u˜) = 3, then [O : R[u˜]] 1, 2, 4 , so once (cid:54)∈ Ku˜ O× × again u˜ ker(ρ). This exhaust the list of all nontrivial elements of /R when ≥ O one of the following conditions holds: (i) p 7; (ii) p = 5 and is not maximal. O Hence ρ is injective in these two cases. Moreover, if p = 5 and is maximal, then O× × ρ is injective since /O is the simple group A . Part (1) of the proposition F 5 follows. O× × (cid:39) ∈ O× × Suppose that p = 2. Then /O S , and for any element u˜ /O F 4 (cid:39) Z F∈ of order 4, O [u˜] coincides with the maximal order O [ζ ]. Thus ρ(u˜) F (cid:39) F Ku˜ 8 (cid:16) (cid:17) PGL ( ) S is a nontrivial element of order 2. Since ρ does not vanish on any 2 2 3 O× × F order 3 elements as remarked, it maps /O surjectively onto PGL ( ). O Sup×pose that p = 3 and O×/OF× (cid:39) S4. InFthis case wZe √have D =√ 2√−Q(1√,2−31) , an⊂d is D -conjugate to B +B(1+i+j +k)/2, where B = [ 3, (1+ 3)(1+i)/2] ∈ O× × (cid:39) Z F(i). If u˜ /O is an element of order 3, then O [u˜] = B [ 3, ζ ], and (cid:54)∈ F F u˜ O× 6 × hence u˜ ker(ρ). Since ρ is also injective on the Sylow 2-subgroups of /O , it O× × F (cid:39) F induces an isomorphism between /O and PGL ( ) S . OF× × (cid:39) 2 3 4 Lastly, suppose that p = 3 and /O D , whose Sylow 2-subgroups F 12 are isomorphic to D . Note that D has a unique Sylow 3-subgroup, which is 4 (cid:111) 12 cyclic of order 3, and D = C D . Necessarily, ker(ρ) is nontrivial because (cid:54)(cid:39) F 12 3 4 D PGL ( ). Since ρ is injective on the Sylow 2-subgroups, we must have 12 2 3 (cid:39) ker(ρ) = C a√nd img(ρ) D . 3 4 (cid:96) A Let π = q be the Weil q = pa-number with a odd. Recall that denotes F π the set of isomorphism classes in the -isogeny class of A . The goal is to evaluate |A | A A q π . We decompose = Sp(π) 1 as the union of the set Sp(π) of superspecial π Aπ π abelian surfaces and 1 consisting of non-superspecial abelian surfaces. The case π A q = p is done in [XYY14] an(cid:101)d we may assume t(cid:101)hat q > p for w(cid:101)hich 1 is non- ∈ A π empty. Let A 1 be a member with covariant Dieudonn´e module M. There π → is a unique isogeny ϕ : A A(cid:101)of degree p with A superspecial: A is constructed − with the Dieudonn´e module M + F 1V M [Yu06, Lemma 4.4]. Consider the map A → pr : 1 Sp(π) sending A to A. π ∈ Proposition 5.3. (1) Let A Sp(π) be a superspecial member. The ﬁber − A −(cid:80) | 0 | pr 1(A ) in 1 has (q p)/ ρ(Aut(A ) elements, where ρ is the map deﬁned in 0 π 0 (5.1) viewing End(A ) as an order of D. |A | | 0 | − | | (2) = Sp(π) + ∈ (q p)/ ρ(Aut(A ) . π A0 Sp(π) 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JIANGWEI

text: XUE

text: AND

text: CHIA-FU

text: YU

text: O

text: When

text: p

text: there

text: are

text: two

text: maximal

text: orders

text: up

text: to

text: conjugation.

text: We

text: have

text: O×

text: /O

text: S

text: and

text: /O

text: D

text: the

text: dihedral

text: group

text: of

text: order

text: F

text: F

text: O

text: Proposition

text: The

text: morphism

text: ρ

text: is

text: injective

text: for

text: any

text: order

text: in

text: the

text: O

text: genus

text: of

text: with

text: r

text: when

text: p

text: r

text: When

text: p

text: ρ

text: is

text: surjective.

text: O

text: O

text: O

text: When

text: p

text: ρ

text: is

text: an

text: isomorphism

text: for

text: and

text: img(ρ)

text: D

text: for

text: O×

text: Proof.

text: Let

text: u˜

text: /R

text: be

text: a

text: nontrivial

text: element.

text: According

text: to

text: the

text: classiﬁcation

text: in

text: if

text: ord(u˜)

text: is

text: a

text: power

text: of

text: then

text: ord(u˜)

text: or

text: and

text: [O

text: R[u˜]]

text: OK×u˜

text: By

text: Lemma

text: ρ

text: is

text: injective

text: on

text: the

text: Sylow

text: of

text: /R

text: when

text: p

text: is

text: odd.

text: Similarly,

text: if

text: p

text: and

text: ord(u˜)

text: then

text: [O

text: R[u˜]]

text: so

text: once

text: Ku˜

text: O×

text: again

text: u˜

text: ker(ρ).

text: This

text: exhaust

text: the

text: list

text: of

text: all

text: nontrivial

text: elements

text: of

text: /R

text: when

text: O

text: one

text: of

text: the

text: following

text: conditions

text: holds:

text: (i)

text: p

text: (ii)

text: p

text: and

text: is

text: not

text: maximal.

text: O

text: Hence

text: ρ

text: is

text: injective

text: in

text: these

text: two

text: cases.

text: Moreover,

text: if

text: p

text: and

text: is

text: maximal,

text: then

text: O×

text: ρ

text: is

text: injective

text: since

text: /O

text: is

text: the

text: simple

text: group

text: A

text: Part

text: of

text: the

text: proposition

text: F

text: follows.

text: O×

text: O×

text: Suppose

text: that

text: p

text: Then

text: /O

text: S

text: and

text: for

text: any

text: element

text: u˜

text: /O

text: F

text: Z

text: F∈

text: of

text: order

text: O

text: [u˜]

text: coincides

text: with

text: the

text: maximal

text: order

text: O

text: [ζ

text: Thus

text: ρ(u˜)

text: F

text: F

text: Ku˜

text: PGL

text: S

text: is

text: a

text: nontrivial

text: element

text: of

text: order

text: Since

text: ρ

text: does

text: not

text: vanish

text: on

text: any

text: O×

text: F

text: order

text: elements

text: as

text: remarked,

text: it

text: maps

text: /O

text: surjectively

text: onto

text: PGL

text: O

text: Sup×pose

text: that

text: p

text: and

text: O×/OF×

text: InFthis

text: case

text: wZe

text: √have

text: D

text: an⊂d

text: is

text: D

text: -conjugate

text: to

text: B

text: where

text: B

text: O×

text: Z

text: F(i).

text: If

text: u˜

text: /O

text: is

text: an

text: element

text: of

text: order

text: then

text: O

text: [u˜]

text: B

text: ζ

text: and

text: F

text: F

text: u˜

text: O×

text: hence

text: u˜

text: ker(ρ).

text: Since

text: ρ

text: is

text: also

text: injective

text: on

text: the

text: Sylow

text: of

text: /O

text: it

text: O×

text: F

text: F

text: induces

text: an

text: isomorphism

text: between

text: /O

text: and

text: PGL

text: S

text: OF×

text: Lastly,

text: suppose

text: that

text: p

text: and

text: /O

text: D

text: whose

text: Sylow

text: F

text: are

text: isomorphic

text: to

text: D

text: Note

text: that

text: D

text: has

text: a

text: unique

text: Sylow

text: which

text: is

text: cyclic

text: of

text: order

text: and

text: D

text: C

text: D

text: Necessarily,

text: ker(ρ)

text: is

text: nontrivial

text: because

text: F

text: D

text: PGL

text: Since

text: ρ

text: is

text: injective

text: on

text: the

text: Sylow

text: we

text: must

text: have

text: ker(ρ)

text: C

text: a√nd

text: img(ρ)

text: D

text: A

text: Let

text: π

text: q

text: be

text: the

text: Weil

text: q

text: pa-number

text: with

text: a

text: odd.

text: Recall

text: that

text: denotes

text: F

text: π

text: the

text: set

text: of

text: isomorphism

text: classes

text: in

text: the

text: -isogeny

text: class

text: of

text: A

text: The

text: goal

text: is

text: to

text: evaluate

text: |A

text: A

text: A

text: q

text: π

text: We

text: decompose

text: Sp(π)

text: as

text: the

text: union

text: of

text: the

text: set

text: Sp(π)

text: of

text: superspecial

text: π

text: Aπ

text: π

text: abelian

text: surfaces

text: and

text: consisting

text: of

text: non-superspecial

text: abelian

text: surfaces.

text: The

text: case

text: π

text: A

text: q

text: p

text: is

text: done

text: in

text: we

text: may

text: assume

text: q

text: p

text: for

text: is

text: non-

text: A

text: π

text: empty.

text: Let

text: A

text: be

text: a

text: member

text: with

text: covariant

text: Dieudonn´e

text: module

text: M.

text: There

text: π

text: is

text: a

text: unique

text: isogeny

text: ϕ

text: A

text: degree

text: p

text: with

text: A

text: superspecial:

text: A

text: is

text: constructed

text: with

text: the

text: Dieudonn´e

text: module

text: M

text: F

text: M

text: Lemma

text: Consider

text: the

text: map

text: A

text: pr

text: Sp(π)

text: sending

text: A

text: to

text: A.

text: π

text: Proposition

text: Let

text: A

text: Sp(π)

text: be

text: a

text: superspecial

text: member.

text: The

text: ﬁber

text: A

text: pr

text: in

text: has

text: (q

text: p)/

text: ρ(Aut(A

text: elements,

text: where

text: ρ

text: is

text: the

text: map

text: deﬁned

text: in

text: π

text: viewing

text: End(A

text: as

text: an

text: order

text: of

text: D.

text: |A

text: Sp(π)

text: (q

text: p)/

text: ρ(Aut(A

text: π

text: Sp(π)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: COUNTING ABELIAN VARIETIES OVER FINITE FIELDS 19 [Zag76] Don Zagier, On the values at negative integers of the zeta-function of a real quadratic ﬁeld, Enseignement Math. (2) 22 (1976), no. 1-2, 55–95. MR 0406957 (53 #10742) (Xue) Collaborative Innovation Center of Mathematics, School of Mathematics and Statistics, Wuhan University, Luojiashan, 430072, Wuhan, Hubei, P.R. China Email address: xue j@whu.edu.cn (Yu) Institute of Mathematics, Academia Sinica and NCTS (Taipei Office), 6th Floor, Astronomy Mathematics Building, No. 1, Roosevelt Rd. Sec. 4, Taipei, Taiwan, 10617 Email address: chiafu@math.sinica.edu.tw ##LTLine##

text: COUNTING

text: ABELIAN

text: VARIETIES

text: OVER

text: FINITE

text: FIELDS

label: reference

text: Don

text: Zagier,

text: On

text: the

text: values

text: at

text: negative

text: integers

text: of

text: the

text: zeta-function

text: of

text: a

text: real

text: quadratic

text: ﬁeld,

text: Enseignement

text: Math.

text: no.

text: MR

text: (Xue)

text: Collaborative

text: Innovation

text: Center

text: of

text: Mathematics,

text: School

text: of

text: Mathematics

text: and

text: Statistics,

text: Wuhan

text: University,

text: Luojiashan,

text: Wuhan,

text: Hubei,

text: P.R.

text: China

text: Email

text: address:

text: xue

text: j@whu.edu.cn

text: (Yu)

text: Institute

text: of

text: Mathematics,

text: Academia

text: Sinica

text: and

text: NCTS

text: (Taipei

text: Office),

text: Floor,

text: Astronomy

text: Mathematics

text: Building,

text: No.

text: Roosevelt

text: Rd.

text: Sec.

text: Taipei,

text: Taiwan,

text: Email

text: address:

text: chiafu@math.sinica.edu.tw

text: ##LTLine##

font: default


text: COUNTING ABELIAN VARIETIES OVER FINITE FIELDS 3 where the vertical maps are natural surjective maps. It follows that the map Φ : A → X is also surjective. We consider the ﬁbers of this map. π π Q Let G be the algebraic group over associated to the multiplicative group × π E . By Tate’s homomorphism theorem on abelian varieties, one has canonical π isomorphisms Q Q (2.1) G ( ) = Aut (V (A )), and G ( ) = Aut (N(A )). π (cid:96) F(cid:96) (cid:96) π π p DM π A Q Let be the ﬁnite adele ring of , and X = ([M], [T ] (cid:54) ) an element of X . f ⊂ A (cid:89) (cid:96) (cid:96)=p π Deﬁne a compact subgroup U G ( ) by X π f × U := Aut (M) Aut (T ); X DM (cid:54) R(cid:96) (cid:96) (cid:96)=p (cid:101) A A this(cid:101)is uniquely determined by X up to conjugation by G ( ). The group G ( ) π f π f has a(cid:101)natural left action on the set X of Dieudonn´e an(cid:101)d Tate modules. It also acts π on Aπ(cid:101) from thAe left through th(cid:101)e isomorphAisms (2.1)A(cf. Lemma 3.2) so that tAhe map Φ is G ( )-equivariant. Denote by and the ﬁbers over X in A π f A π,X A π,X π and , respectively. The set is a single G ( )-orbit and it is isomorphic to A π π,X π f G ( )/U after choosing a base point. The projection map prA is simply modulo π f X Q the left action of G ( ). Therefore, we obtain a bijection π A −∼→ Q \ A G ( ) G ( )/U . π,X π π f X Running over all elements X in X , we obtain the following result. π Theorem 2.1. There is a bij(cid:97)ection A −∼→ Q \ A (2.2) G ( ) G ( )/U . π π π f X ∈ X Xπ ∈ Z For each X = ([M], [T ] (cid:54) ) X , there is an -order O in E such that ⊗Z (cid:39) (cid:96) (cid:96)=p⊗Z (cid:39)π (cid:54) X π O End (M) and O End (T ) for all (cid:96) = p. We may choose the X p DM X (cid:96) R(cid:96) (cid:96) orders O so that they are contained in a common maximal order O , because X max any two maximal orders are locally conj(cid:88)ugate. Theorem 2.1 gives |A | (2.3) = h(O ), π X ∈ X Xπ where h(O ) denotes the class number of locally principal right ideal classes X of O . The calculation of these class numbers diﬀers whether or not the endomor- X phism algebra E satisﬁes the Eichler condition with respect to the archimedean π places of F. For our case, it simply depends on whether or not F is a CM ﬁeld.

text: COUNTING

text: ABELIAN

text: VARIETIES

text: OVER

text: FINITE

text: FIELDS

text: where

text: the

text: vertical

text: maps

text: are

text: natural

text: surjective

text: maps.

text: It

text: follows

text: that

text: the

text: map

text: Φ

text: A

text: X

text: is

text: also

text: surjective.

text: We

text: consider

text: the

text: ﬁbers

text: of

text: this

text: map.

text: π

text: π

text: Q

text: Let

text: G

text: be

text: the

text: algebraic

text: group

text: over

text: associated

text: to

text: the

text: multiplicative

text: group

text: π

text: E

text: By

text: Tate’s

text: homomorphism

text: theorem

text: on

text: abelian

text: varieties,

text: one

text: has

text: canonical

text: π

text: isomorphisms

text: Q

label: equation

text: Q

text: G

text: Aut

text: (V

text: (A

text: and

text: G

text: Aut

text: (N(A

text: π

text: π

text: π

text: p

text: DM

text: π

text: A

text: Q

text: Let

text: be

text: the

text: ﬁnite

text: adele

text: ring

text: of

text: and

text: X

text: ([M],

text: [T

text: an

text: element

text: of

text: X

text: f

text: A

text: π

text: Deﬁne

text: a

text: compact

text: subgroup

text: U

text: G

text: by

text: X

text: π

text: f

text: U

text: Aut

text: (M)

text: Aut

text: (T

text: X

text: DM

text: A

text: A

text: uniquely

text: determined

text: by

text: X

text: up

text: to

text: conjugation

text: by

text: G

text: The

text: group

text: G

text: π

text: f

text: π

text: f

text: has

text: left

text: action

text: on

text: the

text: set

text: X

text: of

text: Dieudonn´e

text: Tate

text: modules.

text: It

text: also

text: acts

text: π

text: on

text: from

text: thAe

text: left

text: through

text: isomorphAisms

text: Lemma

text: so

text: that

text: tAhe

text: map

text: Φ

text: is

text: G

text: )-equivariant.

text: Denote

text: by

text: and

text: the

text: ﬁbers

text: over

text: X

text: in

text: A

text: π

text: f

text: A

text: π,X

text: A

text: π,X

text: π

text: and

text: respectively.

text: The

text: set

text: is

text: a

text: single

text: G

text: )-orbit

text: and

text: it

text: is

text: isomorphic

text: to

text: A

text: π

text: π,X

text: π

text: f

text: G

text: )/U

text: after

text: choosing

text: a

text: base

text: point.

text: The

text: projection

text: map

text: prA

text: is

text: simply

text: modulo

text: π

text: f

text: X

text: Q

text: the

text: left

text: action

text: of

text: G

text: Therefore,

text: we

text: obtain

text: a

text: bijection

text: π

text: A

text: Q

text: A

text: G

text: G

text: )/U

text: π,X

text: π

text: π

text: f

text: X

text: Running

text: over

text: all

text: elements

text: X

text: in

text: X

text: we

text: obtain

text: the

text: following

text: result.

text: π

text: Theorem

text: There

text: is

text: a

text: A

text: Q

text: A

text: G

text: G

text: )/U

text: π

text: π

text: π

text: f

text: X

text: X

text: Xπ

text: Z

text: For

text: each

text: X

text: ([M],

text: [T

text: X

text: there

text: is

text: an

text: -order

text: O

text: in

text: E

text: such

text: that

text: ⊗Z

text: X

text: π

text: O

text: End

text: (M)

text: and

text: O

text: End

text: (T

text: for

text: all

text: p.

text: We

text: may

text: choose

text: the

text: X

text: p

text: DM

text: X

text: orders

text: O

text: so

text: that

text: they

text: are

text: contained

text: in

text: a

text: common

text: maximal

text: order

text: O

text: because

text: X

text: max

text: any

text: two

text: maximal

text: orders

text: are

text: locally

text: Theorem

text: gives

text: |A

text: h(O

text: π

text: X

text: X

text: Xπ

text: where

text: h(O

text: denotes

text: the

text: class

text: number

text: of

text: locally

text: principal

text: right

text: ideal

text: classes

text: X

text: of

text: O

text: The

text: calculation

text: of

text: these

text: class

text: numbers

text: diﬀers

text: whether

text: or

text: not

text: the

text: endomor-

text: X

text: phism

text: algebra

text: E

text: satisﬁes

text: the

text: Eichler

text: condition

text: with

text: respect

text: to

text: the

text: archimedean

text: π

text: places

text: of

text: F.

text: For

text: our

text: case,

text: it

text: simply

text: depends

text: on

text: whether

text: or

text: not

text: F

text: is

text: a

text: CM

text: ﬁeld.


text: IDEMPOTENTS IN TANGLE CATEGORIES SPLIT 5 Figure 3. An example of compressing a 4-punctured sphere to obtain two boundary-parallel 2-punctured spheres. embedding φ is transverse to α for all ﬁxed values of t. Isotopies of t surfaces in this paper will always be proper isotopies that are transverse to the relevant 1-manifolds. ∼ × If α is a 1-manifold properly embedded in M = D2 I and F is a properly embedded k-punctured surface in M, F is boundary-parallel either if F is a 2-punctured 2-sphere bounding a 3-ball that meets α in an unknotted arc or if there is a proper, transverse isotopy of F in M which takes F to a punctured subsurface contained in ∂M. Otherwise, we say F is non-boundary parallel. A loop γ embedded in F is essential if it does not bound a 0-punctured disk in F . The k-punctured surface F is compressible in (M, α) (or just compressible when context is understood) if F is a 0-punctured 2-sphere bounding a ∩ 3-ball or if there exists a disk D embedded in M such that D F = ∂D, ∂D is essential in F and D is disjoint from α. Such a disk is called a compressing disk. Otherwise, we say F is incompressible. A punctured surface F is essential if F is incompressible and non∼-boundary parallel. × Given a 1-manifold α properly embedded in M = D2 I and F a properly embedded k-punctured surface with compressing disk D, we can compress F along D to form a new embedded k-punctured surface ∗ × F . See Figure 3. Let D2 I be a ﬁbered neighborhood of D in M × { } × × such that D = D2 1 , D2 I is disjoint from α, and ∂(D2) I is an 2 embedded annulus in F that is disjoint from the punctures of F . Then ∗ \ × ∪ we deﬁne F to be a surface transversely isotopic to (F (∂(D2) I)) × { } (D2 0, 1 ). Although we take the point of view of incompressible and non- boundary parallel punctured surfaces in this paper, we could have ##LTFigure## ##LTLine##

text: IDEMPOTENTS

text: IN

text: TANGLE

text: CATEGORIES

text: SPLIT

text: Figure

text: An

text: example

text: of

text: compressing

text: a

text: sphere

text: to

text: obtain

text: two

text: boundary-parallel

text: spheres.

text: embedding

text: φ

text: is

text: transverse

text: to

text: α

text: for

text: all

text: ﬁxed

text: values

text: of

text: t.

text: Isotopies

text: of

text: t

text: surfaces

text: in

text: this

text: paper

text: will

text: always

text: be

text: proper

text: isotopies

text: that

text: are

text: transverse

text: to

text: the

text: relevant

text: If

text: α

text: is

text: a

text: properly

text: embedded

text: in

text: M

text: I

text: and

text: F

text: is

text: a

text: properly

text: embedded

text: k-punctured

text: surface

text: in

text: M,

text: F

text: is

text: boundary-parallel

text: either

text: if

text: F

text: is

text: a

text: bounding

text: a

text: that

text: meets

text: α

text: in

text: an

text: unknotted

text: arc

text: or

text: if

text: there

text: is

text: a

text: proper,

text: transverse

text: isotopy

text: of

text: F

text: in

text: M

text: which

text: takes

text: F

text: to

text: a

text: punctured

text: subsurface

text: contained

text: in

text: ∂M.

text: Otherwise,

text: we

text: say

text: F

text: is

text: non-boundary

text: parallel.

text: A

text: loop

text: γ

text: embedded

text: in

text: F

text: is

text: essential

text: if

text: it

text: does

text: not

text: bound

text: a

text: disk

text: in

text: F

text: The

text: k-punctured

text: surface

text: F

text: is

text: compressible

text: in

text: (M,

text: α)

text: (or

text: just

text: compressible

text: when

text: context

text: is

text: understood)

text: if

text: F

text: is

text: a

text: bounding

text: a

text: or

text: if

text: there

text: exists

text: a

text: disk

text: D

text: embedded

text: in

text: M

text: such

text: that

text: D

text: F

text: ∂D,

text: ∂D

text: is

text: essential

text: in

text: F

text: and

text: D

text: is

text: disjoint

text: from

text: α.

text: Such

text: a

text: disk

text: is

text: called

text: a

text: compressing

text: disk.

text: Otherwise,

text: we

text: say

text: F

text: is

text: incompressible.

text: A

text: punctured

text: surface

text: F

text: is

text: essential

text: if

text: F

text: is

text: incompressible

text: and

text: non∼-boundary

text: parallel.

text: Given

text: a

text: α

text: properly

text: embedded

text: in

text: M

text: I

text: and

text: F

text: a

text: properly

text: embedded

text: k-punctured

text: surface

text: with

text: compressing

text: disk

text: D,

text: we

text: can

text: compress

text: F

text: along

text: D

text: to

text: form

text: a

text: new

text: embedded

text: k-punctured

text: surface

text: F

text: See

text: Figure

text: Let

text: I

text: be

text: a

text: ﬁbered

text: neighborhood

text: of

text: D

text: in

text: M

text: such

text: that

text: D

text: I

text: is

text: disjoint

text: from

text: α,

text: and

text: I

text: is

text: an

text: embedded

text: annulus

text: in

text: F

text: that

text: is

text: disjoint

text: from

text: the

text: punctures

text: of

text: F

text: Then

text: we

text: deﬁne

text: F

text: to

text: be

text: a

text: surface

text: transversely

text: isotopic

text: to

text: (F

text: I))

text: Although

text: we

text: take

text: the

text: point

text: of

text: view

text: of

text: incompressible

text: and

text: non-

text: boundary

text: parallel

text: punctured

text: surfaces

text: in

text: this

text: paper,

text: we

text: could

text: have

text: ##LTFigure##

text: ##LTLine##


text: IDEMPOTENTS IN TANGLE CATEGORIES SPLIT 6 equivalently adopted the point of view of studying incompressible and non-boundary parallel meridional surfaces properly embedded in the exterior of α in M. In particular, if F is a properly embedded non- × boundary parallel punctured surface in (D2 I, α) and η(α) is a small × \ open regular neighborhood of α in D2 I, then F η(α) is non-boundary × \ parallel in (D2 I) η(α). Similarly, if F is a properly embedded in- × compressible punctured surface in (D2 I, α) and η(α) is a small open × \ regular neighborhood of α in D2 I, then F η(α) is incompressible × \ in (D2 I) η(α). In particular, we will make use of the following Theorem of Freedman and Freedman. Theorem 4.1. [3] Let M be a compact 3-manifold with boundary and b an integer greater than zero. There is a constant c(M, b) so that if F , ..., F , k > c, is a collection of incompressible surfaces such that all 1 k ≤ ≤ ≤ ≤ the Betti numbers b (F ) < b, 1 i k, and no F , 1 i k, is a 1 i i boundary parallel annulus or a boundary parallel disk, then at least two members F and F are parallel. i j Note that Freedman and Freedman deﬁne two disjoint properly em- bedded surfaces F and F in a compact 3-manifold M to be parallel ∪ i j × × ⊂ if F F cobound a product F I in M such that ∂F I ∂M, i ×j { } × { } F = F 0 and F = F 1 . i j 5. Decomposing Disks and Braid Equivalence T Decomposing a morphism in as a composition of two morphisms involves some amount of choice. This choice can be captured via the notion of braid-equivalence. × × Deﬁnition 5.1. Two tangles (D2 I, A) and (D2 I, B) are braid- ◦ ◦ equivalent if there exist braids C and C such that A = C B C . 1 2 1 2 × × Proposition 5.2. Tangles (D2 I, T ) and (D2 I, T ) are braid- 1 × 2 × equivalent if and only if there is an isotopy of D2 I which ﬁxes (∂D2) I and which takes T to T . 1 2 Proof. This follows from a nearly identical adaptation of the proof of (cid:3) Proposition 3.6 of [2]. × Deﬁnition 5.3. A decomposing disk for a tangle (D2 I, A) is a × punctured disk which is properly embedded in D2 I, whose boundary × × is isotopic in ∂(D2 I) to ∂(∂ (D2 I)). See the disk F in Figure 1. + × × A decomposing disk F for a tangle (D2 I, A) separates D2 I into × two connected components, one containing ∂−(D2 I) and the other × containing ∂ (D2 I). The closure of each component is homeomorphic × + × to D2 I, so F decomposes (D2 I, A) into two tangles, each of which

text: IDEMPOTENTS

text: IN

text: TANGLE

text: CATEGORIES

text: SPLIT

text: equivalently

text: adopted

text: the

text: point

text: of

text: view

text: of

text: studying

text: incompressible

text: and

text: non-boundary

text: parallel

text: meridional

text: surfaces

text: properly

text: embedded

text: in

text: the

text: exterior

text: of

text: α

text: in

text: M.

text: In

text: particular,

text: if

text: F

text: is

text: a

text: properly

text: embedded

text: non-

text: boundary

text: parallel

text: punctured

text: surface

text: in

text: I,

text: α)

text: and

text: η(α)

text: is

text: a

text: small

text: open

text: regular

text: neighborhood

text: of

text: α

text: in

text: I,

text: then

text: F

text: η(α)

text: is

text: non-boundary

text: parallel

text: in

text: I)

text: η(α).

text: Similarly,

text: if

text: F

text: is

text: a

text: properly

text: embedded

text: in-

text: compressible

text: punctured

text: surface

text: in

text: I,

text: α)

text: and

text: η(α)

text: is

text: a

text: small

text: open

text: regular

text: neighborhood

text: of

text: α

text: in

text: I,

text: then

text: F

text: η(α)

text: is

text: incompressible

text: in

text: I)

text: η(α).

text: In

text: particular,

text: we

text: will

text: make

text: use

text: of

text: the

text: following

text: Theorem

text: of

text: Freedman

text: and

text: Freedman.

text: Theorem

text: Let

text: M

text: be

text: a

text: compact

text: with

text: boundary

text: and

text: b

text: an

text: integer

text: greater

text: than

text: zero.

text: There

text: is

text: a

text: constant

text: c(M,

text: b)

text: so

text: that

text: if

text: F

text: F

text: k

text: c,

text: is

text: a

text: collection

text: of

text: incompressible

text: surfaces

text: such

text: that

text: all

text: k

text: the

text: Betti

text: numbers

text: b

text: (F

text: b,

text: i

text: k,

text: and

text: no

text: F

text: i

text: k,

text: is

text: a

text: i

text: i

text: boundary

text: parallel

text: annulus

text: or

text: a

text: boundary

text: parallel

text: disk,

text: then

text: at

text: least

text: two

text: members

text: F

text: and

text: F

text: are

text: parallel.

text: i

text: j

text: Note

text: that

text: Freedman

text: and

text: Freedman

text: deﬁne

text: two

text: disjoint

text: properly

text: em-

text: bedded

text: surfaces

text: F

text: and

text: F

text: in

text: a

text: compact

text: M

text: to

text: be

text: parallel

text: i

text: j

text: if

text: F

text: F

text: cobound

text: a

text: product

text: F

text: I

text: in

text: M

text: such

text: that

text: ∂F

text: I

text: ∂M,

text: i

text: ×j

text: F

text: F

text: and

text: F

text: F

text: i

text: j

text: Decomposing

text: Disks

text: and

text: Braid

text: Equivalence

text: T

text: Decomposing

text: a

text: morphism

text: in

text: as

text: a

text: composition

text: of

text: two

text: morphisms

text: involves

text: some

text: amount

text: of

text: choice.

text: This

text: choice

text: can

text: be

text: captured

text: via

text: the

text: notion

text: of

text: braid-equivalence.

text: Deﬁnition

text: Two

text: tangles

text: I,

text: A)

text: and

text: I,

text: B)

text: are

text: braid-

text: equivalent

text: if

text: there

text: exist

text: braids

text: C

text: and

text: C

text: such

text: that

text: A

text: C

text: B

text: C

text: Proposition

text: Tangles

text: I,

text: T

text: and

text: I,

text: T

text: are

text: braid-

text: equivalent

text: if

text: and

text: only

text: if

text: there

text: is

text: an

text: isotopy

text: of

text: I

text: which

text: ﬁxes

text: I

text: and

text: which

text: takes

text: T

text: to

text: T

text: Proof.

text: This

text: follows

text: from

text: a

text: nearly

text: identical

text: adaptation

text: of

text: the

text: proof

text: of

text: Proposition

text: of

text: Deﬁnition

text: A

text: decomposing

text: disk

text: for

text: a

text: tangle

text: I,

text: A)

text: is

text: a

text: punctured

text: disk

text: which

text: is

text: properly

text: embedded

text: in

text: I,

text: whose

text: boundary

text: is

text: isotopic

text: in

text: I)

text: to

text: I)).

text: See

text: the

text: disk

text: F

text: in

text: Figure

text: A

text: decomposing

text: disk

text: F

text: for

text: a

text: tangle

text: I,

text: A)

text: separates

text: I

text: into

text: two

text: connected

text: components,

text: one

text: containing

text: I)

text: and

text: the

text: other

text: containing

text: I).

text: The

text: closure

text: of

text: each

text: component

text: is

text: homeomorphic

text: to

text: I,

text: so

text: F

text: decomposes

text: I,

text: A)

text: into

text: two

text: tangles,

text: each

text: of

text: which